Learn Extracted exam questions AP Physics 1 2021 Free Response
2021 Free Response
Source PDF on the left, extracted YAML on the right. Compare numbering, marks, options and text.
[Figure: A stunt cyclist rides down a curved ramp from height $H_0$ at the top, reaching a launch point at angle $\theta_0$ from the horizontal at the bottom of the ramp. The cyclist becomes airborne, flying over six parked cars lined up in a row, covering a horizontal distance $X_0$ while in the air (shown as a dotted arc trajectory), and lands on a second ramp on the far side of the cars. Note: Figure not drawn to scale.]
A stunt cyclist builds a ramp that will allow the cyclist to coast down the ramp and jump over several parked cars, as shown above. To test the ramp, the cyclist starts from rest at the top of the ramp, then leaves the ramp, jumps over six cars, and lands on a second ramp.
$H_0$ is the vertical distance between the top of the first ramp and the launch point. $\theta_0$ is the angle of the ramp at the launch point from the horizontal. $X_0$ is the horizontal distance traveled while the cyclist and bicycle are in the air. $m_0$ is the combined mass of the stunt cyclist and bicycle.
Derive an expression for the distance $X_0$ in terms of $H_0$, $\theta_0$, $m_0$, and physical constants, as appropriate.
If the vertical distance between the top of the first ramp and the launch point were $2H_0$ instead of $H_0$, with no other changes to the first ramp, what is the maximum number of cars that the stunt cyclist could jump over? Justify your answer, using the expression you derived in part (a).
On the axes below, sketch a graph of the vertical component of the stunt cyclist's velocity as a function of time from immediately after the cyclist leaves the ramp to immediately before the cyclist lands on the second ramp. On the vertical axis, clearly indicate the initial and final vertical velocity components in terms of $H_0$, $\theta_0$, $m_0$, and physical constants, as appropriate. Take the positive direction to be upward.
[Graph grid: vertical axis labeled "Vertical Component of Stunt Cyclist's Velocity" (unlabeled numeric scale, dashed gridlines); horizontal axis labeled "Time", origin marked $O$; empty dashed grid for the student to sketch on.]
(12 points, suggested time 25 minutes)
A group of students is investigating how the thickness of a plastic rod affects the maximum force $F_{\max}$ with which the rod can be pulled without breaking. Two students are discussing models to represent how $F_{\max}$ depends on rod thickness.
Student A claims that $F_{\max}$ is directly proportional to the radius of the rod.
Student B claims that $F_{\max}$ is directly proportional to the cross-sectional area of the rod—the area of the base of the cylinder, shaded gray in the figure above.
[Figure: A cylindrical rod shown lengthwise, with one circular end face shaded gray to indicate the cross-sectional area.]
The students have a collection of many rods of the same material. The rods are all the same length but come in a range of six different thicknesses. Design an experimental procedure to determine which student's model, if either, correctly represents how $F_{\max}$ depends on rod thickness.
In the table below, list the quantities that would be measured in your experiment. Define a symbol to represent each quantity, and also list the equipment that would be used to measure each quantity. You do not need to fill in every row. If you need additional rows, you may add them to the space just below the table.
| Quantity to be Measured | Symbol for Quantity | Equipment for Measurement |
|---|---|---|
Describe the overall procedure to be used, referring to the table. Provide enough detail so that another student could replicate the experiment, including any steps necessary to reduce experimental uncertainty. As needed, use the symbols defined in the table and/or include a simple diagram of the setup.
For a rod of radius $r_0$, it is determined that $F_{\max}$ is $F_0$, as indicated by the dot on the grid below. On the grid, draw and label graphs corresponding to the two students' models of the dependence of $F_{\max}$ on rod radius. Clearly label each graph "A" or "B," corresponding to the appropriate model.
[Graph grid: vertical axis labeled with gridlines at $2F_0, 4F_0, 6F_0, 8F_0, 10F_0$ (origin $0$); horizontal axis labeled with gridlines at $1r_0, 2r_0, 3r_0, 4r_0, 5r_0$ (origin $0$); a single data point plotted at $(1r_0, F_0)$.]
The table below shows results of measurements taken by another group of students for rods of different thicknesses.
| Rod radius (mm) | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 |
|---|---|---|---|---|---|
| $F_{\max}$ (N) | 40 | 120 | 320 | 520 | 900 |
On the grid below, plot the data points from the table. Clearly scale and label all axes, including units. Draw either a straight line or a curve that best represents the data.
[Graph: large blank grid (approximately 20 columns by 18 rows of gridlines) for the student to scale, label, plot the five data points, and draw a best-fit line/curve.]
Which student's model is more closely represented by the evidence shown in the graph you drew in part (c)?
____ Student A's model: $F_{\max}$ is directly proportional to the radius of the rod.
____ Student B's model: $F_{\max}$ is directly proportional to the cross-sectional area of the rod.
Explain your reasoning.
(12 points, suggested time 25 minutes)
A student of mass $M_S$, standing on a smooth surface, uses a stick to push a disk of mass $M_D$. The student exerts a constant horizontal force of magnitude $F_H$ over the time interval from time $t = 0$ to $t = t_f$ while pushing the disk. Assume there is negligible friction between the disk and the surface.
Assuming the disk begins at rest, determine an expression for the final speed $v_D$ of the disk relative to the surface. Express your answer in terms of $F_H$, $t_f$, $M_S$, $M_D$, and physical constants, as appropriate.
Assume there is negligible friction between the student's shoes and the surface. After time $t_f$, the student slides with speed $v_S$. Derive an equation for the ratio $v_D / v_S$. Express your answer in terms of $M_S$, $M_D$, and physical constants, as appropriate.
Assume that the student's mass is greater than that of the disk ($M_S > M_D$). On the grid below, sketch graphs of the speeds of both the student and the disk as functions of time $t$ between $t = 0$ and $t = 2t_f$. Assume that neither the disk nor the student collides with anything after $t = t_f$. On the vertical axis, label $v_D$ and $v_S$. Label the graphs "S" and "D" for the student and the disk, respectively.
[Graph grid: vertical axis labeled "Speed" (dashed gridlines, unlabeled numeric scale); horizontal axis labeled "$t$" with marked points at $0$, $t_f$, and $2t_f$; empty dashed grid for the student to sketch on.]
[Figure: A disk (shown as a cylinder with motion lines indicating it moves to the right) and a rectangular block, shown side by side and labeled "Disk" and "Block" respectively.]
The disk is now moving at a constant speed $v_1$ on the surface toward a block of mass $M_B$, which is at rest on the surface, as shown above. The disk and block collide head-on and stick together, and the center of mass of the disk-block system moves with speed $v_{cm}$.
Suppose the mass of the disk is much greater than the mass of the block. Estimate the velocity of the center of mass of the disk-block system. Explain how you arrived at your prediction without deriving it mathematically.
Suppose the mass of the disk is much less than the mass of the block. Estimate the velocity of the center of mass of the disk-block system. Explain how you arrived at your prediction without deriving it mathematically.
Now suppose that neither object's mass is much greater than the other but that they are not necessarily equal. Derive an equation for $v_{cm}$. Express your answer in terms of $v_1$, $M_D$, $M_B$, and physical constants, as appropriate.
Consider the scenario from part (c)(i), where the mass of the disk was much greater than the mass of the block. Does your equation for $v_{cm}$ from part (c)(iii) agree with your reasoning from part (c)(i)?
____ Yes ____ No
Explain your reasoning by addressing why, according to your equation, $v_{cm}$ becomes (or approaches) a certain value when $M_D$ is much greater than $M_B$.
[Figure: A cylinder of mass $m_0$ sits at the top of an incline of length $L_0$ and height $H_0$, which descends to a horizontal surface. The horizontal distance from the base of the incline is also marked $L_0$.]
(7 points, suggested time 13 minutes)
A cylinder of mass $m_0$ is placed at the top of an incline of length $L_0$ and height $H_0$, as shown above, and released from rest. The cylinder rolls without slipping down the incline and then continues rolling along a horizontal surface.
On the grid below, sketch a graph that represents the total kinetic energy of the cylinder as a function of the distance traveled by the cylinder as it rolls down the incline and continues to roll across the horizontal surface.
[Graph grid: vertical axis labeled "Total Kinetic Energy" (dashed gridlines, unlabeled numeric scale); horizontal axis labeled "Distance Traveled" with marked points at $O$, $L_0$, and $2L_0$; empty dashed grid for the student to sketch on.]
[Figure: A block of mass $m_0$ sits at the top of a separate incline of length $L_0$ and height $H_0$, which descends to a horizontal surface. The horizontal distance from the base of the incline is also marked $L_0$.]
The cylinder is again placed at the top of the incline. A block, also of mass $m_0$, is placed at the top of a separate rough incline of length $L_0$ and height $H_0$, as shown above. When the cylinder and block are released at the same instant, the cylinder begins to roll without slipping while the block begins to accelerate uniformly. The cylinder and the block reach the bottoms of their respective inclines with the same translational speed.
In terms of energy, explain why the two objects reach the bottom of their respective inclines with the same translational speed. Provide your answer in a clear, coherent paragraph-length response that may also contain figures and/or equations.
[Figure: Two pulleys of different radii are mounted concentrically on a common horizontal axle. The larger pulley (radius $2r_0$) is on the outside, the smaller pulley (radius $r_0$) is on the inside, both centered on the "Axle at Center of Pulleys." Object 1 (mass $m_0$) hangs from a light string wrapped around the larger pulley on the left side. Object 2 (mass $1.5m_0$) hangs from a light string wrapped around the smaller pulley on the right side.]
(7 points, suggested time 13 minutes)
Two pulleys with different radii are attached to each other so that they rotate together about a horizontal axle through their common center. There is negligible friction in the axle. Object 1 hangs from a light string wrapped around the larger pulley, while object 2 hangs from another light string wrapped around the smaller pulley, as shown in the figure above.
$m_0$ is the mass of object 1. $1.5m_0$ is the mass of object 2. $r_0$ is the radius of the smaller pulley. $2r_0$ is the radius of the larger pulley.
At time $t = 0$, the pulleys are released from rest and the objects begin to accelerate.
Derive an expression for the magnitude of the net torque exerted on the objects-pulleys system about the axle after the pulleys are released. Express your answer in terms of $m_0$, $r_0$, and physical constants, as appropriate.
Object 1 accelerates downward after the pulleys are released. Briefly explain why.
At a later time $t = t_C$, the string of object 1 is cut while the objects are still moving and the pulley is still rotating. Immediately after the string is cut, how do the directions of the angular velocity and angular acceleration of the pulley compare to each other?
____ Same direction ____ Opposite directions
Briefly explain your reasoning.
On the axes below, sketch a graph of the angular velocity $\omega$ of the system consisting of the two pulleys as a function of time $t$. Include the entire time interval shown. The pulleys are released at $t = 0$, and the string is cut at $t = t_C$.
[Graph grid: vertical axis labeled "$\omega$" (dashed gridlines, unlabeled numeric scale); horizontal axis labeled "$t$" with marked points at $O$ and $t_C$; empty dashed grid for the student to sketch on.]