AP Physics 2 Mathematical Routines FRQ Practice
77 Mathematical Routines questions written in the AP format. Pick one, read the full question, and write your response.
Your first scored response is included. A plan unlocks unlimited scoring.
77 questions
sorted by unit
Monatomic ideal gas isothermal expansion entropy change
Unit 9: Thermodynamics
A sealed cylinder contains of a monatomic ideal gas and is fitted with a frictionless, movable piston of cross-sectional area . The cylinder is initially in thermal equilibrium with a large reservoir at temperature . The gas is initially at pressure and volume . A schematic of the apparatus is shown in Figure 1.
Figure 1. Sealed cylinder with movable frictionless piston in thermal contact with an adjustable-temperature reservoir (initial state at T₀ and P₀).
Figure 2. Microscopic model of gas particles at state 0 (T₀) and state 1 (T₁) for comparing average speed and momentum transfer rate at the same volume.
Figure 3. Blank P–V diagram with states 0 and 1 at the same pressure P₀; students sketch the constant-pressure path from 0 to 1.
The reservoir temperature is increased so that the gas reaches a new equilibrium temperature while the piston is free to move.
Complete the following tasks in Figure 2.
Indicate in Panel (b) whether the average speed of the particles is greater than, less than, or equal to that in Panel (a).
Indicate in Panel (b) whether the average rate of momentum transfer to the cylinder wall (and thus the pressure) is greater than, less than, or equal to that in Panel (a) at the same volume.
The gas is heated from to in a quasistatic process at constant pressure . Using Figure 3, sketch the process path from state 0 to state 1.
Derive an expression for the final volume in terms of , , and . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Figure 4. Heat transfer through a metal plate of thickness L between a hot reservoir and the gas; plate area equals the cylinder cross-sectional area.
Indicate whether the change in entropy of the gas is positive, negative, or zero. The gas undergoes the constant-pressure process from state 0 to state 1, where the final temperature is and the pressure remains . The amount of gas is , and the gas is monatomic and ideal. Assume the process is quasistatic and that no phase changes occur. The heat transfer mechanism is shown in Figure 4.
Positive
Negative
Zero
Justify your answer.
What the Mathematical Routines FRQ asks
The Mathematical Routines question is FRQ 1 on the AP Physics 2 exam. You analyze one physical scenario by completing a diagram and deriving an expression, then indicate an answer in part B and justify it.
- 20–25 min
- suggested on the exam
- 10 points
- across parts A and B
- 95 min
- for all four FRQs
Part A: Analysis
Complete a diagram, then derive an expression
7 pts
Part B: Claim
Indicate an answer, such as whether a claim is correct, and justify it
3 pts
How Mathematical Routines FRQ practice works
Each question follows the exam’s format, from the full prompt to a score on every part.
Read the full question
The scenario, its figures, and parts A and B, laid out the way the exam shows them.
Write on a 23-minute timer
Inside the 20 to 25 minutes the exam suggests for this question. Pause it or turn it off, and your response saves as you go.
Submit for a score out of 10
Your response is scored against the scoring guidelines written for that question. Your first score is included.
Feedback on every part
A summary at the top tells you what to work on next. Below it, each part shows whether you earned the point and what the scoring guidelines were looking for.
Detailed Feedback
Part A-i-1: 0/1 point
Did not earn the point. You mentioned that the arrows will have a greater magnitude, but you needed to describe how the distribution of speeds changes. At a higher temperature, the distribution broadens, meaning there is a wider spread of speeds with more fast molecules, but still some slow ones.
Part A-i-2: 1/1 point
Earned the point. You correctly indicated that the net force on the wall is a vector pointing to the right.
Part A-ii-1: 1/1 point
Earned the point. You successfully started your derivation by applying the ideal gas law.
Questions about the Mathematical Routines FRQ
How many responses can I get scored?
You can read every question without a plan. Your first scored response is included. A plan unlocks unlimited scoring.
How do I complete the diagram in part A?
Open the drawing tool in the editor and draw it there. You can also switch to handwrite and add a photo of your page.
How do I type equations and symbols?
Use the subscript, superscript, and special-character buttons in the editor toolbar. Or write your derivations on paper, switch to handwrite, and add a photo.
How is the Mathematical Routines question different from the Qualitative/Quantitative Translation (QQT) question?
Both ask for a derivation. The Mathematical Routines question also asks for a diagram and is worth 10 points. The QQT starts with reasoning in words and is worth 8 points.
Are these real College Board questions?
No. We wrote them in the AP format, with the same parts and points, and wrote scoring guidelines for each one.
More AP Physics 2 practice
Representations FRQ practice
Translation Between Representations questions with the full scenario and all four parts.
Experimental Design FRQ practice
Experimental Design and Analysis questions with the full lab setup and data table.
Qualitative/Quantitative FRQ practice
Qualitative/Quantitative Translation questions with the full scenario and all three parts.
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