Learn Extracted exam questions AP Physics 1 2026 Free Response
2026 Free Response
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Water exits the nozzle of a fountain at an angle $\theta_0$ above the horizontal.
At time $t = 0$, a droplet of water exits the nozzle and follows the path shown in Figure 1. The droplet reaches a maximum height $h_1$ above the nozzle.
[Figure 1: Side-view diagram of a fountain. A curved gray arc labeled "Path" rises from the nozzle to a peak and comes back down. A vertical double-headed arrow labeled $h_1$ marks the height from the dashed horizontal line at the nozzle up to the peak of the path. At the base, "Fountain" labels the fountain body, "Nozzle" labels the opening, and the angle $\theta_0$ is marked between the nozzle direction and the dashed horizontal line. A small coordinate axis to the right shows $+y$ (up) and $+x$ (right). Note: Figure not drawn to scale.]
At $t = t_f$, the droplet returns to the same height at which the droplet exited the nozzle.
On the axes shown in Figure 2, sketch graphs of the horizontal and vertical components of the velocity of the water droplet as functions of $t$ from $t = 0$ to $t = t_f$.
[Figure 2: Two blank coordinate-axis grids side by side for sketching. Left graph: y-axis labeled "Horizontal Velocity Component", x-axis labeled $t$ with origin $O$ and a tick mark at $t_f$. Right graph: y-axis labeled "Vertical Velocity Component", x-axis labeled $t$ with origin $O$ and a tick mark at $t_f$. Both axes have arrows indicating positive directions upward and to the right, and each also extends slightly below the horizontal axis.]
Derive an expression for the speed of the water exiting the nozzle. Express your final answer in terms of $\theta_0$, $h_1$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
The nozzle has a circular cross-section with radius $r_0$. Derive an expression for the volume flow rate of the water exiting the nozzle. Express your final answer in terms of $\theta_0$, $h_1$, $r_0$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
The fountain nozzle is replaced with a new nozzle with a radius smaller than $r_0$. The water exits the new nozzle at the same angle $\theta_0$ above the horizontal. The volume flow rate of the water exiting the new nozzle is equal to the volume flow rate of the water exiting the original nozzle. A water droplet exiting the new nozzle reaches a maximum height $h_2$ above the nozzle.
Indicate whether $h_2$ is greater than, less than, or equal to $h_1$ by writing one of the following.
- $h_2 > h_1$
- $h_2 < h_1$
- $h_2 = h_1$
Justify your answer. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.
Two small disks, R and S, are on a straight, horizontal track.
At time $t = 0$, Disk R of mass $m_0$ is located at position $x = 0$. Disk R is initially moving with speed $v_0$ in the $+x$-direction. Disk S of mass $3m_0$ is initially at rest, as shown in the top view in Figure 1. Frictional forces are negligible.
At $t = t_1$, the disks collide. Immediately after the collision, Disk R moves with speed $\dfrac{1}{2}v_0$ in the $-x$-direction.
[Figure 1 (Top View): A horizontal dashed track with two disks. On the left, a smaller shaded disk labeled $m_0$ ("Disk R") has an arrow above it labeled $v_0$ pointing right (in the $+x$-direction, with a $+x$ axis arrow shown above). On the right, a larger open disk labeled $3m_0$ ("Disk S") is labeled $v = 0$ above it. Note: Figure not drawn to scale.]
The momentum-vector diagram in Figure 2 represents the momentums of Disk R and Disk S before the collision.
[Figure 2: Two rows, each with a horizontal dashed number-line-style axis (light vertical tick marks along it) crossing a vertical axis at a central dot. Top row labeled "Disk R": a solid arrow starts at the central dot and points right, labeled with the heading "Momentum Before Collision" above both rows. Bottom row labeled "Disk S": just a dot with no arrow, labeled "$p = 0$" above it.]
Draw arrows on Figure 3 to represent the momentum vectors of Disk R and Disk S after the collision.
- Each arrow must start on, and point away from, each dot.
- If the momentum of either disk is zero, write "$p = 0$" next to the dot.
- Draw the length of each arrow to represent the magnitude of each momentum, consistent with the scale used in Figure 2.
[Figure 3: Same style as Figure 2 but blank — two rows each with a horizontal dashed axis and a central dot, headed "Momentum After Collision", one row labeled "Disk R" and one row labeled "Disk S", for the student to draw arrows on.]
Starting with conservation of linear momentum, derive an expression for the kinetic energy of Disk S immediately after the collision. Express your final answer in terms of $m_0$, $v_0$, and physical constants, as appropriate. Begin your derivation by writing the fundamental physics principle or an equation from the reference information.
The graph shown in Figure 4 represents the positions $x$ as functions of time $t$ from $t = 0$ until $t = t_1$ for each of the following:
- Disk R
- Disk S
- The center of mass of the two-disk system
On the graph shown in Figure 4, draw three lines that represent the positions $x$ of Disk R, Disk S, and the center of mass of the two-disk system as functions of $t$ from $t = t_1$ to $t = 2t_1$. Distinctly label each line.
[Figure 4: A position ($x$, vertical axis) vs. time ($t$, horizontal axis) grid. From $t=0$ to $t=t_1$, three straight lines rise from the origin $O$: the steepest line (largest slope) is labeled "Disk S", a shallower line is labeled "Center of Mass" (shown as a dashed line), and the shallowest line is labeled "Disk R". The horizontal axis is marked with tick values $t_1$ and $2t_1$ (blank beyond $t_1$ for the student to continue the lines).]
During the collision, the magnitudes of the changes in momentum of Disk R and Disk S are $|\Delta p_R|$ and $|\Delta p_S|$, respectively.
Indicate whether $|\Delta p_R|$ is greater than, less than, or equal to $|\Delta p_S|$ by writing one of the following.
- $|\Delta p_R| > |\Delta p_S|$
- $|\Delta p_R| < |\Delta p_S|$
- $|\Delta p_R| = |\Delta p_S|$
Briefly justify your response by referencing a fundamental physics principle.
The following information applies to parts A and B.
A group of students are investigating friction using the following procedure. The students release a block of unknown mass near the top of a curved ramp. The block slides down the ramp. The block then transitions to a horizontal surface, as shown in Figure 1. Frictional forces are negligible between the block and the curved ramp, but there is friction between the block and the horizontal surface.
[Figure 1: A side-view diagram showing a curved ramp descending from upper left to a horizontal surface at the bottom. A small shaded square labeled "Block" sits at the top of the curve. The curve smoothly transitions into a horizontal line extending to the right, on which another small shaded square (the block, later position) sits partway along.]
The students are asked to perform an experiment in which a single quantity is varied in order to collect data that could be graphed to determine the value $\mu_k$ of the coefficient of kinetic friction between the block and the horizontal surface. The students have access to only a meterstick.
Indicate quantities that could be measured by the students that would allow them to determine $\mu_k$ using a linear graph.
Briefly describe a method to reduce experimental uncertainty for the measured quantities.
Indicate what quantities the students could graph on the horizontal and vertical axes to create a linear graph that can be used to determine $\mu_k$. Clearly state which quantity will be graphed on each axis.
Briefly describe the relationship between $\mu_k$ and a feature of the graph from part B (i). Your answer may include an equation that relates $\mu_k$ and the chosen feature of the graph.
The following information applies to parts C and D.
In a different experiment, the students release a block from rest on a rough ramp. The block is released a distance $d$ from a photogate. The ramp is inclined at an angle $\theta$ above the horizontal, as shown in Figure 2. The block slides down the ramp and passes through the photogate, which is positioned near the bottom of the ramp. The photogate determines the speed $v$ of the block as it passes through the photogate. The unknown coefficient of kinetic friction between the block and the ramp is $\mu_k$.
[Figure 2: A side-view diagram of a ramp inclined at angle $\theta$ above the horizontal (a right-triangle shape). A block sits on the incline near the top, with a dashed double-headed arrow labeled $d$ running down the slope from the block to a photogate device (shown as a small hatched rectangle) positioned further down the incline near the bottom. The angle $\theta$ is marked at the base of the incline.]
The experiment is repeated several times with different release distances $d$ for the same block. Table 1 shows the measured values of $d$ and $v$.
Table 1
| $d$ (m) | $v$ (m/s) |
|---|---|
| 0.20 | 0.29 |
| 0.30 | 0.38 |
| 0.40 | 0.41 |
| 0.50 | 0.49 |
| 0.60 | 0.52 |
The students correctly determine that the relationship between $d$ and $v$ is given by
where $\theta = 30°$. The students want to determine $\mu_k$. The students create a graph with $d$ plotted on the horizontal axis.
Label the vertical 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 $\mu_k$.
On the grid provided in Figure 3, create a graph that can be used to determine $\mu_k$.
- Clearly label the vertical axis with a numerical scale.
- Plot the corresponding data points on the grid.
- Table 2 is provided in your booklet for scratch work and will not be scored.
[Figure 3: A blank grid for plotting, with the vertical axis unlabeled (headed "Quantity (units, if appropriate)" with two blank lines above it for the student to write the quantity and units) and the horizontal axis labeled $d$ (m), with tick marks at 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6.]
Draw a best-fit line for the data plotted in part C (ii).
Using the best-fit line that you drew in part C (iii), calculate an experimental value for $\mu_k$.
Toys X and Y are free to rotate with negligible friction about a vertical axle through the center of each toy. The upper portion of each toy has a radius $r_0$, as shown in the side view in Figure 1. A string of length $\ell_0$ is completely wrapped around the upper portion of each toy. The rotational inertia of Toy X is $I_X$, and the rotational inertia of Toy Y is $I_Y = \dfrac{1}{2}I_X$.
[Figure 1 (Side View): Two spinning-top-style toys shown side by side. Toy X (left): a vertical axle labeled "Axle" at the top with radius $r_0$ marked at the top of the "Upper Portion" cylindrical section; "String" labels the wrapped string near the top of the upper portion; an arrow labeled $F_0$ points horizontally to the right from the upper portion; below the upper portion is a wide, ridged, bulbous body tapering to a point at the bottom, with a small curved arrow near the point indicating rotation. Toy Y (right): the same axle/radius $r_0$/"Upper Portion"/"String"/$F_0$ labeling at the top, but the lower body is a flatter shape with a wide horizontal disk (like a ring) partway down, tapering to a point at the bottom with a rotation arrow, similar to Toy X but with different mass distribution.]
Toys X and Y are initially at rest. A constant horizontal force of magnitude $F_0$ is then exerted on the string of each toy, which causes the toys to rotate, as shown in the overhead view in Figure 2. The axles of the toys remain vertical as the force is exerted on the strings. The strings do not slip as the force is exerted on the strings.
When the string loses contact with the axle of Toy X, the angular speed of Toy X is $\omega_X$.
When the string loses contact with the axle of Toy Y, the angular speed of Toy Y is $\omega_Y$.
[Figure 2 (Overhead View): A large shaded circle (the upper portion of a toy, viewed from above) with a small labeled circle at its center marked $r_0$, and an arrow labeled $F_0$ pointing to the right from the center, representing the string being pulled tangentially. A curved arrow above and to the left of the circle indicates the direction of rotation (counterclockwise).]
Indicate whether $\omega_Y$ is greater than, less than, or equal to $\omega_X$ by writing one of the following.
- $\omega_Y > \omega_X$
- $\omega_Y < \omega_X$
- $\omega_Y = \omega_X$
Justify your answer using qualitative reasoning beyond referencing equations.
Starting with the work-energy theorem or Newton's second law in rotational form, derive an expression for the angular speed $\omega_X$ of Toy X at the instant the string loses contact with the axle. Express your final answer in terms of $\ell_0$, $I_X$, $F_0$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Justify how your derived equation in part B is or is not consistent with your reasoning in part A.