Learn Extracted exam questions AP Physics 2 2026 Free Response
2026 Free Response
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Part A
A sample of $n$ moles of a monatomic, ideal gas is in a large, sealed, thermally conducting container with fixed volume. A small sphere of mass $m_{\text{S}}$ is in the container. The volume of the sphere is much less than the volume of the container.
The gas is initially in State X with pressure $P$ and volume $V$, as shown in Figure 1.
[Figure 1: A sealed square container labelled "Gas $n, P, V$" with a small sphere labelled "Sphere $m_{\text{S}}$" inside near the bottom, connected by an arrow from the "Sphere" label to a dot inside the container. Caption below the figure: "State X". Note: Figure not drawn to scale.]
The sphere is initially in thermal equilibrium with the gas. The gas undergoes a heating process until the gas is in State Y with pressure $3P$ and the sphere is again in thermal equilibrium with the gas. The total energy transferred to the sphere during the heating process is $Q_{\text{S}}$.
The graph in Figure 2 represents the number of atoms per unit speed as a function of atom speed for the gas when the gas is in State X. On Figure 3, sketch a curve that could represent the number of atoms per unit speed as a function of atom speed for the gas when the gas is in State Y.
[Figure 2: Axes labelled "Number of Atoms per Unit Speed" (vertical) vs. "Atom Speed" (horizontal), captioned "State X". A Maxwell-Boltzmann-type curve starts at the origin, rises to a single rounded peak (marked with a dashed horizontal line to the peak height and a dashed vertical line to the corresponding atom speed), then decreases back toward the horizontal axis at higher speeds.]
[Figure 3: Blank axes labelled "Number of Atoms per Unit Speed" (vertical) vs. "Atom Speed" (horizontal), captioned "State Y", with the same dashed reference lines (horizontal at the State X peak height, vertical at the State X peak speed) shown for reference, but no curve drawn — to be sketched by the student.]
Derive an expression for the change $\Delta T$ in temperature of the gas for the heating process that the gas undergoes as the gas changes from State X to State Y. Express your answer in terms of $n$, $P$, $V$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Derive an expression for the specific heat $c_{\text{S}}$ of the sphere. Express your answer in terms of $n$, $m_{\text{S}}$, $P$, $V$, $Q_{\text{S}}$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Part B
An insulated container is filled with a liquid of mass $m_{\text{L}}$ and specific heat $c_{\text{L}}$. The original sphere is submerged in the liquid. The sphere has mass $m_{\text{S}}$, where $m_{\text{S}} < m_{\text{L}}$, and specific heat $c_{\text{S}}$, where $c_{\text{S}} < c_{\text{L}}$. The initial temperature of the sphere is greater than the initial temperature of the liquid. Later, the liquid and the sphere reach thermal equilibrium. The absolute values of the changes in the temperatures of the liquid and the sphere are $|\Delta T_{\text{L}}|$ and $|\Delta T_{\text{S}}|$, respectively.
Indicate whether $|\Delta T_{\text{S}}|$ is greater than, less than, or equal to $|\Delta T_{\text{L}}|$. Include one of the following relationships in your response.
- $|\Delta T_{\text{S}}| > |\Delta T_{\text{L}}|$
- $|\Delta T_{\text{S}}| < |\Delta T_{\text{L}}|$
- $|\Delta T_{\text{S}}| = |\Delta T_{\text{L}}|$
Justify your answer. Your justification may include expressions/equations but must include conceptual reasoning beyond algebraic solutions.
Figure 1 represents the energy levels and their corresponding states for a hypothetical atom.
[Figure 1: An energy-level diagram with two columns, "Energy" and "State". Three horizontal levels are drawn: the topmost level is labelled $-2E_0$ on the left and $n=3$ on the right; the middle level is labelled $-3E_0$ on the left and $n=2$ on the right; the bottom level (drawn noticeably lower, with extra vertical spacing below $n=2$) is labelled $-5E_0$ on the left and $n=1$ on the right.]
Part A
On Figure 1, draw arrows to represent all possible atomic transitions that could result in the emission of a photon.
Part B
Derive an expression for the wavelength of the highest-energy photon that can be emitted from the atom. Express your answer in terms of $E_0$ and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
The following information applies to parts C and D.
A device can emit monochromatic electromagnetic radiation. The wavelength $\lambda$ of the radiation can be varied over a continuous range of wavelengths.
Part C
On Figure 2, sketch the energy $E$ of emitted photons from the device as a function of $\lambda$ for $\lambda_0 < \lambda < 4\lambda_0$. Photons with wavelength $\lambda_0$ have energy $4E_0$. Sketches made in the shaded region will not be scored.
[Figure 2: A set of axes with vertical axis $E$ labelled with gridlines at $2E_0$ and $4E_0$, and horizontal axis $\lambda$ labelled with gridlines at $0$, $\lambda_0$, $2\lambda_0$, $3\lambda_0$, $4\lambda_0$. The region from $\lambda = 0$ to $\lambda = \lambda_0$ is shaded gray (sketches there are not scored); dashed gridlines mark the intersections at $(\lambda_0, 4E_0)$ and other grid points across the unshaded region from $\lambda_0$ to $4\lambda_0$. No curve is drawn — to be sketched by the student.]
Part D
A student claims that, based on the energy states shown in Figure 1, the atom can emit a photon of wavelength $\lambda_0$ and energy $4E_0$ like the device described in part C.
Indicate whether the claim is correct or incorrect. Briefly justify your answer by referencing the representation in part A.
The following information applies to parts A and B.
In Experiment 1, scientists want to collect data that can be graphed to determine the magnitude $B_0$ of an external, uniform magnetic field. The field is directed into Figure 1 in the region between identical, parallel, conducting plates that are connected to a power supply of variable emf $\mathcal{E}$. The plates are a known distance $d$ apart, where $d$ is much smaller than the dimensions of the plates.
[Figure 1: A power supply (with terminals $+$ and $-$) connected by wires to a rectangular "Device" box, which emits particles with speed $v$ toward the region between two horizontal parallel plates (upper plate labelled at right as "Plate", lower plate unlabeled) separated by a known distance $d$. Between the plates, an array of "×" symbols indicates a uniform magnetic field $B_0$ directed into the page. A "Motion Detector" (shaded square) is positioned to the right of the plates, in line with the gap between them. Note: Figure not drawn to scale.]
In each trial, a device emits small, charged spheres with speed $v$ through the region between the plates. $\mathcal{E}$ is varied until the spheres move undeflected toward a motion detector. The scientists can vary $v$ between trials. Gravitational effects are negligible.
In addition to the equipment shown in Figure 1, the scientists also have access to a voltmeter. The scientists do not have access to other measuring tools, devices, sensors, or equipment.
Part A
Indicate quantities that could be measured using the available equipment that would allow the scientists to determine $B_0$ by using a linear graph.
Briefly describe a method to reduce experimental uncertainty for the measured quantities indicated in part A (i).
Part B
Indicate what quantities the scientists could graph on the horizontal and vertical axes to create a linear graph that can be used to determine $B_0$. Clearly indicate which quantity corresponds to each axis.
Briefly describe the relationship between $B_0$ and one feature of the graph from part B (i). Your answer may include an equation that relates $B_0$ and the indicated feature of the graph.
The following information applies to parts C and D.
In Experiment 2, scientists want to use a graph to determine the mass $m$ of identical, charged particles that are emitted from a device. Each particle has charge $Q = +6.4 \times 10^{-19}\ \text{C}$.
In each trial, a particle is emitted with speed $v_0 = 3.0 \times 10^{6}\ \text{m/s}$ toward a region of external, uniform magnetic field of variable magnitude $B$. The radius $r$ of the path along which the particle moves while in the field, shown in Figure 2, is recorded.
[Figure 2: A particle of mass $m$ and charge $Q$ enters from the left with velocity $v_0$ (dashed horizontal arrow) into a rectangular region filled with dots representing a uniform magnetic field $B$ directed out of the page. Inside the field region, a dashed curved arc (a semicircular path) curves the particle's trajectory, with a double-headed arrow labelled $r$ marking the radius of the circular path.]
In subsequent trials, $B$ is increased and the resulting $r$ is recorded. Gravitational effects are negligible. Collected data are provided in Table 1.
Table 1
| $B$ (T) | $r$ (m) |
|---|---|
| 0.04 | 1.8 |
| 0.06 | 1.2 |
| 0.14 | 0.5 |
| 0.16 | 0.4 |
| 0.20 | 0.3 |
Part C
Label the axes of the grid provided with measured or calculated quantities. Include units, as appropriate. The graphed quantities should yield a linear graph that can be used to determine $m$.
On the grid, create a graph of the quantities indicated in part C (i).
- Label the vertical and horizontal axes with numerical scales.
- Plot the corresponding data points on the grid.
- Table 2 is provided in your booklet for scratch work, but the table will not be scored.
[A blank labelled grid (square grid ruling) is provided for the student to label axes, choose numerical scales, and plot the data points from Table 1, with blank lines above and below the grid for writing the "Quantity (units, if appropriate)" for each axis.]
Draw a best-fit line for the data graphed in part C (ii).
Part D
Calculate $m$ from the best-fit line drawn on the grid.
Equipotential lines for a region with an electric field are shown.
[Figure: Five curved, roughly vertical equipotential lines are drawn, each concave toward the right, labelled from left to right along the top: $-3V_0$ (leftmost, unlabeled at top but visible at the line itself), $-2V_0$, $-V_0$, $0$, $V_0$, $2V_0$. Point S is marked with a dot on the $-3V_0$ line; Point T is marked with a dot on the $2V_0$ line. An arrow at the bottom right labelled "$+x$" indicates the positive x-direction, pointing right (in the direction of increasing potential).]
A small sphere (not shown) with positive charge $+Q$ has initial speed $v_{\text{S}}$ in the $+x$-direction and kinetic energy $K_{\text{S}}$ when the sphere is at Point S. The sphere moves to Point T. The force from the electric field is the only force that is exerted on the sphere as the sphere moves between Points S and T.
The sphere has final speed $v_{\text{T}}$ and kinetic energy $K_{\text{T}}$ when the sphere is at Point T.
Part A
Indicate whether $v_{\text{T}}$ is greater than, less than, or equal to $v_{\text{S}}$. Include one of the following relationships in your response.
- $v_{\text{T}} > v_{\text{S}}$
- $v_{\text{T}} < v_{\text{S}}$
- $v_{\text{T}} = v_{\text{S}}$
Justify your answer. Your justification may include expressions/equations but must include conceptual reasoning beyond algebraic solutions.
Part B
Derive an expression for $K_{\text{T}}$ in terms of $V_0$, $Q$, $K_{\text{S}}$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Part C
A new, small sphere of the same mass as the original sphere but with negative charge $-Q$ has the same initial speed $v_{\text{S}}$ as the original sphere in the $+x$-direction when the sphere is at Point S. The force from the electric field is the only force that is exerted on the new sphere as the new sphere moves between Points S and T. The new sphere has kinetic energy $K_{\text{new}}$ when the new sphere is at Point T.
Indicate whether $K_{\text{new}}$ is greater than, less than, or equal to $K_{\text{T}}$. Include one of the following relationships in your response.
- $K_{\text{new}} > K_{\text{T}}$
- $K_{\text{new}} < K_{\text{T}}$
- $K_{\text{new}} = K_{\text{T}}$
Briefly justify your answer by referencing your derivation in part B.