Linear Momentum & Impulse (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

A student conducts an experiment using a rubber ball of mass and a clay ball of the same mass. At time , each ball is thrown horizontally with speed toward two identical blocks that are fixed in place, as shown in Figure 1. At time , the balls collide with their respective blocks. The forces exerted by the blocks on the balls last for the same amount of time. In Experiment A, the rubber ball bounces off of the block and rebounds with the same speed. In Experiment B, the clay ball sticks to the block.

The arrow in Figure 2 represents the momentum of the rubber ball immediately before the collision in Experiment A.

Figure 2

The dots in Figure 3 represent the rubber ball in Experiment A and the clay ball in Experiment B respectively.

Figure 3

On the dots in Figure 3, draw arrows to represent the momentum of the rubber ball and the clay ball immediately after their respective collisions.

• If the momentum is zero, write “zero” next to the dot.

• The momentum, if it is not zero, must be represented by an arrow starting on, and pointing away from, the dot.

• The length of the arrows, if not zero, should reflect the magnitude of the momentum relative to the arrow in Figure 2.

i) Derive an expression for the magnitude of the impulse exerted by the block on the rubber ball in terms of and . Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

ii) Determine an expression for the magnitude of the impulse exerted by the block on the clay ball in terms of and .

The momentum bar charts in Figure 4 represent the momentum of the rubber ball and clay ball before the collision at time , during the collision in the time interval between and , and after the collision at time . The bar chart at is complete. Draw shaded rectangles to complete the momentum bar charts in Figure 4 in the time interval and at time .

• Positive values of momentum are above the zero line (“0”), and negative values of momentum are below the zero line.

• Shaded regions should start at the dashed line representing zero momentum.

• Represent any momentum that is equal to zero with a distinct line on the zero line.

• The relative height of each shaded region should reflect the magnitude of the respective momentum consistent with the scale shown.

Figure 4

Figure 5

A student sketches the free-body diagrams in Figure 5, and makes the following claim:

“The free-body diagrams show the average forces exerted on each ball during the time interval ”.

Justify why the student’s sketch (Figure 5) and claim are or are not consistent with the momentum bar charts you drew in part c).

Figure 1

A cart of mass is attached to an ideal spring, whose other end is fixed to a force sensor which is attached to a wall. The student places a motion detector to the right of the cart, as shown in Figure 1, and pulls the cart to the right a small distance so that the spring is stretched. The student releases the cart from rest, and the cart-spring system oscillates with period . While the cart is oscillating, it has a maximum speed .

Estimate the total impulse exerted on the cart by the spring over one full oscillation cycle. Justify your estimate using qualitative reasoning beyond referencing equations.

The student determines the mass of the cart is . The graphs in Figure 2 show the velocity of the cart and the force exerted on the cart by the spring as functions of time .

Figure 2

Does the data from the graphs in Figure 2 agree with your estimate from part a)? Justify your answer.

Ball 1 has mass and moves horizontally with speed towards a wall. During the collision, Ball 1 is in contact with the wall for time . After the collision, Ball 1 rebounds at half its initial speed.

i) Determine an expression for the impulse exerted by the wall on Ball 1 in terms of and .

ii) Derive an expression for the average force exerted by the wall on Ball 1 in terms of , and . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

A different ball, Ball 2, moves horizontally towards the wall. Ball 2 has the same mass and initial speed as Ball 1. During the collision, Ball 2 is in contact with the wall for a shorter time than Ball 1. After the collision, Ball 2 rebounds with speed . The average force exerted by the wall is the same in both collisions.

Figure 1

On the dots in Figure 1, draw arrows to represent the impulses exerted on Ball 1 and Ball 2 by the wall during their respective collisions.

• If the impulse is zero, write “zero” next to the dot.

• The impulse, if it is not zero, must be represented by an arrow starting on, and pointing away from, the dot.

• The length of the vectors, if not zero, should reflect the relative magnitude of the impulses exerted on each ball.

Indicate whether the final speed of Ball 2 is greater than, equal to, or less than the final speed of Ball 1.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning using your answer to part b).

Figure 1

A projectile of mass is fired horizontally from a launching device, exiting with a speed , as shown in Figure 1. While the projectile is in the launching device, the impulse given to it is , and the average force exerted on it is . Assume the force becomes zero just as the projectile reaches the end of the launching device.

i) Determine an expression for the time required for the projectile to travel the length of the launching device, in terms of and .

ii) Determine an expression for the mass of the projectile, in terms of and .

The mass of the projectile is . The graph in Figure 2 shows the force exerted on the projectile by the launcher as a function of time .

Figure 2

i) Using the graph in Figure 2, calculate the impulse given to the projectile.

ii) Calculate the launch speed of the projectile.

A groups of students have a cart of unknown mass and a straight horizontal track. The students are asked to take measurements to create a graph that could be used to determine the mass of the cart. The students have access to a force sensor, a motion detector, and other standard laboratory equipment. They do not have access to a scale.

Describe an experimental procedure the students could use to collect data needed to determine the mass of the cart. Include any steps necessary to reduce experimental uncertainty. If needed, you may include a simple diagram of the setup with your procedure.

Describe how the data collected in part a) could be plotted to create a linear graph and how that graph would be analyzed to determine the mass of the cart.

Figure 1

Block 1 of mass is held at rest while an ideal spring of spring constant is compressed by . Block 2 has mass where . At time , Block 1 is released. At time , the spring is no longer compressed and Block 1 immediately collides with and sticks to Block 2. The two-block system moves with constant speed , as shown in Figure 1. Friction between the blocks and the surface is negligible.

The impulse on Block 1 from the spring during the time interval is . The impulse on Block 1 from Block 2 during the collision is .

Indicate whether the magnitude of is greater than, less than, or equal to the magnitude of.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.

On the axes provided in Figure 2, sketch graphs of the magnitude of the momentum of each block as functions of time from to after . The collision occurs in a negligible amount of time. The grid lines on each graph are drawn to the same scale.

Figure 2

A student has a rubber ball of known mass which they release from rest from height . The student is asked to verify the relationship between the impulse exerted on the ball upon impact with the ground and the change in its momentum. The student has access to a force sensor, a motion sensor, and other standard laboratory equipment.

Describe an experimental procedure the student could use to collect data that would allow them to verify the relationship between impulse and change in momentum. Include any steps necessary to reduce experimental uncertainty. If needed, you may include a simple diagram of the setup with your procedure.

Describe how the collected data should be analyzed to verify the relationship between impulse and change in momentum.

The experiment is repeated using a clay ball of mass 50 g which is dropped from different heights above the ground. The clay ball sticks to the force sensor on the ground upon impact. The student's measurements are shown in Table 1.

Table 1

i) Indicate two quantities that when graphed produce a straight line that could be used to calculate an experimental value for the acceleration due to gravity. You may use the blank columns in the table for any quantities you graph other than the given data. Use the blank columns in the table to list any calculated quantities (including units) you will graph other than the data provided.

Vertical Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ Horizontal Axis: ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

ii) Plot the data points for the quantities indicated in part c)i) on the following graph. Clearly scale and label all axes, including units.

iii) Draw a best fit line to the data graphed in part c)(ii).

Calculate an experimental value for the acceleration due to gravity using the best-fit line that you drew in part c)iii).

Figure 1

Blocks 1 and 2, each of mass , are placed on a horizontal surface at points A and E, respectively, as shown in Figure 1. The surface is frictionless except for the region between points C and D.

In the time interval to , a mechanical plunger pushes Block 1 with a constant horizontal force of magnitude . At point B, Block 1 loses contact with the plunger and continues moving to the right with speed . In the time interval to , Block 1 moves over the rough surface where the coefficient of kinetic friction between the block and the surface is . At point D, Block 1 is moving to the right with speed .

The magnitude of the impulse exerted on Block 1 as it moves over the rough surface is equal to half the magnitude of the impulse exerted on it by the plunger.

i) Determine an expression for the magnitude of the force exerted on Block 1 by the plunger, in terms of , , , , , , and physical constants as appropriate.

ii) Derive an expression for the speed of Block 1 in terms of . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 1

Carts 1 and 2 are placed on a track at positions A and G, respectively, as shown in Figure 1. Each point on the track is separated by a distance . Cart 1 of mass is held at rest a vertical distance equal to above the bottom of an incline. Cart 2 of mass is held at rest while compressing an ideal spring of spring constant by an amount . At time , both carts are released. At time , Cart 1 reaches point B at the bottom of the incline, and Cart 2 reaches point F, at which point the spring is no longer compressed. Friction between the carts and the track is negligible.

During the time interval , the magnitudes of the average forces on Cart 1 and Cart 2 are equal.

If Cart 2 is more massive than Cart 1 , predict the approximate position along the track where the two carts meet. Justify your prediction using qualitative reasoning beyond referencing equations.

Derive the relationship between and to show that . Begin your derivation by writing a fundamental physics principle or an equation from the reference material.

Figure 1

Two identical carts A and B, each of mass , are moving along a frictionless horizontal track. A force sensor is positioned to record the motion of the carts along the track, as shown in Figure 1. The carts travel toward the force sensor and collide with it. Before the collision, Cart A travels at speed and Cart B travels at speed . The graphs in Figure 2 show the forces exerted on the carts during the collisions as functions of time.

Figure 2

i) Using the data from the graphs in Figure 2, determine the ratio of the change in Cart A's momentum to the change in Cart B's momentum.

The dots in Figure 3 represent Cart A and B respectively. The arrow labeled represents the impulse exerted on Cart A.

Figure 3

ii) On the dot in Figure 3 representing Cart B, draw an arrow to represent the impulse exerted on Cart B.

• The impulse must be represented by a distinct arrow starting on, and pointing away from, the dot.

• The length of the arrow should reflect the relative magnitude of the impulse exerted on Cart A.

After the collision, Cart A rebounds with the same speed.

Derive an expression for the final speed of Cart B in terms of .

On the axes provided in Figure 4, sketch a graph of the momentum of the carts as a function of time.

Figure 4

A spring is attached to the force sensor and the experiment is repeated. The carts collide with the spring, with both Cart A and Cart B having the same initial and final velocities as in the first collision. In the original collision without the spring, the magnitude of the average net force exerted on the carts is . In the collision with the spring, the magnitude of the average net force exerted on the carts is .

Indicate whether the magnitude of is greater than, less than, or equal to the magnitude of .

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.