Learn Extracted exam questions AP Physics C: Mechanics 2025 Free Response
2025 Free Response
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Two blocks, 1 and 2, slide toward each other on a horizontal surface. Block 1 has mass $m$ and slides in the $+x$-direction with constant speed $2v_0$. Block 2 has mass $6m$ and slides in the $-x$-direction with constant speed $v_0$, as shown in Figure 1. The blocks then collide and stick together. The collision occurs from time $t=0$ to $t=t_c$. After the collision, where $t>t_c$, the blocks move together with the same constant speed.
[Figure 1: Horizontal surface with a $+x$ arrow pointing right. Block 1 (mass $m$) on the left, moving in the $+x$-direction with speed $2v_0$ (arrow pointing right labeled $2v_0$). Block 2 (mass $6m$) on the right, moving in the $-x$-direction with speed $v_0$ (arrow pointing left labeled $v_0$).]
The diagrams in Figure 2 can be used to represent the momentum of blocks 1 and 2 before and after the collision. The momentum vector diagram for Block 1 before the collision is shown.
[Figure 2: A table titled "Momentum Before Collision" / "Momentum After Collision" with rows for Block 1, Block 2, and Two-Block System. Each cell shows a horizontal zero-momentum grid line labeled "0". The "Block 1, Before Collision" cell already shows an arrow starting at 0 and pointing right (+x-direction), representing the momentum of Block 1 before the collision. All other cells (Block 2 before collision; Block 1 after collision [shaded, not applicable]; Block 2 after collision [shaded, not applicable]; Two-Block System before collision; Two-Block System after collision) are blank grids to be filled in.]
Draw arrows on the grids to represent the momentum vectors of Block 2 before the collision and the two-block system before and after the collision.
- Arrows should start at the zero-momentum line.
- The length of the arrows should be proportional to the relative magnitudes of the vectors.
- Represent an arrow of zero length by drawing a dot at zero.
During the time interval $0 \le t \le t_c$, a force $F$ is exerted on Block 2 by Block 1 along the $x$-direction as a function of $t$ that is modeled by $F(t) = F_{\max}\sin(At)$, where $A$ is a positive constant and $F_{\max}$ is the magnitude of the maximum force exerted on Block 2 by Block 1 during the collision.
Derive an expression for $F_{\max}$. Express your answer in terms of $m$, $v_0$, $A$, $t_c$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Consider a new scenario where Block 1 initially slides in the $+x$-direction with a new constant speed $v_1$ and Block 2 again initially slides in the $-x$-direction with constant speed $v_0$. The blocks collide and stick together. In this new scenario, the two-block system has constant speed $v_0$ after the collision.
Derive an expression for $v_1$ in terms of $v_0$. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
In Scenario 1, a system composed of two springs, A and B, and a block of mass $m$ is at rest on a horizontal surface. Friction between the block and the surface is negligible. Each spring is attached to a fixed wall and the block, as shown in Figure 1. Spring A has a spring constant $k$ and Spring B has a spring constant $2k$. Each spring is at its relaxed length when the block is at position $x=0$, as shown.
[Figure 1: Spring A (spring constant $k$) attached to a wall on the left, connected to a block; Spring B (spring constant $2k$) attached to the block on the right, connected to a wall on the right. The block's position is marked $x=0$.]
The block is moved to $x=x_1$ and held at rest, as shown in Figure 2.
[Figure 2: Same spring-block system as Figure 1, now with the block displaced to the right to position $x=x_1$ (Spring A stretched, Spring B compressed). A coordinate axis is shown at top right with $+y$ (up) and $+x$ (right) directions labeled. The positions $x=0$ and $x=x_1$ are marked on the horizontal surface.]
An energy bar chart can be used to represent the elastic potential energy $U_A$ of Spring A, the elastic potential energy $U_B$ of Spring B, and the kinetic energy $K_{\text{block}}$ of the block. On the energy bar chart in Figure 3, draw shaded bars to represent the energy of the system for when the block is at $x=x_1$.
- The height of the shaded bars should be proportional to the relative values of $U_A$, $U_B$, and $K_{\text{block}}$.
- Any energy that is equal to zero should be represented by a distinct line on the zero-energy line.
[Figure 3: A blank bar-chart grid titled "Energy" (vertical axis) with three labeled columns: $U_A$, $U_B$, $K_{\text{block}}$, and a horizontal "0" line partway up the grid, with gridlines above and below for plotting bar heights.]
The block is released from rest at $x=x_1$ and begins to oscillate. Derive an expression for the speed $v$ of the block as the block passes through $x=\dfrac{1}{2}x_1$. Express your answer in terms of $m$, $k$, $x_1$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
In Scenario 1, the block oscillates with period $T$. The position $x$ of the block in Scenario 1 as a function of time $t$ is shown in Figure 4.
[Figure 4: Graph titled "Scenario 1" of position $x$ vs. time $t$. Vertical axis $x$ with values $+x_1$ marked above 0 and $-x_1$ marked below 0; horizontal axis $t$ with gridmarks at $\tfrac{1}{2}T$, $T$, $\tfrac{3}{2}T$, $2T$, $\tfrac{5}{2}T$, $3T$. The curve is a cosine-like oscillation: starts at $+x_1$ at $t=0$, crosses zero at $\tfrac{1}{2}T$ going down to $-x_1$ at around $\tfrac{1}{2}T$ to $T$, back up through zero, reaching $+x_1$ again at $T$, and repeating this pattern periodically through $3T$.]
In Scenario 2, the block-springs system is placed on a new surface. There is friction between the block and the new surface. The block is again moved to the same position $x=x_1$ and released from rest. The block completes multiple oscillations with the same period as in Scenario 1 before coming to rest.
On the axes shown in Figure 5, sketch a graph of the kinetic energy $K$ of the block as a function of $t$ for Scenario 2.
[Figure 5: Blank axes titled "Scenario 2" with vertical axis $K$ and horizontal axis $t$ with gridmarks at $\tfrac{1}{2}T$, $T$, $\tfrac{3}{2}T$, $2T$, $\tfrac{5}{2}T$, $3T$, origin labeled $O$.]
In Scenario 3, the block is replaced with a new block of larger mass. The coefficient of kinetic friction between the new block and the surface in Scenario 3 is the same as the coefficient of kinetic friction between the original block and the surface in Scenario 2.
The new block is moved to position $x=x_1$ and released from rest. The kinetic energy of the new block is plotted as a function of time.
Describe how one feature of the graph of $K$ as a function of $t$ in Scenario 3 would differ from the graph you drew in Figure 5 for Scenario 2.
Briefly justify your answer.
A box is connected to one end of a rigid rod. Both the box and the rod have negligible mass. The other end of the rod is connected to a pivot. The box is open on one side, and a block is placed inside the box.
The center of mass of the block is displaced a vertical distance $h$, as shown in Figure 1. The block-box system is then released from rest and swings downward. There is negligible friction about the pivot. When the system is at the lowest point of its swing, the rod collides with a rigid stopper, as shown in Figure 2. The box comes to rest, and the block is launched horizontally out of the box. The block moves across a horizontal surface toward a motion sensor that measures the speed of the block. All frictional forces are negligible.
[Figure 1: A pivot at top connected by a rod to a box (containing a block) displaced to the left and up, with the center of mass of the block shown a vertical distance $h$ above the box's lowest-point position (dashed horizontal line). A stopper is shown attached near the pivot.]
[Figure 2: The rod now hangs vertically straight down from the pivot, with the box (and block) at its lowest point, aligned with the dashed horizontal line.]
Students are asked to experimentally determine the acceleration due to gravity $g$ using a linear graph. To determine $g$, the students are permitted to use measurements from only a meterstick and the motion sensor.
Describe an experimental procedure using the described setup to collect data that would allow the students to determine an experimental value of $g$ using a linear graph. Include any steps necessary to reduce experimental uncertainty.
Describe how the data collected in part A could be graphed and how that graph would be analyzed to determine the value of $g$.
The experiment is repeated, but the horizontal surface on which the block slides is replaced with a new rough surface, as shown in Figure 3. The coefficient of kinetic friction between the block and the new surface is $\mu$.
[Figure 3: A rod hanging vertically from a pivot with the box/block at its lowest point on a rough (wavy-line) horizontal surface at position $x=0$; a dashed square outline is shown further right at position $x=x_{\max}$, indicating where the block comes to rest.]
The block-box system is pulled aside so that the center of mass of the block is displaced various vertical distances $h$ and then released from rest. For each vertical distance, students measure the position $x=x_{\max}$ at which the block comes to rest.
The students' measurements of $h$ and $x_{\max}$ are shown in Table 1.
Table 1
| $h$ (m) | $x_{\max}$ (m) |
|---|---|
| 0.30 | 0.76 |
| 0.45 | 1.10 |
| 0.60 | 1.40 |
| 0.75 | 1.90 |
| 0.90 | 2.30 |
Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine $\mu$.
Vertical axis: ______ Horizontal axis: ______
On the grid provided, create a graph of the quantities indicated in part C (i).
- Use Table 2 to record the measured or calculated quantities that you will plot.
- Clearly label the axes, including units as appropriate.
- Plot the points you recorded in Table 2.
[A blank graphing grid (square-ruled) is provided for plotting the chosen quantities.]
Draw a best-fit line to the data graphed in part C (ii).
Using the best-fit line that you drew in part C (iii), calculate an experimental value for $\mu$.
A uniform disk and ring, each of mass $M$ and radius $R$, roll without slipping along a horizontal surface, as shown in Figure 1. The outer edges of the disk and ring are made of the same material. The center of mass of the disk and the center of mass of the ring each initially move with the same constant speed $v$.
The disk and the ring then smoothly transition to a ramp that is inclined at an angle $\theta$ above the horizontal. Both the disk and the ring continue to roll without slipping as they move up the ramp, as shown in Figure 2.
The ring travels a greater distance along the ramp than the disk travels before each momentarily comes to rest.
[Figure 1: A horizontal surface with a Disk (solid gray circle, radius $R$ labeled) on the left and a Ring (gray annulus/hoop) to its right, both moving to the right (motion lines shown behind each) toward a ramp inclined at angle $\theta$.]
[Figure 2: The disk and ring shown together partway up the ramp inclined at angle $\theta$, moving up-slope (motion lines behind them), with the ring positioned slightly behind/around the disk.]
While the disk and the ring are rolling on the ramp without slipping, the magnitudes of the static frictional force exerted on the disk and on the ring by the ramp are $f_D$ and $f_R$, respectively.
Indicate whether $f_D$ is greater than, less than, or equal to $f_R$ by writing one of the following.
- $f_D > f_R$
- $f_D < f_R$
- $f_D = f_R$
Justify your answer using qualitative reasoning beyond referencing equations.
A cylinder has mass $M$, radius $R$, and rotational inertia $I$ about its central axis. The cylinder rolls without slipping up a ramp that is inclined at an angle $\theta$ above the horizontal.
Derive an expression for the magnitude of the static frictional force $f$ exerted on the cylinder by the ramp. Express your answer in terms of $M$, $R$, $I$, $\theta$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
In a different scenario, the centers of mass of the original disk and ring each have the same initial speed $v$ as they did in the original scenario. The ramp is replaced by a new ramp on which the disk and the ring initially slip as they roll up the new ramp.
Indicate whether the magnitude of the kinetic frictional force exerted on the disk by the new ramp is greater than, less than, or equal to the magnitude of the kinetic frictional force exerted on the ring by the new ramp while both are slipping.
Briefly justify your answer.