Translation Between Representations (College Board AP® Physics 1: Algebra-Based): Exam Questions

A projectile is fired from level ground with speed $v_0$ at an angle $\theta$ to the ground, where $\theta >45°$. The projectile is fired from a few centimeters before position $x_1$, reaches its maximum height at position $x_2$, and lands on the ground at position $x_{3.}$

The energy bar charts in Figure 1 represent the gravitational potential energy $U_g$ and the kinetic energy $K$ of the projectile as it passes through positions $x_1$, $x_2$ and $x_3$. The bar chart at position $x_1$ is complete. Draw shaded rectangles to complete the energy bar charts in Figure 1 for positions $x_2$ and $x_3$.

• Positive energy values are above the zero-energy line.

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

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

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

Figure 1

i) Derive an expression for the maximum height of the projectile in terms of $v_0$, $g$, $\theta$ and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

ii) Determine an expression for the mass of the projectile in terms of maximum gravitational potential energy $U_{gmax}$, $v_0$, $\theta$ and physical constants as appropriate.

On the axes provided in Figure 2 sketch a graph of the horizontal and vertical components of the acceleration of the projectile at $t=0$ where $y=0$. Label the lines $a_x$ and $a_y$ respectively. Add any relevant values to the axes.

Figure 2

Figure 3

A student sketches the free-body diagram shown in Figure 3, and makes the following claim:

"The free-body diagram shows the forces acting on the projectile at position $x_1$. "

Justify why the student's sketch (Figure 3) and claim are or are not consistent with the graph you sketched in part c).

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.

On the axes provided in Figure 2 sketch and label graphs of the horizontal and vertical components of the velocity of the clay ball as a function of time from the time the ball is launched to the time it strikes the block.

Figure 2

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 information.

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 information.

Figure 5 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 3 and Figure 4 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 5.

Figure 3‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎Figure 4‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎Figure 5

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

Two ideal springs, 1 and 2, of spring constant $k_1$ and $k_2$ respectively, are connected end to end. A block of mass $m$ is attached to the end of Spring 2 and the other end of Spring 1 is fixed to a wall. The block is displaced to the left of the spring's equilibrium position, $x=0$, and held stationary at position $x=-A$, as shown in Figure 1.

On the diagram in Figure 1, draw and label arrows that represent the forces (not components) that are exerted on Spring 2 as the block is held at position $x=-A$.

• Each force in your diagram must be represented by a distinct arrow starting on, and pointing away from, the point at which the force is exerted on Spring 2.

• The length of arrows should represent the relative magnitudes of the forces.

At time $t=0$, the block is released. At time $t=t_0$, the block's position is $x=\frac{1}{2}A$ and it is travelling to the right.

i) Derive an expression for the effective spring constant of Spring 1 and Spring 2 in terms of $k_1$, $k_2$ and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

ii) Starting with the equation for the position of an object in simple harmonic motion, derive an expression for time $t_0$ in terms of $k_1$, $k_2$, $m$ and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

The energy bar chart in Figure 2 represents the spring potential energy $U_s$ of the block-spring system and the kinetic energy $K$ of the block at time $t=t_0$. Draw shaded rectangles to complete the energy bar charts in Figure 2 for the block-spring system at time $t=t_0$.

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

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

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

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

Figure 2

Figure 3

A student sketches a graph of the block's displacement as a function of time, as shown in Figure 3.

Indicate whether the student's graph (Figure 3) is consistent with the energy bar chart you drew in part c). Justify your answer.

Figure 1

A student conducts an experiment using a rubber ball of mass $m$ and a clay ball of the same mass. At time $t=0$, each ball is thrown horizontally with speed $v_0$ toward two identical blocks that are fixed in place, as shown in Figure 1. At time $t=t_0$, 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 $m$ and $v_0$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

ii) Determine an expression for the magnitude of the impulse exerted by the block on the clay ball in terms of $m$ and $v_0$.

The momentum bar charts in Figure 4 represent the momentum of the rubber ball and clay ball before the collision at time $t=0$, during the collision in the time interval between $t=t_0$ and $t=2t_0$, and after the collision at time $t=2t_0$. The bar chart at $t=0$ is complete. Draw shaded rectangles to complete the momentum bar charts in Figure 4 in the time interval $t_0<t<2t_0$ and at time $t=2t_0$.

• 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 $t_0<t<2t_0$”.

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

Two identical carts A and B, each of mass $m$, 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 $v_A$ and Cart B travels at speed $v_B=2v_A$. 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 $J_A$ 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 $v_B$. Begin your derivation by writing a fundamental physics principle or an equation from the reference book.

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 $F_0$. In the collision with the spring, the magnitude of the average net force exerted on the carts is $F_S$.

Indicate whether the magnitude of $F_S$ is greater than, less than, or equal to the magnitude of $F_0$ by writing one of the following:

• $F_S>F_0$‎

• $F_S=F_0$

• $F_S<F_0$

Justify your reasoning.

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

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.

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 $F_p$ exerted on Block 1 by the plunger, in terms of $\mu$, $m$, $t_A$, $t_B$, $t_C$, $t_D$, and physical constants as appropriate.

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

On the axes provided in Figure 4, 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 4

Figure 5

A uniform solid sphere is placed at point A, as shown in Figure 5. The surface is frictionless except for the region between points C and D, where the surface is rough. The sphere is pushed by the plunger from point A to point B with a constant horizontal force that is directed toward the sphere’s center of mass. The sphere loses contact with the plunger at point B and continues moving across the horizontal surface toward point E.

In which interval(s), if any, does the sphere’s angular momentum about its center of mass change? Justify your reasoning.

Figure 1

A block of mass $m$ is attached to an ideal spring, whose other end is fixed to a wall. The block is displaced a distance $x_0$ to the left of the spring’s equilibrium position, as shown in Figure 1. The block is then released from rest and oscillates with negligible friction along the horizontal surface. While the block is oscillating, it has a maximum speed $v_{max}$ .

The energy bar charts in Figure 2 represent the spring potential energy $U_s$ of the block-spring system, and the kinetic energy $K$ of the block, as the block passes through positions $x=-x_0$, $x=0$ and $x=+x_0$ while the block oscillates. The bar chart at $x=-x_0$ is complete. Draw shaded rectangles to complete the energy bar charts in Figure 2 for positions $x=0$ and $x=+x_0$.

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

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

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

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

Figure 2

Figure 3 shows the position of the block as a function of time. Figure 4 shows the force exerted by the spring on the block as a function of time.

Figure 3

Figure 4

i) Using figures 3 and 4, determine an expression for the spring constant of the spring.

ii) Starting with the equation for the period of a mass on a spring, derive an expression for the mass of the block. Express your answer in terms of $t_0$, $x_0$, $F_{max}$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

On the axes provided in Figure 5 sketch a graph of the velocity of the block as a function of time.

Figure 5

Figure 6

A student sketches the free-body diagram in Figure 6, and makes the following claim:

“The free-body diagram shows the forces exerted on the block at time $t=1.5t_0$”.

Justify why the student’s sketch (Figure 6) and claim are or are not consistent with the graph of velocity as a function of time you sketched in part c).