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

Figure 1

Block 1 of mass $M_1$ slides to the right along a horizontal surface with speed $v_0$ toward Block 2 of mass $M_2$, which is initially at rest, as shown in Figure 1. When Block 1 collides with Block 2, the blocks stick together. Friction between the blocks and the surface is negligible.

If Block 2 is much less massive than Block 1 $\left(M_2\ll M_1\right)$, estimate the approximate speed of the two-block system after the collision in terms of $v_0$. Justify your estimate using qualitative reasoning beyond referencing equations.

Starting with conservation of momentum, derive an equation for $v_f$, the speed of the two–block system after the collision, in terms of $M_1$, $M_2$, and $v_0$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

If Block 2 is much less massive than Block 1, does the equation you derived in part b) agree with your qualitative reasoning from part a)? Justify why or why not.

Figure 1

At time $t=0$, Block A slides along a horizontal surface toward Block B, which is initially at rest, as shown in Figure 1. The masses of blocks A and B are $6\mathrm{kg}$ and $2\mathrm{kg}$, respectively. The blocks collide elastically at $t=1.0\mathrm{s}$ and, as a result, the magnitude of the change in kinetic energy of Block B is $9\mathrm{J}$. All frictional forces are negligible.

i) Determine the speed of Block B immediately after the collision.

ii) Determine the speed of Block A immediately after the collision.

The arrow in Figure 2 represents the velocity of the center of mass of the two-block system immediately after the collision.

Figure 2

The dots in Figure 3 represent Block A and B respectively.

Figure 3

On the dots in Figure 3, draw arrows to represent the velocities of Block A and Block B immediately after the collision.

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

• The velocity, 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 velocity relative to the arrow in Figure 2.

Figure 4

The graph shown in Figure 4 represents the positions $x$ of Block A, Block B, and the center of mass of the two-block system as functions of $t$ between $t=0$ and $t=1.0\mathrm{s}$.

On the graph in Figure 4, draw and label three lines to represent the positions of Block A, Block B, and the center of mass of the two-block system as functions of $t$ between $t=1.0\mathrm{s}$ and $t=2.0\mathrm{s}$. Each line should be distinctly labeled.

A student considers whether the line drawn for the center of mass in part c) would change if the blocks stuck together after the collision rather than colliding elastically. They make the following claim:

“The slope of the line drawn for the center of mass of the two-block system would be less steep”.

Justify whether or not the student’s claim is correct.

Figure 1

Block 1 of mass $M$ is oscillating with amplitude $A_1$ at the end of a spring with spring constant $k$. The other end of the spring is attached to a wall, as shown in Figure 1. Friction between the block and the floor is negligible.

Figure 2

Block 2 of mass $2M$ is dropped from rest and sticks to Block 1 as it passes through its equilibrium position, as shown in Figure 2. The two-block system oscillates with amplitude $A_2$.

i) Derive an expression for the velocity $v$ of Block 1 at the equilibrium position before Block 2 is dropped, in terms of $k$, $M$, and $A_1$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

ii) Starting with conservation of momentum, derive an expression for the velocity of the two-block system immediately after Block 2 sticks to Block 1 in terms of $v$.

Indicate whether amplitude $A_2$ is greater than, less than, or equal to amplitude $A_1$ by writing one of the following:

• $A_2>A_1$

• $A_2=A_1$

• $A_2<A_1$

Justify your answer using qualitative reasoning beyond mathematical derivations or expressions.

Block 1 of mass $m_1$ and Block 2 of mass $m_2$ are sliding along a horizontal frictionless surface when they collide at time $t_c$. Before the collision, the blocks slide toward each other. After the collision, the blocks slide together.

Indicate whether the kinetic energy of the two-block system after the collision is greater than, equal to, or less than the kinetic energy before the collision. Justify your answer using qualitative reasoning beyond referencing equations.

Figure 1

The graph in Figure 1 shows the velocities of the blocks as a function of time.

Using the graph in Figure 1, derive an expression for the ratio of the mass of Block 2 to the mass of Block 1, $\frac{m_2}{m_1}$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Does the graph in Figure 1 agree with your qualitative reasoning from part a)? Use the expression derived in part b) to justify your answer.

Figure 1

A block of mass $M_b$ is suspended from the ceiling using a string with negligible mass. A projectile of mass $M_p$ is fired horizontally from a launching device with speed $v_x$ into the block, as shown in Figure 1. While the projectile is in the launching device, the impulse given to it is $J_p$. When the projectile strikes the block, the impulse given to the block is$J_b$. After the collision, the projectile becomes embedded in the block.

If the projectile is much less massive than the block $\left(M_p\ll M_b\right)$, estimate the approximate magnitude of the impulse given to the block in terms of $J_p$. Justify your estimate using physical principles.

Figure 1

A student conducts an experiment using a clay ball of mass $m$ and a rubber ball of the same mass. Each ball is thrown horizontally with speed $v_0$ toward two identical blocks of mass $M$ that are at rest at the edge of identical tables, as shown in Figure 1. After each collision, the blocks fall to the floor. Friction between the blocks and the table is negligible.

In Experiment A, the clay ball sticks to Block A. In Experiment B, the rubber ball bounces off of Block B at a speed of $v_r$. After the collisions, Block A lands a horizontal distance $d_A$, and Block B lands a horizontal distance $d_B$ from the table's edge.

Indicate whether $d_A$ is greater than, less than, or equal to $d_B$ by writing one of the following:

• $d_A>d_B$

• $d_A=d_B$

• $d_A<d_B$

Justify your answer using qualitative reasoning beyond referencing equations.

Starting with conservation of momentum, derive an equation for the ratio of the speed of Block A after the collision to the speed of Block B after the collision, $\frac{v_A}{v_B}$. Express your answer in terms of $m$, $M$, $v_0$ and $v_r$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

If the balls are much less massive than the blocks $\left(m\ll M\right)$, does the equation you derived in part b) agree with your qualitative reasoning from part a)? Justify why or why not.

Figure 1

Block 1 of mass $m_1$ is held at rest while an ideal spring of spring constant $k$ is compressed by $∆x$. Block 2 has mass $m_2$ where $m_2<m_1$. At time $t=0$, Block 1 is released. At time $t=t_C$ , 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 $v$, as shown in Figure 1. Friction between the blocks and the surface is negligible.

Derive an expression for the velocity of the two-block system after the collision in terms of $m_1$, $m_2$, $k$ and $∆x$. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

In another scenario with the same equipment, Block 1 is used to compress the spring by the same amount as before, and Block 2 is placed in the same initial position. When the spring is released, Block 1 remains attached to the spring as it collides with Block 2. The velocity of Block 2 immediately after the collision is $v_2$.

Indicate whether $v_2$ is greater than, less than, or equal to $v_f$ by writing one of the following:

• $v_2>v_f$

• $v_2=v_f$

• $v_2<v_f$

Briefly justify your answer. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.

Figure 1

A clay ball of mass $m$ is launched at an angle $\varphi$ above the horizontal with initial speed $v_0$. At the moment it reaches the highest point in its trajectory and is moving horizontally, it collides with and sticks to a wooden block of mass $M$, as shown in Figure 1. The block is suspended from a light string of length $L$. The block and the clay then swing up to a maximum height $h_0$ above the block’s initial position and make an angle $\theta$ with the vertical.

i) Starting with conservation of momentum, derive an expression for the velocity of the block-clay system immediately after the collision. Express your answer in terms of $m$, $M$, $v_0$ and $\varphi$. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

ii) Starting with conservation of energy, derive an expression for the speed of the clay ball immediately before it strikes the block. Express your answer in terms of $m$, $M$, $L$, $\theta$ and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

Figure 4 shows a momentum diagram that represents the velocity and mass of the clay-block system immediately after the clay ball strikes the block.

Draw shaded rectangles to complete the momentum diagrams in Figure 2 and Figure 3 to represent the velocity and mass of the clay ball and block, respectively, immediately before the clay ball strikes the block.

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

• Shaded regions should start at the lines representing zero velocity and mass.

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

• The relative height and width of each shaded region should reflect the magnitude of the velocity and mass, respectively, consistent with the scale used in Figure 4.

Figure 2‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎Figure 3‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎Figure 4

In a second experiment, a steel ball is used instead of a clay ball. The steel ball is launched at the same speed and angle, but instead collides elastically with the block. The block swings up to a maximum height $h_0^{\text{'}}$.

Indicate whether $h_0^{\text{'}}$ is greater than, less than, or equal to $h_0$ by writing one of the following:

• $h_0^{\text{'}}>h_0$‎

• $h_0^{\text{'}}=h_0$

• $h_0^{\text{'}}<h_0$

Justify your answer.

Figure 1

Blocks 1 and 2, each of mass $m$, 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 $t=t_A$ to $t=t_B$, a mechanical plunger pushes Block 1 with a constant horizontal force of magnitude $F_p$. At point B, Block 1 loses contact with the plunger and continues moving to the right with speed $v_B$. In the time interval $t=t_C$ to $t=t_D$, Block 1 moves over the rough surface where the coefficient of kinetic friction between the block and the surface is $\mu$. At point D, Block 1 is moving to the right with speed $v_D$. At point E, Block 1 collides with and sticks to Block 2, after which the two-block system continues moving across the surface, eventually passing point F at time $t_F$.

Figure 2

Figure 2 shows the location of the center of mass of the two-block system just before and during the collision.

The dots in Figure 3 represent the blocks while they are in contact during the collision, and the center of mass of the two-block system.

Figure 3

On each of the dots in Figure 3, draw arrows to indicate the direction of the net force, if any, exerted on each block, and the two-block system.

• If the net force is zero for either block or the two-block system, write “zero” next to the dot.

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

• The length of the arrows, if not zero, should reflect the magnitude of the net force relative to the other arrows.

On the axes provided below, sketch and label a graph of the velocity of the center of mass of the two-block system as a function of time, from time $t_A$ until the blocks pass point F at time $t_F$. The times at which Block 1 reaches points A through F are indicated on the time axis.

Figure 1

A group of students have two carts, Cart 1 and Cart 2, each of identical, unknown mass $M$. The carts are initially at rest and placed next to each other on a horizontal track. Cart 2 contains a compressed spring which can be released by pressing a switch on top of the cart. When the switch is pressed, the spring expands and pushes a plunger outward, causing the two carts to recoil, as shown in Figure 1. Blocks of different known masses can be attached to each cart.

The group of students is asked to determine whether the total momentum of the system is conserved. The students have access to equipment that can be found in a typical school physics laboratory.

i) Indicate quantities that could be measured by the students that would allow them to determine whether the total momentum of the system is conserved during the recoil.

ii) Briefly describe a method to reduce experimental uncertainty for the measured quantities indicated in part a)i).

i) Indicate what quantities the students could graph on the horizontal and vertical axes to create a linear graph that can be used to determine whether the total momentum of the system is conserved.

ii) Briefly describe how the graph will be analyzed to determine whether the total momentum of the system is conserved.

Figure 2

In a later experiment, Cart 2 is placed next to a wall. When the switch is pressed, the spring expands and the plunger pushes on the wall, causing Cart 2 to move towards Cart 1, as shown in Figure 2. Cart 1 is initially at rest. After the collision, the two carts stick together and move along the track with speed $v$. Blocks of identical, known mass can be attached to Cart 1 and Cart 2, as long as the total mass of the system remains constant. The spring constant $k$ of the spring in Cart 2 is unknown. When the spring is contained within the cart, it is compressed by a fixed distance $x$ from its equilibrium position.

The students are asked to determine the value of the spring constant of the spring. The students measure the combined mass $M_{1,B}$ of Cart 1 and the blocks, the combined mass $M_{2,B}$ of Cart 2 and the blocks, and the final speed of the two-cart-block system.

The students measure the fixed value $x=0.062\mathrm{m}$. The students repeat the experiment using different numbers of blocks on each cart and collect the data shown in Table 1.

Table 1

The students create a graph with $v^2$ plotted on the vertical axis.

i) Label the horizontal axis of Figure 3 with a measured or calculated quantity. Include units, as appropriate. The graphed quantities should yield a linear graph that can be used to determine the spring constant $k$ of the spring.

ii) On the grid provided in Figure 3, create a graph of the quantities indicated in part c)i).

• Clearly label the horizontal axis with a numerical scale

• Plot the corresponding data points on the grid

• Any columns added to Table 1 for scratch work will not be scored

Figure 3

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

Using the line drawn in part c)iii) and the measured value $x=0.062\mathrm{m}$ as needed, calculate the spring constant $k$ of the spring.