AP Physics 1 Mathematical Routines FRQ Practice
68 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.
68 questions
sorted by unit
Horizontal velocity magnitude across different reference frames
Unit 1: Kinematics
A student stands on a platform that is moving horizontally to the right with constant speed vp = 3.0 m/s. The student throws a ball with an initial speed of vb = 5.0 m/s at an angle of 37° above the horizontal, as measured in the platform's reference frame, as shown in Figure 1.
Figure 1. Student throwing ball on platform
Figure 2. Ball vertical velocity vs. time
Consider the motion of the ball as observed by a stationary observer standing on the ground.
On the axes shown in Figure 2, sketch a graph of the y-component of the velocity vy of the ball as a function of time t from t = 0 until the ball strikes the ground at t = t2. Explicitly label any intercepts, maximum values, or minimum values with numerical values or algebraic expressions, as appropriate.
Derive an expression for the x-component of the velocity vx of the ball as measured by the stationary observer on the ground at t = 0 in terms of vp, vb, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Calculate the horizontal distance the ball travels from the moment it is released until it strikes the ground, as measured by the stationary observer on the ground. Express your answer in meters to two significant figures.
Indicate whether the magnitude of the horizontal component of the ball's velocity, as measured by Student 1 on the platform, is greater than, less than, or equal to the magnitude of the horizontal component of the ball's velocity as measured by Student 2 on the ground. Student 1 is standing on the moving platform and observes the motion of the ball. Student 2 is a stationary observer standing on the ground and observes the motion of the same ball.
Greater than
Less than
Equal to
Justify your response using principles of reference frames and vector addition.
What the Mathematical Routines FRQ asks
The Mathematical Routines question is FRQ 1 on the AP Physics 1 exam. You analyze one physical scenario by sketching a graph and deriving or calculating quantities, 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
Sketch a graph, then derive or calculate in two more subparts
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): 1/1 point
Earned the point. You correctly sketched a constant horizontal line for the momentum.
Part A(i): 1/1 point
Earned the point. You correctly drew the line continuously from t=0 to past t_2, showing that momentum is conserved throughout the entire time interval.
Part A(iii): 0/1 point
Did not earn the point. You did not clearly indicate that the initial moving mass is m1 and the final moving mass is (m1 + m2). In your final expression for K_change, you incorrectly associated (m1 + m2) with the initial velocity and m with the final velocity term.
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 sketch the graph in part A?
Open the drawing tool in the editor and sketch 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 graph 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 1 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.
Write your first Mathematical Routines FRQ
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