Learning how to solve physics problems is a big part of learning physics. Here’s a collection of example physics problems and solutions to help you tackle problems sets and understand concepts and how to work with formulas:
Physics Homework Tips Physics homework can be challenging! Get tips to help make the task a little easier.
Unit Conversion Examples
There are now too many unit conversion examples to list in this space. This Unit Conversion Examples page is a more comprehensive list of worked example problems.
Newton’s Equations of Motion Example Problems
Equations of Motion – Constant Acceleration Example This equations of motion example problem consist of a sliding block under constant acceleration. It uses the equations of motion to calculate the position and velocity of a given time and the time and position of a given velocity.
Equations of Motion Example Problem – Constant Acceleration This example problem uses the equations of motion for constant acceleration to find the position, velocity, and acceleration of a breaking vehicle.
Equations of Motion Example Problem – Interception
This example problem uses the equations of motion for constant acceleration to calculate the time needed for one vehicle to intercept another vehicle moving at a constant velocity.
Vertical Motion Example Problem – Coin Toss Here’s an example applying the equations of motion under constant acceleration to determine the maximum height, velocity and time of flight for a coin flipped into a well. This problem could be modified to solve any object tossed vertically or dropped off a tall building or any height. This type of problem is a common equation of motion homework problem.
Projectile Motion Example Problem This example problem shows how to find different variables associated with parabolic projectile motion.
Accelerometer and Inertia Example Problem Accelerometers are devices to measure or detect acceleration by measuring the changes that occur as a system experiences an acceleration. This example problem uses one of the simplest forms of an accelerometer, a weight hanging from a stiff rod or wire. As the system accelerates, the hanging weight is deflected from its rest position. This example derives the relationship between that angle, the acceleration and the acceleration due to gravity. It then calculates the acceleration due to gravity of an unknown planet.
Weight In An Elevator Have you ever wondered why you feel slightly heavier in an elevator when it begins to move up? Or why you feel lighter when the elevator begins to move down? This example problem explains how to find your weight in an accelerating elevator and how to find the acceleration of an elevator using your weight on a scale.
Equilibrium Example Problem This example problem shows how to determine the different forces in a system at equilibrium. The system is a block suspended from a rope attached to two other ropes.
Equilibrium Example Problem – Balance This example problem highlights the basics of finding the forces acting on a system in mechanical equilibrium.
Force of Gravity Example This physics problem and solution shows how to apply Newton’s equation to calculate the gravitational force between the Earth and the Moon.
Coupled Systems Example Problems
Coupled systems are two or more separate systems connected together. The best way to solve these types of problems is to treat each system separately and then find common variables between them. Atwood Machine The Atwood Machine is a coupled system of two weights sharing a connecting string over a pulley. This example problem shows how to find the acceleration of an Atwood system and the tension in the connecting string. Coupled Blocks – Inertia Example This example problem is similar to the Atwood machine except one block is resting on a frictionless surface perpendicular to the other block. This block is hanging over the edge and pulling down on the coupled string. The problem shows how to calculate the acceleration of the blocks and the tension in the connecting string.
Friction Example Problems
These example physics problems explain how to calculate the different coefficients of friction.
Friction Example Problem – Block Resting on a Surface Friction Example Problem – Coefficient of Static Friction Friction Example Problem – Coefficient of Kinetic Friction Friction and Inertia Example Problem
Momentum and Collisions Example Problems
These example problems show how to calculate the momentum of moving masses.
Momentum and Impulse Example Finds the momentum before and after a force acts on a body and determine the impulse of the force.
Elastic Collision Example Shows how to find the velocities of two masses after an elastic collision.
It Can Be Shown – Elastic Collision Math Steps Shows the math to find the equations expressing the final velocities of two masses in terms of their initial velocities.
Simple Pendulum Example Problems
These example problems show how to use the period of a pendulum to find related information.
Find the Period of a Simple Pendulum Find the period if you know the length of a pendulum and the acceleration due to gravity.
Find the Length of a Simple Pendulum Find the length of the pendulum when the period and acceleration due to gravity is known.
Find the Acceleration due to Gravity Using A Pendulum Find ‘g’ on different planets by timing the period of a known pendulum length.
Harmonic Motion and Waves Example Problems
These example problems all involve simple harmonic motion and wave mechanics.
Energy and Wavelength Example This example shows how to determine the energy of a photon of a known wavelength.
Hooke’s Law Example Problem An example problem involving the restoring force of a spring.
Wavelength and Frequency Calculations See how to calculate wavelength if you know frequency and vice versa, for light, sound, or other waves.
Heat and Energy Example Problems
Heat of Fusion Example Problem Two example problems using the heat of fusion to calculate the energy required for a phase change.
Specific Heat Example Problem This is actually 3 similar example problems using the specific heat equation to calculate heat, specific heat, and temperature of a system.
Heat of Vaporization Example Problems Two example problems using or finding the heat of vaporization.
Ice to Steam Example Problem Classic problem melting cold ice to make hot steam. This problem brings all three of the previous example problems into one problem to calculate heat changes over phase changes.
Charge and Coulomb Force Example Problems
Electrical charges generate a coulomb force between themselves proportional to the magnitude of the charges and inversely proportional to the distance between them. Coulomb’s Law Example This example problem shows how to use Coulomb’s Law equation to find the charges necessary to produce a known repulsive force over a set distance. Coulomb Force Example This Coulomb force example shows how to find the number of electrons transferred between two bodies to generate a set amount of force over a short distance.
Answer: v i = 5.03 m/s and hang time = 1.03 s (except for in sports commericals)
Answer: a = 1.62*10 5 m/s/s
Answer: d = 48.0 m
Answer: t = 8.69 s
Answer: a = -1.08*10^6 m/s/s
Answer: d = -57.0 m (57.0 meters deep)
Answer: v i = 47.6 m/s
Answer: a = 2.86 m/s/s and t = 30. 8 s
Answer: a = 15.8 m/s/s
Answer: v i = 94.4 mi/hr
Solutions to Above Problems
t = 32.8 s
v = 0 m/s
d = (0 m/s)*(32.8 s)+ 0.5*(3.20 m/s 2 )*(32.8 s) 2
Return to Problem 1
t = 5.21 s
v = 0 m/s
110 m = (0 m/s)*(5.21 s)+ 0.5*(a)*(5.21 s) 2
110 m = (13.57 s 2 )*a
a = (110 m)/(13.57 s 2 )
a = 8.10 m/ s 2
Return to Problem 2
t = 2.6 s
v = 0 m/s
d = (0 m/s)*(2.60 s)+ 0.5*(-9.8 m/s 2 )*(2.60 s) 2
d = -33.1 m (- indicates direction)
v f = v i + a*t
v f = 0 + (-9.8 m/s 2 )*(2.60 s)
v f = -25.5 m/s (- indicates direction)
Return to Problem 3
v = 18.5 m/s
v = 46.1 m/s
t = 2.47 s
a = (46.1 m/s - 18.5 m/s)/(2.47 s)
a = 11.2 m/s 2
d = v i *t + 0.5*a*t 2
d = (18.5 m/s)*(2.47 s)+ 0.5*(11.2 m/s 2 )*(2.47 s) 2
d = 45.7 m + 34.1 m
(Note: the d can also be calculated using the equation v f 2 = v i 2 + 2*a*d)
Return to Problem 4
v = 0 m/s
d = -1.40 m
-1.40 m = (0 m/s)*(t)+ 0.5*(-1.67 m/s 2 )*(t) 2
-1.40 m = 0+ (-0.835 m/s 2 )*(t) 2
(-1.40 m)/(-0.835 m/s 2 ) = t 2
1.68 s 2 = t 2
Return to Problem 5
v = 0 m/s
v = 444 m/s
a = (444 m/s - 0 m/s)/(1.83 s)
a = 243 m/s 2
d = (0 m/s)*(1.83 s)+ 0.5*(243 m/s 2 )*(1.83 s) 2
d = 0 m + 406 m
Return to Problem 6
v = 0 m/s
v = 7.10 m/s
(7.10 m/s) 2 = (0 m/s) 2 + 2*(a)*(35.4 m)
50.4 m 2 /s 2 = (0 m/s) 2 + (70.8 m)*a
(50.4 m 2 /s 2 )/(70.8 m) = a
a = 0.712 m/s 2
Return to Problem 7
v = 0 m/s
v = 65 m/s
(65 m/s) 2 = (0 m/s) 2 + 2*(3 m/s 2 )*d
4225 m 2 /s 2 = (0 m/s) 2 + (6 m/s 2 )*d
(4225 m 2 /s 2 )/(6 m/s 2 ) = d
Return to Problem 8
v = 22.4 m/s
v = 0 m/s
d = (22.4 m/s + 0 m/s)/2 *2.55 s
d = (11.2 m/s)*2.55 s
Return to Problem 9
a = -9.8 m/s
v = 0 m/s
(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(2.62 m)
0 m 2 /s 2 = v i 2 - 51.35 m 2 /s 2
51.35 m 2 /s 2 = v i 2
v i = 7.17 m/s
Return to Problem 10
(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(1.29 m)
0 m 2 /s 2 = v i 2 - 25.28 m 2 /s 2
25.28 m 2 /s 2 = v i 2
v i = 5.03 m/s
To find hang time, find the time to the peak and then double it.
0 m/s = 5.03 m/s + (-9.8 m/s 2 )*t up
-5.03 m/s = (-9.8 m/s 2 )*t up
(-5.03 m/s)/(-9.8 m/s 2 ) = t up
t up = 0.513 s
hang time = 1.03 s
Return to Problem 11
v = 0 m/s
v = 521 m/s
(521 m/s) 2 = (0 m/s) 2 + 2*(a)*(0.840 m)
271441 m 2 /s 2 = (0 m/s) 2 + (1.68 m)*a
(271441 m 2 /s 2 )/(1.68 m) = a
a = 1.62*10 5 m /s 2
Return to Problem 12
(NOTE: the time required to move to the peak of the trajectory is one-half the total hang time - 3.125 s.)
First use: v f = v i + a*t
0 m/s = v i + (-9.8 m/s 2 )*(3.13 s)
0 m/s = v i - 30.7 m/s
v i = 30.7 m/s (30.674 m/s)
Now use: v f 2 = v i 2 + 2*a*d
(0 m/s) 2 = (30.7 m/s) 2 + 2*(-9.8 m/s 2 )*(d)
0 m 2 /s 2 = (940 m 2 /s 2 ) + (-19.6 m/s 2 )*d
-940 m 2 /s 2 = (-19.6 m/s 2 )*d
(-940 m 2 /s 2 )/(-19.6 m/s 2 ) = d
Return to Problem 13
v = 0 m/s
d = -370 m
-370 m = (0 m/s)*(t)+ 0.5*(-9.8 m/s 2 )*(t) 2
-370 m = 0+ (-4.9 m/s 2 )*(t) 2
(-370 m)/(-4.9 m/s 2 ) = t 2
75.5 s 2 = t 2
Return to Problem 14
v = 367 m/s
v = 0 m/s
(0 m/s) 2 = (367 m/s) 2 + 2*(a)*(0.0621 m)
0 m 2 /s 2 = (134689 m 2 /s 2 ) + (0.1242 m)*a
-134689 m 2 /s 2 = (0.1242 m)*a
(-134689 m 2 /s 2 )/(0.1242 m) = a
a = -1.08*10 6 m /s 2
(The - sign indicates that the bullet slowed down.)
Return to Problem 15
t = 3.41 s
v = 0 m/s
d = (0 m/s)*(3.41 s)+ 0.5*(-9.8 m/s 2 )*(3.41 s) 2
d = 0 m+ 0.5*(-9.8 m/s 2 )*(11.63 s 2 )
d = -57.0 m
(NOTE: the - sign indicates direction)
Return to Problem 16
a = -3.90 m/s
v = 0 m/s
(0 m/s) 2 = v i 2 + 2*(- 3.90 m/s 2 )*(290 m)
0 m 2 /s 2 = v i 2 - 2262 m 2 /s 2
2262 m 2 /s 2 = v i 2
v i = 47.6 m /s
Return to Problem 17
v = 0 m/s
v = 88.3 m/s
( 88.3 m/s) 2 = (0 m/s) 2 + 2*(a)*(1365 m)
7797 m 2 /s 2 = (0 m 2 /s 2 ) + (2730 m)*a
7797 m 2 /s 2 = (2730 m)*a
(7797 m 2 /s 2 )/(2730 m) = a
a = 2.86 m/s 2
88.3 m/s = 0 m/s + (2.86 m/s 2 )*t
(88.3 m/s)/(2.86 m/s 2 ) = t
t = 30. 8 s
Return to Problem 18
v = 0 m/s
v = m/s
( 112 m/s) 2 = (0 m/s) 2 + 2*(a)*(398 m)
12544 m 2 /s 2 = 0 m 2 /s 2 + (796 m)*a
12544 m 2 /s 2 = (796 m)*a
(12544 m 2 /s 2 )/(796 m) = a
a = 15.8 m/s 2
Return to Problem 19
v f 2 = v i 2 + 2*a*d
(0 m/s) 2 = v i 2 + 2*(-9.8 m/s 2 )*(91.5 m)
0 m 2 /s 2 = v i 2 - 1793 m 2 /s 2
1793 m 2 /s 2 = v i 2
v i = 42.3 m/s
Now convert from m/s to mi/hr:
v i = 42.3 m/s * (2.23 mi/hr)/(1 m/s)
v i = 94.4 mi/hr
Return to Problem 20
6.1 Solving Problems with Newton’s Laws
Learning objectives.
By the end of this section, you will be able to:
Apply problem-solving techniques to solve for quantities in more complex systems of forces
Use concepts from kinematics to solve problems using Newton’s laws of motion
Solve more complex equilibrium problems
Solve more complex acceleration problems
Apply calculus to more advanced dynamics problems
Success in problem solving is necessary to understand and apply physical principles. We developed a pattern of analyzing and setting up the solutions to problems involving Newton’s laws in Newton’s Laws of Motion ; in this chapter, we continue to discuss these strategies and apply a step-by-step process.
Problem-Solving Strategies
We follow here the basics of problem solving presented earlier in this text, but we emphasize specific strategies that are useful in applying Newton’s laws of motion . Once you identify the physical principles involved in the problem and determine that they include Newton’s laws of motion, you can apply these steps to find a solution. These techniques also reinforce concepts that are useful in many other areas of physics. Many problem-solving strategies are stated outright in the worked examples, so the following techniques should reinforce skills you have already begun to develop.
Problem-Solving Strategy
Applying newton’s laws of motion.
Identify the physical principles involved by listing the givens and the quantities to be calculated.
Sketch the situation, using arrows to represent all forces.
Determine the system of interest. The result is a free-body diagram that is essential to solving the problem.
Apply Newton’s second law to solve the problem. If necessary, apply appropriate kinematic equations from the chapter on motion along a straight line.
Check the solution to see whether it is reasonable.
Let’s apply this problem-solving strategy to the challenge of lifting a grand piano into a second-story apartment. Once we have determined that Newton’s laws of motion are involved (if the problem involves forces), it is particularly important to draw a careful sketch of the situation. Such a sketch is shown in Figure 6.2 (a). Then, as in Figure 6.2 (b), we can represent all forces with arrows. Whenever sufficient information exists, it is best to label these arrows carefully and make the length and direction of each correspond to the represented force.
As with most problems, we next need to identify what needs to be determined and what is known or can be inferred from the problem as stated, that is, make a list of knowns and unknowns. It is particularly crucial to identify the system of interest, since Newton’s second law involves only external forces. We can then determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See Figure 6.2 (c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated in Newton’s Laws of Motion , the system of interest depends on the question we need to answer. Only forces are shown in free-body diagrams, not acceleration or velocity. We have drawn several free-body diagrams in previous worked examples. Figure 6.2 (c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.
Once a free-body diagram is drawn, we apply Newton’s second law. This is done in Figure 6.2 (d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then the forces can be handled algebraically. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. We do this by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known. Generally, just write Newton’s second law in components along the different directions. Then, you have the following equations:
(If, for example, the system is accelerating horizontally, then you can then set a y = 0 . a y = 0 . ) We need this information to determine unknown forces acting on a system.
As always, we must check the solution. In some cases, it is easy to tell whether the solution is reasonable. For example, it is reasonable to find that friction causes an object to slide down an incline more slowly than when no friction exists. In practice, intuition develops gradually through problem solving; with experience, it becomes progressively easier to judge whether an answer is reasonable. Another way to check a solution is to check the units. If we are solving for force and end up with units of millimeters per second, then we have made a mistake.
There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills. We look first at problems involving particle equilibrium, which make use of Newton’s first law, and then consider particle acceleration, which involves Newton’s second law.
Particle Equilibrium
Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration, but it is important to remember that these conditions are relative. For example, an object may be at rest when viewed from our frame of reference, but the same object would appear to be in motion when viewed by someone moving at a constant velocity. We now make use of the knowledge attained in Newton’s Laws of Motion , regarding the different types of forces and the use of free-body diagrams, to solve additional problems in particle equilibrium .
Example 6.1
Different tensions at different angles.
Thus, as you might expect,
This gives us the following relationship:
Note that T 1 T 1 and T 2 T 2 are not equal in this case because the angles on either side are not equal. It is reasonable that T 2 T 2 ends up being greater than T 1 T 1 because it is exerted more vertically than T 1 . T 1 .
Now consider the force components along the vertical or y -axis:
This implies
Substituting the expressions for the vertical components gives
There are two unknowns in this equation, but substituting the expression for T 2 T 2 in terms of T 1 T 1 reduces this to one equation with one unknown:
which yields
Solving this last equation gives the magnitude of T 1 T 1 to be
Finally, we find the magnitude of T 2 T 2 by using the relationship between them, T 2 = 1.225 T 1 T 2 = 1.225 T 1 , found above. Thus we obtain
Significance
Particle acceleration.
We have given a variety of examples of particles in equilibrium. We now turn our attention to particle acceleration problems, which are the result of a nonzero net force. Refer again to the steps given at the beginning of this section, and notice how they are applied to the following examples.
Example 6.2
Drag force on a barge.
The drag of the water F → D F → D is in the direction opposite to the direction of motion of the boat; this force thus works against F → app , F → app , as shown in the free-body diagram in Figure 6.4 (b). The system of interest here is the barge, since the forces on it are given as well as its acceleration. Because the applied forces are perpendicular, the x - and y -axes are in the same direction as F → 1 F → 1 and F → 2 . F → 2 . The problem quickly becomes a one-dimensional problem along the direction of F → app F → app , since friction is in the direction opposite to F → app . F → app . Our strategy is to find the magnitude and direction of the net applied force F → app F → app and then apply Newton’s second law to solve for the drag force F → D . F → D .
The angle is given by
From Newton’s first law, we know this is the same direction as the acceleration. We also know that F → D F → D is in the opposite direction of F → app , F → app , since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as F → app , F → app , but its magnitude is slightly less than F → app . F → app . The problem is now one-dimensional. From the free-body diagram, we can see that
However, Newton’s second law states that
This can be solved for the magnitude of the drag force of the water F D F D in terms of known quantities:
Substituting known values gives
The direction of F → D F → D has already been determined to be in the direction opposite to F → app , F → app , or at an angle of 53 ° 53 ° south of west.
In Newton’s Laws of Motion , we discussed the normal force , which is a contact force that acts normal to the surface so that an object does not have an acceleration perpendicular to the surface. The bathroom scale is an excellent example of a normal force acting on a body. It provides a quantitative reading of how much it must push upward to support the weight of an object. But can you predict what you would see on the dial of a bathroom scale if you stood on it during an elevator ride? Will you see a value greater than your weight when the elevator starts up? What about when the elevator moves upward at a constant speed? Take a guess before reading the next example.
Example 6.3
What does the bathroom scale read in an elevator.
From the free-body diagram, we see that F → net = F → s − w → , F → net = F → s − w → , so we have
Solving for F s F s gives us an equation with only one unknown:
or, because w = m g , w = m g , simply
No assumptions were made about the acceleration, so this solution should be valid for a variety of accelerations in addition to those in this situation. ( Note: We are considering the case when the elevator is accelerating upward. If the elevator is accelerating downward, Newton’s second law becomes F s − w = − m a . F s − w = − m a . )
We have a = 1.20 m/s 2 , a = 1.20 m/s 2 , so that F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) F s = ( 75.0 kg ) ( 9.80 m/s 2 ) + ( 75.0 kg ) ( 1.20 m/s 2 ) yielding F s = 825 N . F s = 825 N .
Now, what happens when the elevator reaches a constant upward velocity? Will the scale still read more than his weight? For any constant velocity—up, down, or stationary—acceleration is zero because a = Δ v Δ t a = Δ v Δ t and Δ v = 0 . Δ v = 0 . Thus, F s = m a + m g = 0 + m g F s = m a + m g = 0 + m g or F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , F s = ( 75.0 kg ) ( 9.80 m/s 2 ) , which gives F s = 735 N . F s = 735 N .
Thus, the scale reading in the elevator is greater than his 735-N (165-lb.) weight. This means that the scale is pushing up on the person with a force greater than his weight, as it must in order to accelerate him upward. Clearly, the greater the acceleration of the elevator, the greater the scale reading, consistent with what you feel in rapidly accelerating versus slowly accelerating elevators. In Figure 6.5 (b), the scale reading is 735 N, which equals the person’s weight. This is the case whenever the elevator has a constant velocity—moving up, moving down, or stationary.
Check Your Understanding 6.1
Now calculate the scale reading when the elevator accelerates downward at a rate of 1.20 m/s 2 . 1.20 m/s 2 .
The solution to the previous example also applies to an elevator accelerating downward, as mentioned. When an elevator accelerates downward, a is negative, and the scale reading is less than the weight of the person. If a constant downward velocity is reached, the scale reading again becomes equal to the person’s weight. If the elevator is in free fall and accelerating downward at g , then the scale reading is zero and the person appears to be weightless.
Example 6.4
Two attached blocks.
For block 1: T → + w → 1 + N → = m 1 a → 1 T → + w → 1 + N → = m 1 a → 1
For block 2: T → + w → 2 = m 2 a → 2 . T → + w → 2 = m 2 a → 2 .
Notice that T → T → is the same for both blocks. Since the string and the pulley have negligible mass, and since there is no friction in the pulley, the tension is the same throughout the string. We can now write component equations for each block. All forces are either horizontal or vertical, so we can use the same horizontal/vertical coordinate system for both objects
When block 1 moves to the right, block 2 travels an equal distance downward; thus, a 1 x = − a 2 y . a 1 x = − a 2 y . Writing the common acceleration of the blocks as a = a 1 x = − a 2 y , a = a 1 x = − a 2 y , we now have
From these two equations, we can express a and T in terms of the masses m 1 and m 2 , and g : m 1 and m 2 , and g :
Check Your Understanding 6.2
Calculate the acceleration of the system, and the tension in the string, when the masses are m 1 = 5.00 kg m 1 = 5.00 kg and m 2 = 3.00 kg . m 2 = 3.00 kg .
Example 6.5
Atwood machine.
We have For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . For m 1 , ∑ F y = T − m 1 g = m 1 a . For m 2 , ∑ F y = T − m 2 g = − m 2 a . (The negative sign in front of m 2 a m 2 a indicates that m 2 m 2 accelerates downward; both blocks accelerate at the same rate, but in opposite directions.) Solve the two equations simultaneously (subtract them) and the result is ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . ( m 2 − m 1 ) g = ( m 1 + m 2 ) a . Solving for a : a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 . a = m 2 − m 1 m 1 + m 2 g = 4 kg − 2 kg 4 kg + 2 kg ( 9.8 m/s 2 ) = 3.27 m/s 2 .
Observing the first block, we see that T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N . T − m 1 g = m 1 a T = m 1 ( g + a ) = ( 2 kg ) ( 9.8 m/s 2 + 3.27 m/s 2 ) = 26.1 N .
Check Your Understanding 6.3
Determine a general formula in terms of m 1 , m 2 m 1 , m 2 and g for calculating the tension in the string for the Atwood machine shown above.
Newton’s Laws of Motion and Kinematics
Physics is most interesting and most powerful when applied to general situations that involve more than a narrow set of physical principles. Newton’s laws of motion can also be integrated with other concepts that have been discussed previously in this text to solve problems of motion. For example, forces produce accelerations, a topic of kinematics , and hence the relevance of earlier chapters.
When approaching problems that involve various types of forces, acceleration, velocity, and/or position, listing the givens and the quantities to be calculated will allow you to identify the principles involved. Then, you can refer to the chapters that deal with a particular topic and solve the problem using strategies outlined in the text. The following worked example illustrates how the problem-solving strategy given earlier in this chapter, as well as strategies presented in other chapters, is applied to an integrated concept problem.
Example 6.6
What force must a soccer player exert to reach top speed.
We are given the initial and final velocities (zero and 8.00 m/s forward); thus, the change in velocity is Δ v = 8.00 m/s Δ v = 8.00 m/s . We are given the elapsed time, so Δ t = 2.50 s . Δ t = 2.50 s . The unknown is acceleration, which can be found from its definition: a = Δ v Δ t . a = Δ v Δ t . Substituting the known values yields a = 8.00 m/s 2.50 s = 3.20 m/s 2 . a = 8.00 m/s 2.50 s = 3.20 m/s 2 .
Here we are asked to find the average force the ground exerts on the runner to produce this acceleration. (Remember that we are dealing with the force or forces acting on the object of interest.) This is the reaction force to that exerted by the player backward against the ground, by Newton’s third law. Neglecting air resistance, this would be equal in magnitude to the net external force on the player, since this force causes her acceleration. Since we now know the player’s acceleration and are given her mass, we can use Newton’s second law to find the force exerted. That is, F net = m a . F net = m a . Substituting the known values of m and a gives F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N . F net = ( 70.0 kg ) ( 3.20 m/s 2 ) = 224 N .
This is a reasonable result: The acceleration is attainable for an athlete in good condition. The force is about 50 pounds, a reasonable average force.
Check Your Understanding 6.4
The soccer player stops after completing the play described above, but now notices that the ball is in position to be stolen. If she now experiences a force of 126 N to attempt to steal the ball, which is 2.00 m away from her, how long will it take her to get to the ball?
Example 6.7
What force acts on a model helicopter.
The magnitude of the force is now easily found:
Check Your Understanding 6.5
Find the direction of the resultant for the 1.50-kg model helicopter.
Example 6.8
Baggage tractor.
∑ F x = m system a x ∑ F x = m system a x and ∑ F x = 820.0 t , ∑ F x = 820.0 t , so 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . 820.0 t = ( 650.0 + 250.0 + 150.0 ) a a = 0.7809 t . Since acceleration is a function of time, we can determine the velocity of the tractor by using a = d v d t a = d v d t with the initial condition that v 0 = 0 v 0 = 0 at t = 0 . t = 0 . We integrate from t = 0 t = 0 to t = 3 : t = 3 : d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s . d v = a d t , ∫ 0 3 d v = ∫ 0 3.00 a d t = ∫ 0 3.00 0.7809 t d t , v = 0.3905 t 2 ] 0 3.00 = 3.51 m/s .
Refer to the free-body diagram in Figure 6.8 (b). ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N . ∑ F x = m tractor a x 820.0 t − T = m tractor ( 0.7805 ) t ( 820.0 ) ( 3.00 ) − T = ( 650.0 ) ( 0.7805 ) ( 3.00 ) T = 938 N .
Recall that v = d s d t v = d s d t and a = d v d t a = d v d t . If acceleration is a function of time, we can use the calculus forms developed in Motion Along a Straight Line , as shown in this example. However, sometimes acceleration is a function of displacement. In this case, we can derive an important result from these calculus relations. Solving for dt in each, we have d t = d s v d t = d s v and d t = d v a . d t = d v a . Now, equating these expressions, we have d s v = d v a . d s v = d v a . We can rearrange this to obtain a d s = v d v . a d s = v d v .
Example 6.9
Motion of a projectile fired vertically.
The acceleration depends on v and is therefore variable. Since a = f ( v ) , a = f ( v ) , we can relate a to v using the rearrangement described above,
We replace ds with dy because we are dealing with the vertical direction,
We now separate the variables ( v ’s and dv ’s on one side; dy on the other):
Thus, h = 114 m . h = 114 m .
Check Your Understanding 6.6
If atmospheric resistance is neglected, find the maximum height for the mortar shell. Is calculus required for this solution?
Interactive
Explore the forces at work in this simulation when you try to push a filing cabinet. Create an applied force and see the resulting frictional force and total force acting on the cabinet. Charts show the forces, position, velocity, and acceleration vs. time. View a free-body diagram of all the forces (including gravitational and normal forces).
As an Amazon Associate we earn from qualifying purchases.
This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.
Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.
Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction
Authors: William Moebs, Samuel J. Ling, Jeff Sanny
Publisher/website: OpenStax
Book title: University Physics Volume 1
Publication date: Sep 19, 2016
Location: Houston, Texas
Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
PRO Courses Guides New Tech Help Pro Expert Videos About wikiHow Pro Upgrade Sign In
EDIT Edit this Article
EXPLORE Tech Help Pro About Us Random Article Quizzes Request a New Article Community Dashboard This Or That Game Popular Categories Arts and Entertainment Artwork Books Movies Computers and Electronics Computers Phone Skills Technology Hacks Health Men's Health Mental Health Women's Health Relationships Dating Love Relationship Issues Hobbies and Crafts Crafts Drawing Games Education & Communication Communication Skills Personal Development Studying Personal Care and Style Fashion Hair Care Personal Hygiene Youth Personal Care School Stuff Dating All Categories Arts and Entertainment Finance and Business Home and Garden Relationship Quizzes Cars & Other Vehicles Food and Entertaining Personal Care and Style Sports and Fitness Computers and Electronics Health Pets and Animals Travel Education & Communication Hobbies and Crafts Philosophy and Religion Work World Family Life Holidays and Traditions Relationships Youth
Browse Articles
Learn Something New
Quizzes Hot
This Or That Game
Train Your Brain
Explore More
Support wikiHow
About wikiHow
Log in / Sign up
Education and Communications
How to Solve Any Physics Problem
Last Updated: July 21, 2023 Fact Checked
This article was co-authored by Sean Alexander, MS . Sean Alexander is an Academic Tutor specializing in teaching mathematics and physics. Sean is the Owner of Alexander Tutoring, an academic tutoring business that provides personalized studying sessions focused on mathematics and physics. With over 15 years of experience, Sean has worked as a physics and math instructor and tutor for Stanford University, San Francisco State University, and Stanbridge Academy. He holds a BS in Physics from the University of California, Santa Barbara and an MS in Theoretical Physics from San Francisco State University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 329,393 times.
Baffled as to where to begin with a physics problem? There is a very simply and logical flow process to solving any physics problem.
Ask yourself if your answers make sense. If the numbers look absurd (for example, you get that a rock dropped off a 50-meter cliff moves with the speed of only 0.00965 meters per second when it hits the ground), you made a mistake somewhere.
Don't forget to include the units into your answers, and always keep track of them. So, if you are solving for velocity and get your answer in seconds, that is a sign that something went wrong, because it should be in meters per second.
Plug your answers back into the original equations to make sure you get the same number on both sides.
Community Q&A
Many people report that if they leave a problem for a while and come back to it later, they find they have a new perspective on it and can sometimes see an easy way to the answer that they did not notice before. Thanks Helpful 249 Not Helpful 48
Try to understand the problem first. Thanks Helpful 186 Not Helpful 51
Remember, the physics part of the problem is figuring out what you are solving for, drawing the diagram, and remembering the formulae. The rest is just use of algebra, trigonometry, and/or calculus, depending on the difficulty of your course. Thanks Helpful 115 Not Helpful 34
Physics is not easy to grasp for many people, so do not get bent out of shape over a problem. Thanks Helpful 100 Not Helpful 25
If an instructor tells you to draw a free body diagram, be sure that that is exactly what you draw. Thanks Helpful 89 Not Helpful 24
Things You'll Need
A Writing Utensil (preferably a pencil or erasable pen of sorts)
Calculator with all the functions you need for your exam
An understanding of the equations needed to solve the problems. Or a list of them will suffice if you are just trying to get through the course alive.
You Might Also Like
Expert Interview
Thanks for reading our article! If you’d like to learn more about teaching, check out our in-depth interview with Sean Alexander, MS .
Level up your tech skills and stay ahead of the curve
Exam Center
Ticket Center
Flash Cards
Straight Line Motion
Solved Speed, Velocity, and Acceleration Problems
Simple problems on speed, velocity, and acceleration with descriptive answers are presented for the AP Physics 1 exam and college students. In each solution, you can find a brief tutorial.
Speed and velocity Problems:
Problem (1): What is the speed of a rocket that travels $8000\,{\rm m}$ in $13\,{\rm s}$?
Solution : Speed is defined in physics as the total distance divided by the elapsed time, so the rocket's speed is \[\text{speed}=\frac{8000}{13}=615.38\,{\rm m/s}\]
Problem (2): How long will it take if you travel $400\,{\rm km}$ with an average speed of $100\,{\rm m/s}$?
Solution : Average speed is the ratio of the total distance to the total time. Thus, the elapsed time is \begin{align*} t&=\frac{\text{total distance}}{\text{average speed}}\\ \\ &=\frac{400\times 10^{3}\,{\rm m}}{100\,{\rm m/s}}\\ \\ &=4000\,{\rm s}\end{align*} To convert it to hours, it must be divided by $3600\,{\rm s}$ which gives $t=1.11\,{\rm h}$.
Problem (3): A person walks $100\,{\rm m}$ in $5$ minutes, then $200\,{\rm m}$ in $7$ minutes, and finally $50\,{\rm m}$ in $4$ minutes. Find its average speed.
Solution : First find its total distance traveled ($D$) by summing all distances in each section, which gets $D=100+200+50=350\,{\rm m}$. Now, by definition of average speed, divide it by the total time elapsed $T=5+7+4=16$ minutes.
But keep in mind that since the distance is in SI units, so the time traveled must also be in SI units, which is $\rm s$. Therefore, we have\begin{align*}\text{average speed}&=\frac{\text{total distance} }{\text{total time} }\\ \\ &=\frac{350\,{\rm m}}{16\times 60\,{\rm s}}\\ \\&=0.36\,{\rm m/s}\end{align*}
Problem (4): A person walks $750\,{\rm m}$ due north, then $250\,{\rm m}$ due east. If the entire walk takes $12$ minutes, find the person's average velocity.
Solution : Average velocity , $\bar{v}=\frac{\Delta x}{\Delta t}$, is displacement divided by the elapsed time. Displacement is also a vector that obeys the addition vector rules. Thus, in this velocity problem, add each displacement to get the total displacement .
In the first part, displacement is $\Delta x_1=750\,\hat{j}$ (due north) and in the second part $\Delta x_2=250\,\hat{i}$ (due east). The total displacement vector is $\Delta x=\Delta x_1+\Delta x_2=750\,\hat{i}+250\,\hat{j}$ with magnitude of \begin{align*}|\Delta x|&=\sqrt{(750)^{2}+(250)^{2}}\\ \\&=790.5\,{\rm m}\end{align*} In addition, the total elapsed time is $t=12\times 60$ seconds. Therefore, the magnitude of the average velocity is \[\bar{v}=\frac{790.5}{12\times 60}=1.09\,{\rm m/s}\]
Problem (5): An object moves along a straight line. First, it travels at a velocity of $12\,{\rm m/s}$ for $5\,{\rm s}$ and then continues in the same direction with $20\,{\rm m/s}$ for $3\,{\rm s}$. What is its average speed?
Solution: Average velocity is displacement divided by elapsed time, i.e., $\bar{v}\equiv \frac{\Delta x_{tot}}{\Delta t_{tot}}$.
Here, the object goes through two stages with two different displacements, so add them to find the total displacement. Thus,\[\bar{v}=\frac{x_1 + x_2}{t_1 +t_2}\] Again, to find the displacement, we use the same equation as the average velocity formula, i.e., $x=vt$. Thus, displacements are obtained as $x_1=v_1\,t_1=12\times 5=60\,{\rm m}$ and $x_2=v_2\,t_2=20\times 3=60\,{\rm m}$. Therefore, we have \begin{align*} \bar{v}&=\frac{x_1+x_2}{t_1+t_2}\\ \\&=\frac{60+60}{5+3}\\ \\&=\boxed{15\,{\rm m/s}}\end{align*}
Problem (6): A plane flies the distance between two cities in $1$ hour and $30$ minutes with a velocity of $900\,{\rm km/h}$. Another plane covers that distance at $600\,{\rm km/h}$. What is the flight time of the second plane?
Solution: first find the distance between two cities using the average velocity formula $\bar{v}=\frac{\Delta x}{\Delta t}$ as below \begin{align*} x&=vt\\&=900\times 1.5\\&=1350\,{\rm km}\end{align*} where we wrote one hour and a half minutes as $1.5\,\rm h$. Now use again the same kinematic equation above to find the time required for another plane \begin{align*} t&=\frac xv\\ \\ &=\frac{1350\,\rm km}{600\,\rm km/h}\\ \\&=2.25\,{\rm h}\end{align*} Thus, the time for the second plane is $2$ hours and $0.25$ of an hour, which converts to minutes as $2$ hours and ($0.25\times 60=15$) minutes.
Problem (7): To reach a park located south of his jogging path, Henry runs along a 15-kilometer route. If he completes the journey in 1.5 hours, determine his speed and velocity.
Solution: Henry travels his route to the park without changing direction along a straight line. Therefore, the total distance traveled in one direction equals the displacement, i.e, \[\text{distance traveled}=\Delta x=15\,\rm km\]Velocity is displacement divided by the time of travel \begin{align*} \text{velocity}&=\frac{\text{displacement}}{\text{time of travel}} \\\\ &=\frac{15\,\rm km}{1.5\,\rm h} \\\\ &=\boxed{10\,\rm km/h}\end{align*} and by definition, its average speed is \begin{align*} \text{speed}&=\frac{\text{distance covered}}{\text{time interval}}\\\\&=\frac{15\,\rm km}{1.5\,\rm h}\\\\&=\boxed{10\,\rm km/h}\end{align*} Thus, Henry's velocity is $10\,\rm km/h$ to the south, and its speed is $10\,\rm km/h$. As you can see, speed is simply a positive number, with units but velocity specifies the direction in which the object is moving.
Problem (8): In 15 seconds, a football player covers the distance from his team's goal line to the opposing team's goal line and back to the midway point of the field having 100-yard-length. Find, (a) his average speed, and (b) the magnitude of the average velocity.
Solution: The total length of the football field is $100$ yards or in meters, $L=91.44\,\rm m$. Going from one goal's line to the other and back to the midpoint of the field takes $15\,\rm s$ and covers a distance of $D=100+50=150\,\rm yd$.
Distance divided by the time of travel gets the average speed, \[\text{speed}=\frac{150\times 0.91}{15}=9.1\,\rm m/s\] To find the average velocity, we must find the displacement of the player between the initial and final points.
The initial point is her own goal line and her final position is the midpoint of the field, so she has displaced a distance of $\Delta x=50\,\rm yd$ or $\Delta x=50\times 0.91=45.5\,\rm m$. Therefore, her velocity is calculated as follows \begin{align*} \text{velocity}&=\frac{\text{displacement}}{\text{time elapsed}} \\\\ &=\frac{45.5\,\rm m}{15\,\rm s} \\\\&=\boxed{3.03\quad \rm m/s}\end{align*} Contrary to the previous problem, here the motion is not in one direction, hence, the displacement is not equal to the distance traveled. Accordingly, the average speed is not equal to the magnitude of the average velocity.
Problem (9): You begin at a pillar and run towards the east (the positive $x$ direction) for $250\,\rm m$ at an average speed of $5\,\rm m/s$. After that, you run towards the west for $300\,\rm m$ at an average speed of $4\,\rm m/s$ until you reach a post. Calculate (a) your average speed from pillar to post, and (b) your average velocity from pillar to post.
Solution : First, you traveled a distance of $L_1=250\,\rm m$ toward east (or $+x$ direction) at $5\,\rm m/s$. Time of travel in this route is obtained as follows \begin{align*} t_1&=\frac{L_1}{v_1}\\\\ &=\frac{250}{5}\\\\&=50\,\rm s\end{align*} Likewise, traveling a distance of $L_2=300\,\rm m$ at $v_2=4\,\rm m/s$ takes \[t_2=\frac{300}{4}=75\,\rm s\] (a) Average speed is defined as the distance traveled (or path length) divided by the total time of travel \begin{align*} v&=\frac{\text{path length}}{\text{time of travel}} \\\\ &=\frac{L_1+L_2}{t_1+t_2}\\\\&=\frac{250+300}{50+75} \\\\&=4.4\,\rm m/s\end{align*} Therefore, you travel between these two pillars in $125\,\rm s$ and with an average speed of $4.4\,\rm m/s$.
(b) Average velocity requires finding the displacement between those two points. In the first case, you move $250\,\rm m$ toward $+x$ direction, i.e., $L_1=+250\,\rm m$. Similarly, on the way back, you move $300\,\rm m$ toward the west ($-x$ direction) or $L_2=-300\,\rm m$. Adding these two gives us the total displacement between the initial point and the final point, \begin{align*} L&=L_1+L_2 \\\\&=(+250)+(-300) \\\\ &=-50\,\rm m\end{align*} The minus sign indicates that you are generally displaced toward the west.
Finally, the average velocity is obtained as follows: \begin{align*} \text{average velocity}&=\frac{\text{displacement}}{\text{time of travel}} \\\\ &=\frac{-50}{125} \\\\&=-0.4\,\rm m/s\end{align*} A negative average velocity indicating motion to the left along the $x$-axis.
This speed problem better makes it clear to us the difference between average speed and average speed. Unlike average speed, which is always a positive number, the average velocity in a straight line can be either positive or negative.
Problem (10): What is the average speed for the round trip of a car moving uphill at 40 km/h and then back downhill at 60 km/h?
Solution : Assuming the length of the hill to be $L$, the total distance traveled during this round trip is $2L$ since $L_{up}=L_{down}=L$. However, the time taken for going uphill and downhill was not provided. We can write them in terms of the hill's length $L$ as $t=\frac L v$.
Applying the definition of average speed gives us \begin{align*} v&=\frac{\text{distance traveled}}{\text{total time}} \\\\ &=\frac{L_{up}+L_{down}}{t_{up}+t_{down}} \\\\ &=\cfrac{2L}{\cfrac{L}{v_{up}}+\cfrac{L}{v_{down}}} \end{align*} By reorganizing this expression, we obtain a formula that is useful for solving similar problems in the AP Physics 1 exams. \[\text{average speed}=\frac{2v_{up} \times v_{down}}{v_{up}+v_{down}}\] Substituting the numerical values into this, yields \begin{align*} v&=\frac{2(40\times 60)}{40+60} \\\\ &=\boxed{48\,\rm m/s}\end{align*} What if we were asked for the average velocity instead? During this round trip, the car returns to its original position, and thus its displacement, which defines the average velocity, is zero. Therefore, \[\text{average velocity}=0\,\rm m/s\]
Acceleration Problems
Problem (9): A car moves from rest to a speed of $45\,\rm m/s$ in a time interval of $15\,\rm s$. At what rate does the car accelerate?
Solution : The car is initially at rest, $v_1=0$, and finally reaches $v_2=45\,\rm m/s$ in a time interval $\Delta t=15\,\rm s$. Average acceleration is the change in velocity, $\Delta v=v_2-v_1$, divided by the elapsed time $\Delta t$, so \[\bar{a}=\frac{45-0}{15}=\boxed{3\,\rm m/s^2} \]
Problem (10): A car moving at a velocity of $15\,{\rm m/s}$, uniformly slows down. It comes to a complete stop in $10\,{\rm s}$. What is its acceleration?
Solution: Let the car's uniform velocity be $v_1$ and its final velocity $v_2=0$. Average acceleration is the difference in velocities divided by the time taken, so we have: \begin{align*}\bar{a}&=\frac{\Delta v}{\Delta t}\\\\&=\frac{v_2-v_1}{\Delta t}\\\\&=\frac{0-15}{10}\\\\ &=\boxed{-1.5\,{\rm m/s^2}}\end{align*}The minus sign indicates the direction of the acceleration vector, which is toward the $-x$ direction.
Problem (11): A car moves from rest to a speed of $72\,{\rm km/h}$ in $4\,{\rm s}$. Find the acceleration of the car.
Solution: Known: $v_1=0$, $v_2=72\,{\rm km/h}$, $\Delta t=4\,{\rm s}$. Average acceleration is defined as the difference in velocities divided by the time interval between those points \begin{align*}\bar{a}&=\frac{v_2-v_1}{t_2-t_1}\\\\&=\frac{20-0}{4}\\\\&=5\,{\rm m/s^2}\end{align*} In above, we converted $\rm km/h$ to the SI unit of velocity ($\rm m/s$) as \[1\,\frac{km}{h}=\frac {1000\,m}{3600\,s}=\frac{10}{36}\, \rm m/s\] so we get \[72\,\rm km/h=72\times \frac{10}{36}=20\,\rm m/s\]
Problem (12): A race car accelerates from an initial velocity of $v_i=10\,{\rm m/s}$ to a final velocity of $v_f = 30\,{\rm m/s}$ in a time interval of $2\,{\rm s}$. Determine its average acceleration.
Solution: A change in the velocity of an object $\Delta v$ over a time interval $\Delta t$ is defined as an average acceleration. Known: $v_i=10\,{\rm m/s}$, $v_f = 30\,{\rm m/s}$, $\Delta t=2\,{\rm s}$. Applying definition of average acceleration, we get \begin{align*}\bar{a}&=\frac{v_f-v_i}{\Delta t}\\&=\frac{30-10}{2}\\&=10\,{\rm m/s^2}\end{align*}
Problem (13): A motorcycle starts its trip along a straight line with a velocity of $10\,{\rm m/s}$ and ends with $20\,{\rm m/s}$ in the opposite direction in a time interval of $2\,{\rm s}$. What is the average acceleration of the car?
Solution: Known: $v_i=10\,{\rm m/s}$, $v_f=-20\,{\rm m/s}$, $\Delta t=2\,{\rm s}$, $\bar{a}=?$. Using average acceleration definition we have \begin{align*}\bar{a}&=\frac{v_f-v_i}{\Delta t}\\\\&=\frac{(-20)-10}{2}\\\\ &=\boxed{-15\,{\rm m/s^2}}\end{align*}Recall that in the definition above, velocities are vector quantities. The final velocity is in the opposite direction from the initial velocity so a negative must be included.
Problem (14): A ball is thrown vertically up into the air by a boy. After $4$ seconds, it reaches the highest point of its path. How fast does the ball leave the boy's hand?
Solution : At the highest point, the ball has zero speed, $v_2=0$. It takes the ball $4\,\rm s$ to reach that point. In this problem, our unknown is the initial speed of the ball, $v_1=?$. Here, the ball accelerates at a constant rate of $g=-9.8\,\rm m/s^2$ in the presence of gravity.
When the ball is tossed upward, the only external force that acts on it is the gravity force.
Using the average acceleration formula $\bar{a}=\frac{\Delta v}{\Delta t}$ and substituting the numerical values into this, we will have \begin{gather*} \bar{a}=\frac{\Delta v}{\Delta t} \\\\ -9.8=\frac{0-v_1}{4} \\\\ \Rightarrow \boxed{v_1=39.2\,\rm m/s} \end{gather*} Note that $\Delta v=v_2-v_1$.
Problem (15): A child drops crumpled paper from a window. The paper hit the ground in $3\,\rm s$. What is the velocity of the crumpled paper just before it strikes the ground?
Solution : The crumpled paper is initially in the child's hand, so $v_1=0$. Let its speed just before striking be $v_2$. In this case, we have an object accelerating down in the presence of gravitational force at a constant rate of $g=-9.8\,\rm m/s^2$. Using the definition of average acceleration, we can find $v_2$ as below \begin{gather*} \bar{a}=\frac{\Delta v}{\Delta t} \\\\ -9.8=\frac{v_2-0}{3} \\\\ \Rightarrow v_2=3\times (-9.8)=\boxed{-29.4\,\rm m/s} \end{gather*} The negative shows us that the velocity must be downward, as expected!
Problem (16): A car travels along the $x$-axis for $4\,{\rm s}$ at an average velocity of $10\,{\rm m/s}$ and $2\,{\rm s}$ with an average velocity of $30\,{\rm m/s}$ and finally $4\,{\rm s}$ with an average velocity $25\,{\rm m/s}$. What is its average velocity across the whole path?
Solution: There are three different parts with different average velocities. Assume each trip is done in one dimension without changing direction. Thus, displacements associated with each segment are the same as the distance traveled in that direction and is calculated as below: \begin{align*}\Delta x_1&=v_1\,\Delta t_1\\&=10\times 4=40\,{\rm m}\\ \\ \Delta x_2&=v_2\,\Delta t_2\\&=30\times 2=60\,{\rm m}\\ \\ \Delta x_3&=v_3\,\Delta t_3\\&=25\times 4=100\,{\rm m}\end{align*}Now use the definition of average velocity, $\bar{v}=\frac{\Delta x_{tot}}{\Delta t_{tot}}$, to find it over the whole path\begin{align*}\bar{v}&=\frac{\Delta x_{tot}}{\Delta t_{tot}}\\ \\&=\frac{\Delta x_1+\Delta x_2+\Delta x_3}{\Delta t_1+\Delta t_2+\Delta t_3}\\ \\&=\frac{40+60+100}{4+2+4}\\ \\ &=\boxed{20\,{\rm m/s}}\end{align*}
Problem (17): An object moving along a straight-line path. It travels with an average velocity $2\,{\rm m/s}$ for $20\,{\rm s}$ and $12\,{\rm m/s}$ for $t$ seconds. If the total average velocity across the whole path is $10\,{\rm m/s}$, then find the unknown time $t$.
Solution: In this velocity problem, the whole path $\Delta x$ is divided into two parts $\Delta x_1$ and $\Delta x_2$ with different average velocities and times elapsed, so the total average velocity across the whole path is obtained as \begin{align*}\bar{v}&=\frac{\Delta x}{\Delta t}\\\\&=\frac{\Delta x_1+\Delta x_2}{\Delta t_1+\Delta t_2}\\\\&=\frac{\bar{v}_1\,t_1+\bar{v}_2\,t_2}{t_1+t_2}\\\\10&=\frac{2\times 20+12\times t}{20+t}\\\Rightarrow t&=80\,{\rm s}\end{align*}
Note : whenever a moving object, covers distances $x_1,x_2,x_3,\cdots$ in $t_1,t_2,t_3,\cdots$ with constant or average velocities $v_1,v_2,v_3,\cdots$ along a straight-line without changing its direction, then its total average velocity across the whole path is obtained by one of the following formulas
Distances and times are known:\[\bar{v}=\frac{x_1+x_2+x_3+\cdots}{t_1+t_2+t_3+\cdots}\]
Velocities and times are known: \[\bar{v}=\frac{v_1\,t_1+v_2\,t_2+v_3\,t_3+\cdots}{t_1+t_2+t_3+\cdots}\]
Distances and velocities are known:\[\bar{v}=\frac{x_1+x_2+x_3+\cdots}{\frac{x_1}{v_1}+\frac{x_2}{v_2}+\frac{x_3}{v_3}+\cdots}\]
Problem (18): A car travels one-fourth of its path with a constant velocity of $10\,{\rm m/s}$, and the remaining with a constant velocity of $v_2$. If the total average velocity across the whole path is $16\,{\rm m/s}$, then find the $v_2$?
Solution: This is the third case of the preceding note. Let the length of the path be $L$ so \begin{align*}\bar{v}&=\frac{x_1+x_2}{\frac{x_1}{v_1}+\frac{x_2}{v_2}}\\\\16&=\frac{\frac 14\,L+\frac 34\,L}{\frac{\frac 14\,L}{10}+\frac{\frac 34\,L}{v_2}}\\\\\Rightarrow v_2&=20\,{\rm m/s}\end{align*}
Problem (19): An object moves along a straight-line path. It travels for $t_1$ seconds with an average velocity $50\,{\rm m/s}$ and $t_2$ seconds with a constant velocity of $25\,{\rm m/s}$. If the total average velocity across the whole path is $30\,{\rm m/s}$, then find the ratio $\frac{t_2}{t_1}$?
Solution: the velocities and times are known, so we have \begin{align*}\bar{v}&=\frac{v_1\,t_1+v_2\,t_2}{t_1+t_2}\\\\30&=\frac{50\,t_1+25\,t_2}{t_1+t_2}\\\\ \Rightarrow \frac{t_2}{t_1}&=4\end{align*}
Read more related articles:
Kinematics Equations: Problems and Solutions
Position vs. Time Graphs
Velocity vs. Time Graphs
In the following section, some sample AP Physics 1 problems on acceleration are provided.
Problem (20): An object moves with constant acceleration along a straight line. If its velocity at instant of $t_1 = 3\,{\rm s}$ is $10\,{\rm m/s}$ and at the moment of $t_2 = 8\,{\rm s}$ is $20\,{\rm m/s}$, then what is its initial speed?
Solution: Let the initial speed at time $t=0$ be $v_0$. Now apply average acceleration definition in the time intervals $[t_0,t_1]$ and $[t_0,t_2]$ and equate them.\begin{align*}\text{average acceleration}\ \bar{a}&=\frac{\Delta v}{\Delta t}\\\\\frac{v_1 - v_0}{t_1-t_0}&=\frac{v_2-v_0}{t_2-t_0}\\\\ \frac{10-v_0}{3-0}&=\frac{20-v_0}{8-0}\\\\ \Rightarrow v_0 &=4\,{\rm m/s}\end{align*} In the above, $v_1$ and $v_2$ are the velocities at moments $t_1$ and $t_2$, respectively.
Problem (21): For $10\,{\rm s}$, the velocity of a car that travels with a constant acceleration, changes from $10\,{\rm m/s}$ to $30\,{\rm m/s}$. How far does the car travel?
Solution: Known: $\Delta t=10\,{\rm s}$, $v_1=10\,{\rm m/s}$ and $v_2=30\,{\rm m/s}$.
Method (I) Without computing the acceleration: Recall that in the case of constant acceleration, we have the following kinematic equations for average velocity and displacement:\begin{align*}\text{average velocity}:\,\bar{v}&=\frac{v_1+v_2}{2}\\\text{displacement}:\,\Delta x&=\frac{v_1+v_2}{2}\times \Delta t\\\end{align*}where $v_1$ and $v_2$ are the velocities in a given time interval. Now we have \begin{align*} \Delta x&=\frac{v_1+v_2}{2}\\&=\frac{10+30}{2}\times 10\\&=200\,{\rm m}\end{align*}
Method (II) with computing acceleration: Using the definition of average acceleration, first determine it as below \begin{align*}\bar{a}&=\frac{\Delta v}{\Delta t}\\\\&=\frac{30-10}{10}\\\\&=2\,{\rm m/s^2}\end{align*} Since the velocities at the initial and final points of the problem are given so use the below time-independent kinematic equation to find the required displacement \begin{align*} v_2^{2}-v_1^{2}&=2\,a\Delta x\\\\ (30)^{2}-(10)^{2}&=2(2)\,\Delta x\\\\ \Rightarrow \Delta x&=\boxed{200\,{\rm m}}\end{align*}
Problem (22): A car travels along a straight line with uniform acceleration. If its velocity at the instant of $t_1=2\,{\rm s}$ is $36\,{\rm km/s}$ and at the moment $t_2=6\,{\rm s}$ is $72\,{\rm km/h}$, then find its initial velocity (at $t_0=0$)?
Solution: Use the equality of definition of average acceleration $a=\frac{v_f-v_i}{t_f-t_i}$ in the time intervals $[t_0,t_1]$ and $[t_0,t_2]$ to find the initial velocity as below \begin{align*}\frac{v_2-v_0}{t_2-t_0}&=\frac{v_1-v_0}{t_1-t_0}\\\\ \frac{20-v_0}{6-0}&=\frac{10-v_0}{2-0}\\\\ \Rightarrow v_0&=\boxed{5\,{\rm m/s}}\end{align*}
All these kinematic problems on speed, velocity, and acceleration are easily solved by choosing an appropriate kinematic equation. Keep in mind that these motion problems in one dimension are of the uniform or constant acceleration type. Projectiles are also another type of motion in two dimensions with constant acceleration.
Active students, messages sent, images uploaded, free learning, end of free trial.
Unlock faster, more accurate responses + 20 more PRO features.
Phy Pro Trial -
Free but limited access to Phy Pro. Need more power?
NEW - Bookmark Phy.Chat
Need to access this page fast? Just type in Phy.Chat into google.
Snap a picture of the problem straight from your phone.
Smart Options
Phy automatically generates short follow ups. Just click it.
New - Learning Lab
Customize your learning to help Phy adapt to you even quicker.
Coming Soon - Chat Notes
View all chats with Phy, save to notes, & create study guides.
Phy Adaptive Engine®
The more you solve, the better Phy adapts to your learning style.
Learn it. Solve it. Grade it. Explain it. With Phy.
Free Response Question? Upload a image of your working. Phy will grade it.
Teacher didn’t explain it? Take a picture of the board and give it to Phy.
Can’t solve a problem? Phy can. And it will show you the best approach.
Upgrade to Phy Pro.
Phy Version 8 (3.20.24) - Systems Operational
The most advanced version of Phy. Currently 50% off, for early supporters.
Billed Monthly. Cancel Anytime.
Trial –> Phy Pro
Unlimited Messages
Unlimited Image Uploads
Unlimited Smart Actions
Unlimited UBQ Credits
30 --> 300 Word Input
3 --> 15 MB Image Size Limit
1 --> 3 Images per Message
200% Memory Boost
150% Better than GPT
75% More Accurate, 50% Faster
Mobile Snaps
Prof Phy ULTRA
Access will be given out on a rolling basis. You must have an active Phy Pro subscription, for at least 90 days, to automatically join the waitlist.
Features include:
Save To Notes
Personalize Phy
Smart Actions V2
Instant Responses
Phy Adaptive Engine
Share Phy.Chat
Enjoying Phy? Share the 🔗 with friends!
Welcome to Phy Panel.
Here you can customize Phy to your preferences. Currently available to only Ultra users. Pro users will get access on a rolling basis.
Not currently eligible to use Phy Panel.
Report a bug.
What went wrong?
You must be signed in to leave feedback
Discover the world's best Physics resources
Continue with.
By continuing you (1) agree to our Terms of Sale and Terms of Use and (2) consent to sharing your IP and browser information used by this site’s security protocols as outlined in our Privacy Policy .
Figure 1.13 Problem-solving skills are essential to your success in physics. (credit: “scui3asteveo”/Flickr)
Problem-solving skills are clearly essential to success in a quantitative course in physics. More important, the ability to apply broad physical principles—usually represented by equations—to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life.
As you are probably well aware, a certain amount of creativity and insight is required to solve problems. No rigid procedure works every time. Creativity and insight grow with experience. With practice, the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and then progressing to the more difficult. After you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.
Although there is no simple step-by-step method that works for every problem, the following three-stage process facilitates problem solving and makes it more meaningful. The three stages are strategy, solution, and significance. This process is used in examples throughout the book. Here, we look at each stage of the process in turn.
Strategy is the beginning stage of solving a problem. The idea is to figure out exactly what the problem is and then develop a strategy for solving it. Some general advice for this stage is as follows:
Examine the situation to determine which physical principles are involved . It often helps to draw a simple sketch at the outset. You often need to decide which direction is positive and note that on your sketch. When you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless.
Make a list of what is given or can be inferred from the problem as stated (identify the “knowns”) . Many problems are stated very succinctly and require some inspection to determine what is known. Drawing a sketch can be very useful at this point as well. Formally identifying the knowns is of particular importance in applying physics to real-world situations. For example, the word stopped means the velocity is zero at that instant. Also, we can often take initial time and position as zero by the appropriate choice of coordinate system.
Identify exactly what needs to be determined in the problem (identify the unknowns) . In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help identify the unknowns.
Determine which physical principles can help you solve the problem . Since physical principles tend to be expressed in the form of mathematical equations, a list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all the other variables are known—so you can solve for the unknown easily. If the equation contains more than one unknown, then additional equations are needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea of equations. You may have to use two (or more) different equations to get the final answer.
The solution stage is when you do the math. Substitute the knowns (along with their units) into the appropriate equation and obtain numerical solutions complete with units . That is, do the algebra, calculus, geometry, or arithmetic necessary to find the unknown from the knowns, being sure to carry the units through the calculations. This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process.
Significance
After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is not math. Rather, in doing physics, we use mathematics as a tool to help us understand nature. So, after you obtain a numerical answer, you should always assess its significance:
Check your units. If the units of the answer are incorrect, then an error has been made and you should go back over your previous steps to find it. One way to find the mistake is to check all the equations you derived for dimensional consistency. However, be warned that correct units do not guarantee the numerical part of the answer is also correct.
Check the answer to see whether it is reasonable. Does it make sense? This step is extremely important: –the goal of physics is to describe nature accurately. To determine whether the answer is reasonable, check both its magnitude and its sign, in addition to its units. The magnitude should be consistent with a rough estimate of what it should be. It should also compare reasonably with magnitudes of other quantities of the same type. The sign usually tells you about direction and should be consistent with your prior expectations. Your judgment will improve as you solve more physics problems, and it will become possible for you to make finer judgments regarding whether nature is described adequately by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to solve a problem mechanically.
Check to see whether the answer tells you something interesting. What does it mean? This is the flip side of the question: Does it make sense? Ultimately, physics is about understanding nature, and we solve physics problems to learn a little something about how nature operates. Therefore, assuming the answer does make sense, you should always take a moment to see if it tells you something about the world that you find interesting. Even if the answer to this particular problem is not very interesting to you, what about the method you used to solve it? Could the method be adapted to answer a question that you do find interesting? In many ways, it is in answering questions such as these that science progresses.
The three stages of the process for solving physics problems used in this book are as follows:
Strategy : Determine which physical principles are involved and develop a strategy for using them to solve the problem.
Solution : Do the math necessary to obtain a numerical solution complete with units.
Significance : Check the solution to make sure it makes sense (correct units, reasonable magnitude and sign) and assess its significance.
Conceptual Questions
What information do you need to choose which equation or equations to use to solve a problem?
What should you do after obtaining a numerical answer when solving a problem?
Check to make sure it makes sense and assess its significance.
Additional Problems
Consider the equation y = mt +b , where the dimension of y is length and the dimension of t is time, and m and b are constants. What are the dimensions and SI units of (a) m and (b) b ?
Consider the equation [latex] s={s}_{0}+{v}_{0}t+{a}_{0}{t}^{2}\text{/}2+{j}_{0}{t}^{3}\text{/}6+{S}_{0}{t}^{4}\text{/}24+c{t}^{5}\text{/}120, [/latex] where s is a length and t is a time. What are the dimensions and SI units of (a) [latex] {s}_{0}, [/latex] (b) [latex] {v}_{0}, [/latex] (c) [latex] {a}_{0}, [/latex] (d) [latex] {j}_{0}, [/latex] (e) [latex] {S}_{0}, [/latex] and (f) c ?
a. [latex] [{s}_{0}]=\text{L} [/latex] and units are meters (m); b. [latex] [{v}_{0}]={\text{LT}}^{-1} [/latex] and units are meters per second (m/s); c. [latex] [{a}_{0}]={\text{LT}}^{-2} [/latex] and units are meters per second squared (m/s 2 ); d. [latex] [{j}_{0}]={\text{LT}}^{-3} [/latex] and units are meters per second cubed (m/s 3 ); e. [latex] [{S}_{0}]={\text{LT}}^{-4} [/latex] and units are m/s 4 ; f. [latex] [c]={\text{LT}}^{-5} [/latex] and units are m/s 5 .
(a) A car speedometer has a 5% uncertainty. What is the range of possible speeds when it reads 90 km/h? (b) Convert this range to miles per hour. Note 1 km = 0.6214 mi.
A marathon runner completes a 42.188-km course in 2 h, 30 min, and 12 s. There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the percent uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?
a. 0.059%; b. 0.01%; c. 4.681 m/s; d. 0.07%, 0.003 m/s
The sides of a small rectangular box are measured to be 1.80 ± 0.1 cm, 2.05 ± 0.02 cm, and 3.1 ± 0.1 cm long. Calculate its volume and uncertainty in cubic centimeters.
When nonmetric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was used, where 1 lbm = 0.4539 kg. (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?
a. 0.02%; b. 1×10 4 lbm
The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and 3.050 ± 0.005 m. Calculate the area of the room and its uncertainty in square meters.
A car engine moves a piston with a circular cross-section of 7.500 ± 0.002 cm in diameter a distance of 3.250 ± 0.001 cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.
a. 143.6 cm 3 ; b. 0.2 cm 3 or 0.14%
Challenge Problems
The first atomic bomb was detonated on July 16, 1945, at the Trinity test site about 200 mi south of Los Alamos. In 1947, the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G. I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was then able to deduce the amount of energy released in the explosion, which was a closely guarded secret at the time. Because of this, Taylor did not publish his results until 1950. This problem challenges you to recreate this famous calculation. (a) Using keen physical insight developed from years of experience, Taylor decided the radius r of the fireball should depend only on time since the explosion, t , the density of the air, [latex] \rho , [/latex] and the energy of the initial explosion, E . Thus, he made the educated guess that [latex] r=k{E}^{a}{\rho }^{b}{t}^{c} [/latex] for some dimensionless constant k and some unknown exponents a , b , and c . Given that [E] = ML 2 T –2 , determine the values of the exponents necessary to make this equation dimensionally consistent. ( Hint : Notice the equation implies that [latex] k=r{E}^{\text{−}a}{\rho }^{\text{−}b}{t}^{\text{−}c} [/latex] and that [latex] [k]=1. [/latex]) (b) By analyzing data from high-energy conventional explosives, Taylor found the formula he derived seemed to be valid as long as the constant k had the value 1.03. From the film reel, he was able to determine many values of r and the corresponding values of t . For example, he found that after 25.0 ms, the fireball had a radius of 130.0 m. Use these values, along with an average air density of 1.25 kg/m 3 , to calculate the initial energy release of the Trinity detonation in joules (J). ( Hint : To get energy in joules, you need to make sure all the numbers you substitute in are expressed in terms of SI base units.) (c) The energy released in large explosions is often cited in units of “tons of TNT” (abbreviated “t TNT”), where 1 t TNT is about 4.2 GJ. Convert your answer to (b) into kilotons of TNT (that is, kt TNT). Compare your answer with the quick-and-dirty estimate of 10 kt TNT made by physicist Enrico Fermi shortly after witnessing the explosion from what was thought to be a safe distance. (Reportedly, Fermi made his estimate by dropping some shredded bits of paper right before the remnants of the shock wave hit him and looked to see how far they were carried by it.)
The purpose of this problem is to show the entire concept of dimensional consistency can be summarized by the old saying “You can’t add apples and oranges.” If you have studied power series expansions in a calculus course, you know the standard mathematical functions such as trigonometric functions, logarithms, and exponential functions can be expressed as infinite sums of the form [latex] \sum _{n=0}^{\infty }{a}_{n}{x}^{n}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}+\cdots , [/latex] where the [latex] {a}_{n} [/latex] are dimensionless constants for all [latex] n=0,1,2,\cdots [/latex] and x is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency. That is, it actually implies the arguments of standard mathematical functions must be dimensionless, so it is not really necessary to make this latter condition a separate requirement of the definition of dimensional consistency as we have done in this section.
Since each term in the power series involves the argument raised to a different power, the only way that every term in the power series can have the same dimension is if the argument is dimensionless. To see this explicitly, suppose [x] = L a M b T c . Then, [x n ] = [x] n = L an M bn T cn . If we want [x] = [x n ], then an = a, bn = b, and cn = c for all n. The only way this can happen is if a = b = c = 0.
OpenStax University Physics. Authored by : OpenStax CNX. Located at : https://cnx.org/contents/[email protected]:Gofkr9Oy@15 . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
Privacy Policy
In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.
Please ensure that your password is at least 8 characters and contains each of the following:
a special character: @$#!%*?&
Physics Problems with Solutions
Forces in physics, tutorials and problems with solutions.
Free tutorials on forces with questions and problems with detailed solutions and examples. The concepts of forces, friction forces, action and reaction forces, free body diagrams, tension of string, inclined planes, etc. are discussed and through examples, questions with solutions and clear and self explanatory diagrams. Questions to practice for the SAT Physics test on forces are also included with their detailed solutions. The discussions of applications of forces engineering system are also included.
Forces: Tutorials with Examples and Detailed Solutions
Problems on forces with detailed solutions, sat questions on forces with solutions, formulas and constants, popular pages.
General Physics
Physics, 12th Edition
ISBN: 978-1-119-79811-8
December 2021
PREFER A PRINTED BOOK BUT LOOKING FOR SAVINGS?
Rent college textbooks at our lowest prices. Wiley Textbook Rental lets you keep your textbook for a period of 130 days.
Digital Evaluation Copy
John D. Cutnell , Shane Stadler , David Young , Kenneth W. Johnson
Physics, 12th Edition focuses on conceptual understanding, problem solving, and providing real-world applications and relevance. Conceptual examples, Concepts and Calculations problems, and Check Your Understanding questions help students understand physics principles. Math Skills boxes, multi-concept problems, and Examples with reasoning steps help students improve their reasoning skills while solving problems. “The Physics Of” boxes, and new “Physics in Biology, Sports, and Medicine” problems show students how physics principles are relevant to their everyday lives.
A wide array of tools help students navigate through this course, and keep them engaged by encouraging active learning. Animated pre-lecture videos (created and narrated by the authors) explain the basic concepts and learning objectives of each section. Problem-solving strategies are discussed, and common misconceptions and potential pitfalls are addressed. Chalkboard videos demonstrate step-by-step practical solutions to typical homework problems. Finally, tutorials that implement a step-by-step approach are also offered, allowing students to develop their problem-solving skills.
Double the number of Team Problems at the end of every chapter, designed for group problem-solving. These are context-rich, end-of-chapter problems of variable difficulty designed for group cooperation. However, they may also be tackled by students on their own. The authors have added a “worksheet” activity to each new Team Problem, which helps guide the students along the way to the solution. These problems offer a great way to promote peer-to-peer instruction in the classroom, as well as serve as the focal point of a group activity during a breakout session as part of an online course.
Physics in Biology, Sports, and Medicine” problems – these new end-of-chapter problems highlight physics applications in the life sciences for students pursuing careers in the premedical fields, but also to underscore the fact that the human body, like any biological system, is governed by the laws of physics.
300+ bio-inspired examples and problems.
Conceptual Understanding helps students tighten their grasp of physics concepts and how they fit together to provide a coherent description of natural phenomena, including:
Conceptual Examples
Concepts and Calculations problems
Focus on Concepts homework material
Check Your Understanding questions
Problem Solving features help students with their ability to reason in an organized and mathematically correct manner, which is essential to solving problems and improving their reasoning skills:
Math Skills boxes for just-in-time delivery of math support
Explicitly reasoning steps for all examples
Reasoning Strategies for solving certain classes of problems.
Analyzing Multiple-Concept Problems
Problem Solving Insights
“The Physics of…” Examples: The current course version includes 40% more enhanced biological application examples. “The Physics of …” are bio-inspired examples similar to what pre-med students will encounter in the Chemical and Physical Foundations of Biological Systems Passages section of the MCAT.
Information
Author Services
Initiatives
You are accessing a machine-readable page. In order to be human-readable, please install an RSS reader.
All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to https://www.mdpi.com/openaccess .
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.
Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive positive feedback from the reviewers.
Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.
Original Submission Date Received: .
Active Journals
Find a Journal
Proceedings Series
For Authors
For Reviewers
For Editors
For Librarians
For Publishers
For Societies
For Conference Organizers
Open Access Policy
Institutional Open Access Program
Special Issues Guidelines
Editorial Process
Research and Publication Ethics
Article Processing Charges
Testimonials
Preprints.org
SciProfiles
Encyclopedia
Article Menu
Subscribe SciFeed
Recommended Articles
Google Scholar
on Google Scholar
Table of Contents
Find support for a specific problem in the support section of our website.
Please let us know what you think of our products and services.
Visit our dedicated information section to learn more about MDPI.
JSmol Viewer
Multi-step physics-informed deep operator neural network for directly solving partial differential equations.
Wang J, Li Y, Wu A, Chen Z, Huang J, Wang Q, Liu F. Multi-Step Physics-Informed Deep Operator Neural Network for Directly Solving Partial Differential Equations. Applied Sciences . 2024; 14(13):5490. https://doi.org/10.3390/app14135490
Wang, Jing, Yubo Li, Anping Wu, Zheng Chen, Jun Huang, Qingfeng Wang, and Feng Liu. 2024. "Multi-Step Physics-Informed Deep Operator Neural Network for Directly Solving Partial Differential Equations" Applied Sciences 14, no. 13: 5490. https://doi.org/10.3390/app14135490
Article Metrics
Article access statistics, further information, mdpi initiatives, follow mdpi.
Subscribe to receive issue release notifications and newsletters from MDPI journals
Physics problems with solutions and tutorials with full explanations are included. More emphasis on the topics of physics included in the SAT physics subject with hundreds of problems with detailed solutions. Physics concepts are clearly discussed and highlighted. Real life applications are also included as they show how these concepts in ...
Example Physics Problems and Solutions
Learning how to solve physics problems is a big part of learning physics. Here's a collection of example physics problems and solutions to help you tackle problems sets and understand concepts and how to work with formulas: Physics Homework Tips Physics homework can be challenging! Get tips to help make the task a little easier.
Physics Problems with Detailed Solutions and Explanations
Electrostatic Problems with Solutions and Explanations. Gravity Problems with Solutions and Explanations. Projectile Problems with Solutions and Explanations. Velocity and Speed: Problems. Uniform Acceleration Motion: Problems. Free Physics SAT and AP Practice Tests Questions.
Kinematic Equations: Sample Problems and Solutions
A useful problem-solving strategy was presented for use with these equations and two examples were given that illustrated the use of the strategy. Then, the application of the kinematic equations and the problem-solving strategy to free-fall motion was discussed and illustrated. In this part of Lesson 6, several sample problems will be presented.
Projectile Problems with Solutions and Explanations
Problem 8. The trajectory of a projectile launched from ground is given by the equation y = -0.025 x 2 + 0.5 x, where x and y are the coordinate of the projectile on a rectangular system of axes. a) Find the initial velocity and the angle at which the projectile is launched. Solution to Problem 8.
1.8: Solving Problems in Physics
Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life. . Figure 1.8.1 1.8. 1: Problem-solving skills are essential to your success in physics. (credit: "scui3asteveo"/Flickr) As you are probably well aware, a certain amount of creativity and insight is required to solve problems.
1.7 Solving Problems in Physics
This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process. Significance. After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is ...
6.1 Solving Problems with Newton's Laws
Success in problem solving is necessary to understand and apply physical principles. We developed a pattern of analyzing and setting up the solutions to problems involving Newton's laws in Newton's Laws of Motion; in this chapter, we continue to discuss these strategies and apply a step-by-step process.. Problem-Solving Strategies
Physics Problems
Physics Problems. Physics Problems with solutions are provided with full explanations. All solved problems are suitable for physics course of high schools and college students. Tutorials are also presented along with dozens of solvd examples.
PDF 500+ Solved Physics Homework and Exam Problems
Physics Problems and Solutions: Homework and Exam Physexams.com 1 Vectors 1.1 Unit Vectors 1. Find the unit vector in the direction w⃗= (5,2). Solution: A unit vector in physics is defined as a dimensionless vector whose magnitude is exactly 1. A unit vector that points in the direction of A⃗is determined by formula Aˆ = A⃗ |A⃗|
How to Solve Any Physics Problem: 10 Steps (with Pictures)
Calm down. It is just a problem, not the end of the world! 2. Read through the problem once. If it is a long problem, read and understand it in parts till you get even a slight understanding of what is going on. 3. Draw a diagram. It cannot be emphasized enough how much easier a problem will be once it is drawn out.
Solved Speed, Velocity, and Acceleration Problems
Kinematics Equations: Problems and Solutions. Position vs. Time Graphs. Velocity vs. Time Graphs In the following section, some sample AP Physics 1 problems on acceleration are provided. Problem (20): An object moves with constant acceleration along a straight line.
PDF Introductory Physics: Problems solving
This collection of physics problems solutions does not intend to cover the whole Introductory Physics course. Its purpose is to show the right way to solve physics problems. Here some useful tips. 1. Always try to find out what a problem is about, which part of the physics course is in question 2. Drawings are very helpful in most cases.
Phy
Just snap a picture. And yes, Phy understands your hand-writing. 5. Click it. Try Phy. A free to use AI Physics tutor. Solve, grade, and explain problems. Just speak to Phy or upload a screenshot of your working.
Gravity Problems with Solutions and Explanations
Solution to Problem 1: a) Given the distance and the time, we can calculate the acceleration a using the distance formula for the uniform acceleration motion as follows: d = (1/2) a t 2. a = 2 d / t 2 = 2 × 13.5 / 3 2 = 3 m/s 2. b) Let R be the radius and m b be the mass of planet Big Alpha and m o the mass of the object. The acceleration is ...
Collection of Solved Problems in Physics
This collection of Solved Problems in Physics is developed by Department of Physics Education, Faculty of Mathematics and Physics, Charles University in Prague since 2006. The Collection contains tasks at various level in mechanics, electromagnetism, thermodynamics and optics. The tasks mostly do not contain the calculus (derivations and ...
1.7 Solving Problems in Physics
Summary. The three stages of the process for solving physics problems used in this book are as follows: Strategy: Determine which physical principles are involved and develop a strategy for using them to solve the problem.; Solution: Do the math necessary to obtain a numerical solution complete with units.; Significance: Check the solution to make sure it makes sense (correct units, reasonable ...
PDF An Expert's Approach to Solving Physics Problems
An example problem, its solution, and annotations on the process of solving the problem. The solutions to the problems from past exams will help you see what a good solution looks like. But seeing the solution alone may not illustrate the general method that could be used to solve other problems.
Solving Physics Problems
Despite the vast differences between problems and solution methods, there are some commonly used tools that make solving physics problems easier. For example, it is important to identify the units ...
Mathway
Free math problem solver answers your physics homework questions with step-by-step explanations. Mathway. Visit Mathway on the web. Start 7-day free trial on the app. Start 7-day free trial on the app. Download free on Amazon. Download free in Windows Store. Take a photo of your math problem on the app. get Go. Physics. Basic Math. Pre-Algebra ...
1.8: Solving Problems in Physics
Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life. . Figure 1.8.1 1.8. 1: Problem-solving skills are essential to your success in physics. (credit: "scui3asteveo"/Flickr) As you are probably well aware, a certain amount of creativity and insight is required to solve problems.
Physics Solver: Solve Your Physics Problems & Homework with AI
Our physics problem solver is powered by industry-leading AI models and is frequently updated. As such, it ensures approximately 98% precision in answering physics questions. However, as with any AI homework assistant, it's best to review the generated solutions before submitting your assignment. 5.
Forces in Physics, tutorials and Problems with Solutions
The concepts of forces, friction forces, action and reaction forces, free body diagrams, tension of string, inclined planes, etc. are discussed and through examples, questions with solutions and clear and self explanatory diagrams. Questions to practice for the SAT Physics test on forces are also included with their detailed solutions.
Physics, 12th Edition
Physics, 12th Edition focuses on conceptual understanding, problem solving, and providing real-world applications and relevance. Conceptual examples, Concepts and Calculations problems, and Check Your Understanding questions help students understand physics principles. Math Skills boxes, multi-concept problems, and Examples with reasoning steps help students improve their reasoning skills ...
Applied Sciences
This paper establishes a method for solving partial differential equations using a multi-step physics-informed deep operator neural network. The network is trained by embedding physics-informed constraints. Different from traditional neural networks for solving partial differential equations, the proposed method uses a deep neural operator network to indirectly construct the mapping ...
IMAGES
VIDEO
COMMENTS
Physics problems with solutions and tutorials with full explanations are included. More emphasis on the topics of physics included in the SAT physics subject with hundreds of problems with detailed solutions. Physics concepts are clearly discussed and highlighted. Real life applications are also included as they show how these concepts in ...
Learning how to solve physics problems is a big part of learning physics. Here's a collection of example physics problems and solutions to help you tackle problems sets and understand concepts and how to work with formulas: Physics Homework Tips Physics homework can be challenging! Get tips to help make the task a little easier.
Electrostatic Problems with Solutions and Explanations. Gravity Problems with Solutions and Explanations. Projectile Problems with Solutions and Explanations. Velocity and Speed: Problems. Uniform Acceleration Motion: Problems. Free Physics SAT and AP Practice Tests Questions.
A useful problem-solving strategy was presented for use with these equations and two examples were given that illustrated the use of the strategy. Then, the application of the kinematic equations and the problem-solving strategy to free-fall motion was discussed and illustrated. In this part of Lesson 6, several sample problems will be presented.
Problem 8. The trajectory of a projectile launched from ground is given by the equation y = -0.025 x 2 + 0.5 x, where x and y are the coordinate of the projectile on a rectangular system of axes. a) Find the initial velocity and the angle at which the projectile is launched. Solution to Problem 8.
Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life. . Figure 1.8.1 1.8. 1: Problem-solving skills are essential to your success in physics. (credit: "scui3asteveo"/Flickr) As you are probably well aware, a certain amount of creativity and insight is required to solve problems.
This step is clearly important because it produces the numerical answer, along with its units. Notice, however, that this stage is only one-third of the overall problem-solving process. Significance. After having done the math in the solution stage of problem solving, it is tempting to think you are done. But, always remember that physics is ...
Success in problem solving is necessary to understand and apply physical principles. We developed a pattern of analyzing and setting up the solutions to problems involving Newton's laws in Newton's Laws of Motion; in this chapter, we continue to discuss these strategies and apply a step-by-step process.. Problem-Solving Strategies
Physics Problems. Physics Problems with solutions are provided with full explanations. All solved problems are suitable for physics course of high schools and college students. Tutorials are also presented along with dozens of solvd examples.
Physics Problems and Solutions: Homework and Exam Physexams.com 1 Vectors 1.1 Unit Vectors 1. Find the unit vector in the direction w⃗= (5,2). Solution: A unit vector in physics is defined as a dimensionless vector whose magnitude is exactly 1. A unit vector that points in the direction of A⃗is determined by formula Aˆ = A⃗ |A⃗|
Calm down. It is just a problem, not the end of the world! 2. Read through the problem once. If it is a long problem, read and understand it in parts till you get even a slight understanding of what is going on. 3. Draw a diagram. It cannot be emphasized enough how much easier a problem will be once it is drawn out.
Kinematics Equations: Problems and Solutions. Position vs. Time Graphs. Velocity vs. Time Graphs In the following section, some sample AP Physics 1 problems on acceleration are provided. Problem (20): An object moves with constant acceleration along a straight line.
This collection of physics problems solutions does not intend to cover the whole Introductory Physics course. Its purpose is to show the right way to solve physics problems. Here some useful tips. 1. Always try to find out what a problem is about, which part of the physics course is in question 2. Drawings are very helpful in most cases.
Just snap a picture. And yes, Phy understands your hand-writing. 5. Click it. Try Phy. A free to use AI Physics tutor. Solve, grade, and explain problems. Just speak to Phy or upload a screenshot of your working.
Solution to Problem 1: a) Given the distance and the time, we can calculate the acceleration a using the distance formula for the uniform acceleration motion as follows: d = (1/2) a t 2. a = 2 d / t 2 = 2 × 13.5 / 3 2 = 3 m/s 2. b) Let R be the radius and m b be the mass of planet Big Alpha and m o the mass of the object. The acceleration is ...
This collection of Solved Problems in Physics is developed by Department of Physics Education, Faculty of Mathematics and Physics, Charles University in Prague since 2006. The Collection contains tasks at various level in mechanics, electromagnetism, thermodynamics and optics. The tasks mostly do not contain the calculus (derivations and ...
Summary. The three stages of the process for solving physics problems used in this book are as follows: Strategy: Determine which physical principles are involved and develop a strategy for using them to solve the problem.; Solution: Do the math necessary to obtain a numerical solution complete with units.; Significance: Check the solution to make sure it makes sense (correct units, reasonable ...
An example problem, its solution, and annotations on the process of solving the problem. The solutions to the problems from past exams will help you see what a good solution looks like. But seeing the solution alone may not illustrate the general method that could be used to solve other problems.
Despite the vast differences between problems and solution methods, there are some commonly used tools that make solving physics problems easier. For example, it is important to identify the units ...
Free math problem solver answers your physics homework questions with step-by-step explanations. Mathway. Visit Mathway on the web. Start 7-day free trial on the app. Start 7-day free trial on the app. Download free on Amazon. Download free in Windows Store. Take a photo of your math problem on the app. get Go. Physics. Basic Math. Pre-Algebra ...
Such analytical skills are useful both for solving problems in this text and for applying physics in everyday life. . Figure 1.8.1 1.8. 1: Problem-solving skills are essential to your success in physics. (credit: "scui3asteveo"/Flickr) As you are probably well aware, a certain amount of creativity and insight is required to solve problems.
Our physics problem solver is powered by industry-leading AI models and is frequently updated. As such, it ensures approximately 98% precision in answering physics questions. However, as with any AI homework assistant, it's best to review the generated solutions before submitting your assignment. 5.
The concepts of forces, friction forces, action and reaction forces, free body diagrams, tension of string, inclined planes, etc. are discussed and through examples, questions with solutions and clear and self explanatory diagrams. Questions to practice for the SAT Physics test on forces are also included with their detailed solutions.
Physics, 12th Edition focuses on conceptual understanding, problem solving, and providing real-world applications and relevance. Conceptual examples, Concepts and Calculations problems, and Check Your Understanding questions help students understand physics principles. Math Skills boxes, multi-concept problems, and Examples with reasoning steps help students improve their reasoning skills ...
This paper establishes a method for solving partial differential equations using a multi-step physics-informed deep operator neural network. The network is trained by embedding physics-informed constraints. Different from traditional neural networks for solving partial differential equations, the proposed method uses a deep neural operator network to indirectly construct the mapping ...