8.1 Linear Momentum and Force
A boy standing on a frictionless ice rink is initially at rest. He throws a snowball in the +x-direction, and it travels on a ballistic trajectory, hitting the ground some distance away. Which of the following is true about the boy while he is in the act of throwing the snowball?
- He feels an upward force to compensate for the downward trajectory of the snowball.
- He feels a backward force exerted by the snowball he is throwing.
- He feels no net force.
- He feels a forward force, the same force that propels the snowball.
A 150-g baseball is initially moving 80 mi/h in the –x-direction. After colliding with a baseball bat for 20 ms, the baseball moves 80 mi/h in the +x-direction. What is the magnitude and direction of the average force exerted by the bat on the baseball?
8.2 Impulse
A 1.0-kg ball of putty is released from rest and falls vertically 1.5 m until it strikes a hard floor, where it comes to rest in a 0.045-s time interval. What is the magnitude and direction of the average force exerted on the ball by the floor during the collision?
- 33 N, up
- 120 N, up
- 120 N, down
- 240 N, down
A 75-g ball is dropped from rest from a height of 2.2 m. It bounces off the floor and rebounds to a maximum height of 1.7 m. If the ball is in contact with the floor for 0.024 s, what is the magnitude and direction of the average force exerted on the ball by the floor during the collision?
A 2.4-kg ceramic bowl falls to the floor. During the 0.018-s impact, the bowl experiences an average force of 750 N from the floor. The bowl is at rest after the impact. From what initial height did the bowl fall?
- 1.6 m
- 2.8 m
- 3.2 m
- 5.6 m
Whether or not an object (such as a plate, glass, or bone) breaks upon impact depends on the average force exerted on that object by the surface. When a 1.2-kg glass figure hits the floor, it will break if it experiences an average force of 330 N. When it hits a tile floor, the glass comes to a stop in 0.015 s. From what minimum height must the glass fall to experience sufficient force to break? How would your answer change if the figure were falling to a padded or carpeted surface? Explain.
A 2.5-kg block slides across a frictionless table toward a horizontal spring.As the block bounces off the spring, a probe measures the velocity of the block (initially negative, moving away from the probe) over time as follows:
| Velocity (m/s) | Time (s) |
|---|---|
| −12.0 | 0 |
| −10.0 | 0.10 |
| −6.0 | 0.20 |
| 0 | 0.30 |
| 6.0 | 0.40 |
| 10.0 | 0.50 |
| 12.0 | 0.60 |
What is the average force exerted on the block by the spring over the entire 0.60-s time interval of the collision?
- 50 N
- 60 N
- 100 N
- 120 N
During an automobile crash test, the average force exerted by a solid wall on a 2500-kg car that hits the wall is measured to be 740,000 N over a 0.22-s time interval. What was the initial speed of the car prior to the collision, assuming the car is at rest at the end of the time interval?
A test car is driving toward a solid crash-test barrier with a speed of 45 mi/h. Two seconds prior to impact, the car begins to brake, but it is still moving when it hits the wall. After the collision with the wall, the car crumples somewhat and comes to a complete stop. In order to estimate the average force exerted by the wall on the car, what information would you need to collect?
- The (negative) acceleration of the car before it hits the wall and the distance the car travels while braking.
- The (negative) acceleration of the car before it hits the wall and the velocity of the car just before impact.
- The velocity of the car just before impact and the duration of the collision with the wall.
- The duration of the collision with the wall and the distance the car travels while braking.
Design an experiment to verify the relationship between the average force exerted on an object and the change in momentum of that object. As part of your explanation, list the equipment you would use and describe your experimental setup. What would you measure and how? How exactly would you verify the relationship? Explain.
A 22-g puck hits the wall of an air hockey table perpendicular to the wall with an initial speed of 14 m/s.The puck is in contact with the wall for 0.0055 s, and it rebounds from the wall with a speed of 14 m/s in the opposite direction.What is the magnitude of the average force exerted by the wall on the puck?
- 0.308 N
- 0.616 N
- 56 N
- 112 N
A 22-g puck hits the wall of an air hockey table perpendicular to the wall with an initial speed of 7 m/s. The puck is in contact with the wall for 0.011 s, and the wall exerts an average force of 28 N on the puck during that time. Calculate the magnitude and direction of the change in momentum of the puck.
The graph in Figure 8.20 represents the force exerted on a particle during a collision. What is the magnitude of the change in momentum of the particle as a result of the collision?
- 1.2 kg • m/s
- 2.4 kg • m/s
- 3.6 kg • m/s
- 4.8 kg • m/s
The graph in Figure 8.21 represents the force exerted on a particle during a collision. What is the magnitude of the change in momentum of the particle as a result of the collision?
8.3 Conservation of Momentum
Which of the following is an example of an open system?
- Two air cars colliding on a track elastically.
- Two air cars colliding on a track and sticking together.
- A bullet being fired into a hanging wooden block and becoming embedded in the block, with the system then acting as a ballistic pendulum.
- A bullet being fired into a hillside and becoming buried in the earth.
A 40-kg girl runs across a mat with a speed of 5.0 m/s and jumps onto a 120-kg hanging platform initially at rest, causing the girl and platform to swing back and forth like a pendulum together after her jump. What is the combined velocity of the girl and platform after the jump? What is the combined momentum of the girl and platform both before and after the collision?
A 50-kg boy runs across a mat with a speed of 6.0 m/s and collides with a soft barrier on the wall, rebounding off the wall and falling to the ground. The boy is at rest after the collision. What is the momentum of the boy before and after the collision? Is momentum conserved in this collision? Explain. Which of these is an example of an open system and which is an example of a closed system? Explain your answer.
A student sets up an experiment to measure the momentum of a system of two air cars, A and B, of equal mass, moving on a linear, frictionless track. Before the collision, car A has a certain speed, and car B is at rest. Which of the following will be true about the total momentum of the two cars?
- It will be greater before the collision.
- It will be equal before and after the collision.
- It will be greater after the collision.
- The answer depends on whether the collision is elastic or inelastic.
A group of students has two carts, A and B, with wheels that turn with negligible friction. The carts can travel along a straight horizontal track. Cart A has known mass mA. The students are asked to use a one-dimensional collision between the carts to determine the mass of cart B. Before the collision, cart A travels to the right and cart B is initially at rest. After the collision, the carts stick together.
- Describe an experimental procedure to determine the velocities of the carts before and after a collision, including all the additional equipment you would need. You may include a labeled diagram of your setup to help in your description. Indicate what measurements you would take and how you would take them. Include enough detail so that another student could carry out your procedure.
- There will be sources of error in the measurements taken in the experiment, both before and after the collision. For your experimental procedure, will the uncertainty in the calculated value of the mass of cart B be affected more by the error in the measurements taken before the collision or by those taken after the collision, or will it be equally affected by both sets of measurements? Justify your answer.
A group of students took measurements for one collision. A graph of the students’ data is shown below.
- Given mA = 0.50 kg, use the graph to calculate the mass of cart B. Explicitly indicate the principles used in your calculations.
- The students are now asked to Consider the kinetic energy changes in an inelastic collision, specifically whether the initial values of one of the physical quantities affect the fraction of mechanical energy dissipated in the collision. How could you modify the experiment to investigate this question? Be sure to explicitly describe the calculations you would make, specifying all equations you would use (but do not actually do any algebra or arithmetic).
Cart A is moving with an initial velocity +v (in the positive direction) toward cart B, initially at rest. Both carts have equal mass and are on a frictionless surface. Which of the following statements correctly characterizes the velocity of the center of mass of the system before and after the collision?
- before, after
- before, 0 after
- before, after
- 0 before, 0 after
Cart A is moving with a velocity of +10 m/s toward cart B, which is moving with a velocity of +4 m/s. Both carts have equal mass and are moving on a frictionless surface. The two carts have an inelastic collision and stick together after the collision. Calculate the velocity of the center of mass of the system before and after the collision. If there were friction present in this problem, how would this external force affect the center-of-mass velocity both before and after the collision?
8.4 Elastic Collisions in One Dimension
Two cars (A and B) of mass 1.5 kg collide. Car A is initially moving at 12 m/s, and car B is initially moving in the same direction with a speed of 6 m/s. The two cars are moving along a straight line before and after the collision. What will be the change in momentum of this system after the collision?
- −27 kg • m/s
- zero
- +27 kg • m/s
- It depends on whether the collision is elastic or inelastic.
Two cars (A and B) of mass 1.5 kg collide. Car A is initially moving at 24 m/s, and car B is initially moving in the opposite direction with a speed of 12 m/s. The two cars are moving along a straight line before and after the collision. (a) If the two cars have an elastic collision, calculate the change in momentum of the two-car system. (b) If the two cars have a completely inelastic collision, calculate the change in momentum of the two-car system.
Puck A (200 g) slides across a frictionless surface to collide with puck B (800 g), initially at rest. The velocity of each puck is measured during the experiment as follows:
| Time | Velocity A | Velocity B |
|---|---|---|
| 0 | +8.0 m/s | 0 |
| 1.0 s | +8.0 m/s | 0 |
| 2.0 s | −2.0 m/s | +2.5 m/s |
| 3.0 s | −2.0 m/s | +2.5 m/s |
What is the change in momentum of the center of mass of the system as a result of the collision?
- +1.6 kg•m/s
- +0.8 kg•m/s
- 0
- −1.6 kg•m/s
For the table above, calculate the center-of-mass velocity of the system both before and after the collision, then calculate the center-of-mass momentum of the system both before and after the collision. From this, determine the change in the momentum of the system as a result of the collision.
Two cars (A and B) of equal mass have an elastic collision. Prior to the collision, car A is moving at 15 m/s in the +x-direction, and car B is moving at 10 m/s in the –x-direction. Assuming that both cars continue moving along the x-axis after the collision, what will be the velocity of car A after the collision?
- same as the original 15 m/s speed, opposite direction
- equal to car B’s velocity prior to the collision
- equal to the average of the two velocities, in its original direction
- equal to the average of the two velocities, in the opposite direction
Two cars (A and B) of equal mass have an elastic collision. Prior to the collision, car A is moving at 20 m/s in the +x-direction, and car B is moving at 10 m/s in the –x-direction. Assuming that both cars continue moving along the x-axis after the collision, what will be the velocities of each car after the collision?
A rubber ball is dropped from rest at a fixed height. It bounces off a hard floor and rebounds upward, but it only reaches 90% of its original fixed height. What is the best way to explain the loss of kinetic energy of the ball during the collision?
- Energy was required to deform the ball’s shape during the collision with the floor.
- Energy was lost due to work done by the ball pushing on the floor during the collision.
- Energy was lost due to friction between the ball and the floor.
- Energy was lost due to the work done by gravity during the motion.
A tennis ball strikes a wall with an initial speed of 15 m/s. The ball bounces off the wall but rebounds with slightly less speed (14 m/s) after the collision. Explain (a) what else changed its momentum in response to the ball’s change in momentum so that overall momentum is conserved, and (b) how some of the ball’s kinetic energy was lost.
Two objects, A and B, have equal mass. Prior to the collision, mass A is moving 10 m/s in the +x-direction, and mass B is moving 4 m/s in the +x-direction. Which of the following results represents an inelastic collision between A and B?
- After the collision, mass A is at rest, and mass B moves 14 m/s in the +x-direction.
- After the collision, mass A moves 4 m/s in the –x-direction, and mass B moves 18 m/s in the +x-direction.
- After the collision, the two masses stick together and move 7 m/s in the +x-direction.
- After the collision, mass A moves 4 m/s in the +x-direction, and mass B moves 10 m/s in the +x-direction.
Mass A is three times more massive than mass B. Mass A is initially moving 12 m/s in the +x-direction. Mass B is initially moving 12 m/s in the –x-direction. Assuming that the collision is elastic, calculate the final velocity of both masses after the collision. Show that your results are consistent with conservation of momentum and conservation of kinetic energy.
Two objects (A and B) of equal mass collide elastically. Mass A is initially moving 5.0 m/s in the +x-direction prior to the collision. Mass B is initially moving 3.0 m/s in the –x-direction prior to the collision. After the collision, mass A will be moving with a velocity of 3.0 m/s in the –x-direction. What will be the velocity of mass B after the collision?
- 3.0 m/s in the +x-direction
- 5.0 m/s in the +x-direction
- 3.0 m/s in the –x-direction
- 5.0 m/s in the –x-direction
Two objects (A and B) of equal mass collide elastically. Mass A is initially moving 4.0 m/s in the +x-direction prior to the collision. Mass B is initially moving 8.0 m/s in the –x-direction prior to the collision. After the collision, mass A will be moving with a velocity of 8.0 m/s in the –x-direction. (a) Use the principle of conservation of momentum to predict the velocity of mass B after the collision. (b) Use the fact that kinetic energy is conserved in elastic collisions to predict the velocity of mass B after the collision.
Two objects of equal mass collide. Object A is initially moving in the +x-direction with a speed of 12 m/s, and object B is initially at rest. After the collision, object A is at rest, and object B is moving away with some unknown velocity. There are no external forces acting on the system of two masses. What statement can we make about this collision?
- Both momentum and kinetic energy are conserved.
- Momentum is conserved, but kinetic energy is not conserved.
- Neither momentum nor kinetic energy is conserved.
- More information is needed in order to determine which is conserved.
Two objects of equal mass collide. Object A is initially moving with a velocity of 15 m/s in the +x-direction, and object B is initially at rest. After the collision, object A is at rest. There are no external forces acting on the system of two masses. (a) Use momentum conservation to deduce the velocity of object B after the collision. (b) Is this collision elastic? Justify your answer.
Which of the following statements is true about an inelastic collision?
- Momentum is conserved, and kinetic energy is conserved.
- Momentum is conserved, and kinetic energy is not conserved.
- Momentum is not conserved, and kinetic energy is conserved.
- Momentum is not conserved, and kinetic energy is not conserved.
Explain how the momentum and kinetic energy of a system of two colliding objects changes as a result of (a) an elastic collision and (b) an inelastic collision.
This figure shows the positions of two colliding objects measured before, during, and after a collision. Mass A is 1.0 kg. Mass B is 3.0 kg. Which of the following statements is true?
- This is an elastic collision, with a total momentum of 0 kg • m/s.
- This is an elastic collision, with a total momentum of 1.67 kg • m/s.
- This is an inelastic collision, with a total momentum of 0 kg • m/s.
- This is an inelastic collision, with a total momentum of 1.67 kg • m/s.
For the above graph, determine the initial and final momentum for both objects, assuming mass A is 1.0 kg and mass B is 3.0 kg. Also, determine the initial and final kinetic energies for both objects. Based on your results, explain whether momentum is conserved in this collision, and state whether the collision is elastic or inelastic.
Mass A (1.0 kg) slides across a frictionless surface with a velocity of 8 m/s in the positive direction. Mass B (3.0 kg) is initially at rest. The two objects collide and stick together. What will be the change in the center-of-mass velocity of the system as a result of the collision?
- There will be no change in the center-of-mass velocity.
- The center-of-mass velocity will decrease by 2 m/s.
- The center-of-mass velocity will decrease by 6 m/s.
- The center-of-mass velocity will decrease by 8 m/s.
Mass A (1.0 kg) slides across a frictionless surface with a velocity of 4 m/s in the positive direction. Mass B (1.0 kg) slides across the same surface in the opposite direction with a velocity of −8 m/s. The two objects collide and stick together after the collision. Predict how the center-of-mass velocity will change as a result of the collision, and explain your prediction. Calculate the center-of-mass velocity of the system both before and after the collision and explain why it remains the same or why it has changed.
8.5 Inelastic Collisions in One Dimension
Mass A (2.0 kg) has an initial velocity of 4 m/s in the +x-direction. Mass B (2.0 kg) has an initial velocity of 5 m/s in the –x-direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision?
- Both will move 0.5 m/s in the –x-direction.
- Mass A will stop; mass B will move 9 m/s in the +x-direction.
- Mass B will stop; mass A will move 9 m/s in the –x-direction.
- Mass A will move 5 m/s in the –x-direction; mass B will move 4 m/s in the +x-direction.
Mass A has an initial velocity of 22 m/s in the +x-direction. Mass B is three times more massive than mass A and has an initial velocity of 22 m/s in the –x-direction. If the two masses have an elastic collision, what will be the final velocities of the masses after the collision?
Mass A (2.0 kg) is moving with an initial velocity of 15 m/s in the +x-direction, and it collides with mass B (5.0 kg), initially at rest. After the collision, the two objects stick together and move as one. What is the change in kinetic energy of the system as a result of the collision?
- no change
- decrease by 225 J
- decrease by 161 J
- decrease by 64 J
Mass A (2.0 kg) is moving with an initial velocity of 15 m/s in the +x-direction, and it collides with mass B (4.0 kg), initially moving 7.0 m/s in the +x-direction. After the collision, the two objects stick together and move as one. What is the change in kinetic energy of the system as a result of the collision?
Mass A slides across a rough table with an initial velocity of 12 m/s in the +x-direction. By the time mass A collides with mass B (a stationary object with equal mass), mass A has slowed to 10 m/s. After the collision, the two objects stick together and move as one. Immediately after the collision, the velocity of the system is measured to be 5 m/s in the +x-direction, and the system eventually slides to a stop. Which of the following statements is true about this motion?
- Momentum is conserved during the collision, but it is not conserved during the motion before and after the collision.
- Momentum is not conserved at any time during this analysis.
- Momentum is conserved at all times during this analysis.
- Momentum is not conserved during the collision, but it is conserved during the motion before and after the collision.
Mass A is initially moving with a velocity of 12 m/s in the +x-direction. Mass B is twice as massive as mass A and is initially at rest. After the two objects collide, the two masses move together as one with a velocity of 4 m/s in the +x-direction. Is momentum conserved in this collision?
Mass A is initially moving with a velocity of 24 m/s in the +x-direction. Mass B is twice as massive as mass A and is initially at rest. The two objects experience a totally inelastic collision. What is the final speed of both objects after the collision?
- A is not moving; B is moving 24 m/s in the +x-direction.
- Neither A nor B is moving.
- A is moving 24 m/s in the –x-direction. B is not moving.
- Both A and B are moving together 8 m/s in the +x-direction.
Mass A is initially moving with some unknown velocity in the +x-direction. Mass B is twice as massive as mass A and initially at rest. The two objects collide, and after the collision, they move together with a speed of 6 m/s in the +x-direction. (a) Is this collision elastic or inelastic? Explain. (b) Determine the initial velocity of mass A.
Mass A is initially moving with a velocity of 2 m/s in the +x-direction. Mass B is initially moving with a velocity of 6 m/s in the –x-direction. The two objects have equal masses. After they collide, mass A moves with a speed of 4 m/s in the –x-direction. What is the final velocity of mass B after the collision?
- 6 m/s in the +x-direction
- 4 m/s in the +x-direction
- zero
- 4 m/s in the –x-direction
Mass A is initially moving with a velocity of 15 m/s in the +x-direction. Mass B is twice as massive and is initially moving with a velocity of 10 m/s in the –x-direction. The two objects collide, and after the collision, mass A moves with a speed of 15 m/s in the –x-direction. (a) What is the final velocity of mass B after the collision? (b) Calculate the change in kinetic energy as a result of the collision, assuming mass A is 5.0 kg.
8.6 Collisions of Point Masses in Two Dimensions
Two cars of equal mass approach an intersection. Car A is moving east at a speed of 45 m/s. Car B is moving south at a speed of 35 m/s. They collide inelastically and stick together after the collision, moving as one object. Which of the following statements is true about the center-of-mass velocity of this system?
- The center-of-mass velocity will decrease after the collision as a result of lost energy (but not drop to zero).
- The center-of-mass velocity will remain the same after the collision since momentum is conserved.
- The center-of-mass velocity will drop to zero since the two objects stick together.
- The magnitude of the center-of-mass velocity will remain the same, but the direction of the velocity will change.
Car A has a mass of 2000 kg and approaches an intersection with a velocity of 38 m/s directed to the east. Car B has a mass of 3500 kg and approaches the intersection with a velocity of 53 m/s directed 63° north of east. The two cars collide and stick together after the collision. Will the center-of-mass velocity change as a result of the collision? Explain why or why not. Calculate the center-of-mass velocity before and after the collision.