AP Physics C: Mechanics Mathematical Routines FRQ Practice
46 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.
46 questions
sorted by unit
Relative motion and reference frame trajectories
Unit 1: Kinematics
A drone flies horizontally at constant velocity vD = 15.0 m/s in the +x-direction at an altitude h = 45.0 m above level ground. At time t = 0, when the drone is directly above point O on the ground, it releases a package. The package has mass m = 2.0 kg. Air resistance is negligible, and the ground is flat. A coordinate system with origin at point O is shown in Figure 1, where the +x-direction is horizontal and the +y-direction is vertical.
Figure 1
Figure 2
Figure 3
Consider the motion of the package as observed from the ground reference frame.
On the axes in Figure 2, sketch graphs of the x-component of velocity vx and the y-component of velocity vy of the package as functions of time from t = 0 to when the package strikes the ground at t = tf. Clearly label any intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
The position vector of the package as a function of time is given by r⃗(t) = (vD·t)î + (h - (1/2)gt²)ĵ, where g = 10 m/s² and t is measured in seconds.
Derive an expression for the speed v of the package as a function of time. Express your answer in terms of vD, g, h, t, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.
Draw an arrow on Figure 3 representing the velocity vector of the package at time t = tf as observed from the ground reference frame. The length and direction of the arrow should accurately represent the magnitude and direction of the velocity vector.
In the drone's reference frame, the package appears to follow a different trajectory than it does in the ground reference frame. Describe the shape of the trajectory of the package as observed from the drone's reference frame, and explain your reasoning using principles of relative motion and reference frames. Your explanation should include a comparison to the trajectory observed from the ground reference frame. Now consider the motion of the package as observed from the reference frame of the drone, which continues to move horizontally at constant velocity vD = 15.0 m/s in the +x-direction.
What the Mathematical Routines FRQ asks
The Mathematical Routines question is FRQ 1 on the AP Physics C: Mechanics exam. You use calculus to derive expressions, calculate values, and sketch graphs or diagrams for a mechanics scenario, then extend the analysis in a second part.
- 20–25 min
- suggested on the exam
- 10 points
- on every question
- 95 min
- for all four FRQs
Part A: Derive and sketch
A derivation from a fundamental principle, plus calculations, graphs, or diagrams
7 pts
Part B: Extend
A derivation or explanation that builds on part A, often for a changed setup
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 every part, 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: 1/1 point
Earned the point. You correctly drew the launch velocity vectors for 45° and 60° at the appropriate angles and with lengths equal to the 30° launch vector.
Part A: 1/1 point
Earned the point. You correctly drew all three highest-point velocity vectors as purely horizontal arrows pointing to the right, showing that the vertical velocity is zero.
Part A: 0/1 point
Did not earn the point. You correctly derived the unsimplified expression for the maximum height, but you left sin(60°) in your final answer. To earn this point, you needed to evaluate sin(60°) as √3/2 to get the final simplified expression 3v₀²/(8g).
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 type equations and integrals?
Type them in plain text, like v = v0 + a*t. The editor has subscript and superscript buttons and a special characters menu with symbols such as ω, τ, θ, and ∫. You can also switch to handwrite and add a photo of your page.
How do I answer the parts that ask for a graph or diagram?
Use the drawing tool to sketch on a blank graph or on the question’s figure. Your drawing is scored with the rest of your response.
Can I use a calculator and the equation sheet?
Yes. The writing screen has a scientific and a graphing calculator and the AP Physics C: Mechanics equation sheet.
Is the point split the same on every question?
No. Most of our questions give part A 7 points and part B 3. A few split the 10 points differently or add a short part C.
Are these real College Board questions?
No. We wrote them in the AP format, with the same kinds of parts, and wrote scoring guidelines for each one.
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AP Physics C: Mechanics FRQs
Every FRQ type for AP Physics C: Mechanics, with practice for each.
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