--- title: "AP Physics C: Mechanics Mathematical Routines FRQ Practice" description: "46 Mathematical Routines questions written in the AP format, each scored part by part." canonical: "https://fiveable.me/ap-physics-c-mechanics/frqs/mathematical-routines" type: "frq-type-practice" subject: "AP Physics C: Mechanics" --- # AP Physics C: Mechanics Mathematical Routines FRQ Practice ## 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 | What it asks | Points | | --- | --- | --- | | A: Derive and sketch | A derivation from a fundamental principle, plus calculations, graphs, or diagrams | 7 | | B: Extend | A derivation or explanation that builds on part A, often for a changed setup | 3 | ## How Mathematical Routines FRQ practice works 1. **Read the full question.** The scenario, its figures, and every part, laid out the way the exam shows them. 2. **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. 3. **Submit for a score out of 10.** Your response is scored against the scoring guidelines written for that question. Your first score is included. ## All 46 questions ### Unit 1: Kinematics (9) - [Relative motion and reference frame trajectories](/ap-physics-c-mechanics/frq-practice?id=6977cc1aed46cbe9476b03de&type=frq-1-mathematical-routines) - [Projectile motion range in moving reference frame](/ap-physics-c-mechanics/frq-practice?id=698ac76695febec0038391b5&type=frq-1-mathematical-routines) - [Projectile motion: horizontal and vertical velocity components](/ap-physics-c-mechanics/frq-practice?id=698ac87b95febec0038391ba&type=frq-1-mathematical-routines) - [Projectile motion range and launch angle relationship](/ap-physics-c-mechanics/frq-practice?id=698acaaa95febec0038391c5&type=frq-1-mathematical-routines) - [Projectile motion in moving reference frame](/ap-physics-c-mechanics/frq-practice?id=698acef895febec0038391d3&type=frq-1-mathematical-routines) - [Projectile motion velocity components and landing angle](/ap-physics-c-mechanics/frq-practice?id=698ad1ec95febec0038391da&type=frq-1-mathematical-routines) - [Time-varying acceleration, projectile motion, maximum height](/ap-physics-c-mechanics/frq-practice?id=698ad3b995febec0038391e2&type=frq-1-mathematical-routines) - [Projectile motion observed from stationary and moving frames](/ap-physics-c-mechanics/frq-practice?id=6a8fab28d98e07178ef4df4a&type=frq-1-mathematical-routines) - [Projectile motion, cliff launch, time of flight](/ap-physics-c-mechanics/frq-practice?id=6a8fae4f99ccc961a9003237&type=frq-1-mathematical-routines) ### Unit 2: Force and Translational Dynamics (7) - [Spring compression, friction, inclined plane motion](/ap-physics-c-mechanics/frq-practice?id=698aca4a95febec0038391c4&type=frq-1-mathematical-routines) - [Spring compression with applied force and friction](/ap-physics-c-mechanics/frq-practice?id=698acb4395febec0038391c8&type=frq-1-mathematical-routines) - [Spring compression at turning point using energy conservation](/ap-physics-c-mechanics/frq-practice?id=698acc8c95febec0038391cb&type=frq-1-mathematical-routines) - [Tension forces in connected pulley system](/ap-physics-c-mechanics/frq-practice?id=698acfad95febec0038391d4&type=frq-1-mathematical-routines) - [Spring force and circular motion dynamics](/ap-physics-c-mechanics/frq-practice?id=698ad2b595febec0038391de&type=frq-1-mathematical-routines) - [Spring compression, friction, energy dissipation](/ap-physics-c-mechanics/frq-practice?id=698ad43d95febec0038391e3&type=frq-1-mathematical-routines) - [Spring-mass system equilibrium on inclined plane](/ap-physics-c-mechanics/frq-practice?id=698ad77595febec0038391ee&type=frq-1-mathematical-routines) ### Unit 3: Work, Energy, and Power (6) - [Variable force work and kinetic energy relationship](/ap-physics-c-mechanics/frq-practice?id=698ac76e95febec0038391b6&type=frq-1-mathematical-routines) - [Spring-mass system energy dissipation and friction](/ap-physics-c-mechanics/frq-practice?id=698ac9f695febec0038391c1&type=frq-1-mathematical-routines) - [Variable force, kinetic friction, work-energy theorem](/ap-physics-c-mechanics/frq-practice?id=69935ab84a2fbb579da96e8c&type=frq-1-mathematical-routines) - [Block sliding down incline, gravitational power and energy](/ap-physics-c-mechanics/frq-practice?id=69935b2a4a2fbb579da96e8d&type=frq-1-mathematical-routines) - [Block acceleration, friction, spring compression dynamics](/ap-physics-c-mechanics/frq-practice?id=6a8fafcc880cd676f5aae4b8&type=frq-1-mathematical-routines) - [Block acceleration, friction, spring compression](/ap-physics-c-mechanics/frq-practice?id=6a8fb120d100c0396428c5b9&type=frq-1-mathematical-routines) ### Unit 4: Linear Momentum (7) - [Horizontal velocity component of projectile motion](/ap-physics-c-mechanics/frq-practice?id=6977e3b66a3b7673d92b039e&type=frq-1-mathematical-routines) - [Elastic versus inelastic collision analysis](/ap-physics-c-mechanics/frq-practice?id=698aca3095febec0038391c2&type=frq-1-mathematical-routines) - [Elastic versus inelastic collision determination](/ap-physics-c-mechanics/frq-practice?id=698acdd495febec0038391d1&type=frq-1-mathematical-routines) - [Inelastic collision, kinetic energy loss](/ap-physics-c-mechanics/frq-practice?id=69935895c0084858b9af7aea&type=frq-1-mathematical-routines) - [Elastic versus inelastic collision analysis](/ap-physics-c-mechanics/frq-practice?id=6a8fb4f4e654de134193d947&type=frq-1-mathematical-routines) - [Collision forces between objects with different masses](/ap-physics-c-mechanics/frq-practice?id=6a8fb595e654de134193d957&type=frq-1-mathematical-routines) - [Elastic versus inelastic ball-wall collision](/ap-physics-c-mechanics/frq-practice?id=6a8fb643e654de134193d961&type=frq-1-mathematical-routines) ### Unit 5: Torque and Rotational Dynamics (3) - [Rotational dynamics with time-varying torque](/ap-physics-c-mechanics/frq-practice?id=698ad23795febec0038391dc&type=frq-1-mathematical-routines) - [Rotating disk with falling mass and string tension](/ap-physics-c-mechanics/frq-practice?id=698ad50795febec0038391e6&type=frq-1-mathematical-routines) - [Falling block, rotating disk, air resistance torque](/ap-physics-c-mechanics/frq-practice?id=698ad6c795febec0038391ea&type=frq-1-mathematical-routines) ### Unit 6: Energy and Momentum of Rotating Systems (6) - [Rotating disk with time-varying tangential force](/ap-physics-c-mechanics/frq-practice?id=698acc8f95febec0038391cc&type=frq-1-mathematical-routines) - [Rotational dynamics with time-varying friction force](/ap-physics-c-mechanics/frq-practice?id=698acdb095febec0038391d0&type=frq-1-mathematical-routines) - [Rotational kinetic energy from time-varying torque application](/ap-physics-c-mechanics/frq-practice?id=698ad11895febec0038391d9&type=frq-1-mathematical-routines) - [Rotating disk with time-varying torque application](/ap-physics-c-mechanics/frq-practice?id=698ad67995febec0038391e9&type=frq-1-mathematical-routines) - [Angular momentum conservation in rotating disk-clay collision](/ap-physics-c-mechanics/frq-practice?id=698ad76395febec0038391ec&type=frq-1-mathematical-routines) - [Rotational collision and angular momentum conservation](/ap-physics-c-mechanics/frq-practice?id=698ad8a995febec0038391f3&type=frq-1-mathematical-routines) ### Unit 7: Oscillations (8) - [Damped rotational oscillation energy dissipation](/ap-physics-c-mechanics/frq-practice?id=6977ce5669aa266f2fc7cfc8&type=frq-1-mathematical-routines) - [Nonuniform rod oscillation and angular displacement](/ap-physics-c-mechanics/frq-practice?id=698ad22b95febec0038391db&type=frq-1-mathematical-routines) - [Physical pendulum with composite rigid body](/ap-physics-c-mechanics/frq-practice?id=698ad5c495febec0038391e7&type=frq-1-mathematical-routines) - [Physical pendulum oscillation period](/ap-physics-c-mechanics/frq-practice?id=698ad73e95febec0038391eb&type=frq-1-mathematical-routines) - [Time-varying force on spring-mass system oscillation](/ap-physics-c-mechanics/frq-practice?id=698ad85095febec0038391f0&type=frq-1-mathematical-routines) - [Physical pendulum oscillation with added mass](/ap-physics-c-mechanics/frq-practice?id=698ad85395febec0038391f1&type=frq-1-mathematical-routines) - [Spring-mass system with friction and energy dissipation](/ap-physics-c-mechanics/frq-practice?id=698ad8cc95febec0038391f4&type=frq-1-mathematical-routines) - [Damped rotational oscillation energy dissipation](/ap-physics-c-mechanics/frq-practice?id=698ada3895febec0038391f7&type=frq-1-mathematical-routines) ## Sample question **Relative motion and reference frame trajectories** 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. Figures: Type: setup_diagram Caption: Figure 1 URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-mechanics/71842779-e14f-487b-aac8-08f8e675f9b0_fig1.jpg Placement: after-background Figure 2: Type: graph_axes Caption: Figure 2 URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-mechanics/71842779-e14f-487b-aac8-08f8e675f9b0_fig2.jpg Placement: before-part Part Letter: A Figure 3: Type: vector_diagram Caption: Figure 3 URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-mechanics/71842779-e14f-487b-aac8-08f8e675f9b0_fig3.jpg Placement: before-part Part Letter: A Part A (7 points): Context: Consider the motion of the package as observed from the ground reference frame. i. 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. (2 pt) Verb: sketch ii. 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. (3 pt) Verb: derive iii. **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. (2 pt) Verb: draw Part B (3 points): Prompt: 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. Context: 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. Verb: describe and explain Image URLs (3): - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-mechanics/71842779-e14f-487b-aac8-08f8e675f9b0_fig1.jpg - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-mechanics/71842779-e14f-487b-aac8-08f8e675f9b0_fig2.jpg - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-c-mechanics/71842779-e14f-487b-aac8-08f8e675f9b0_fig3.jpg ## Questions **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. ## Related - [Mathematical Routines FRQ exam guide](/ap-physics-c-mechanics/ap-physics-c-mechanics-exam/ap-physics-c-mechanics-ap-phys-mech-frq-mathematical-routines/study-guide/ap-physics-c-mechanics-ap-phys-mech-frq-mathematical-routines) - [AP Physics C: Mechanics Representations FRQ practice](/ap-physics-c-mechanics/frqs/translation-between-representations) - [AP Physics C: Mechanics Experimental Design FRQ practice](/ap-physics-c-mechanics/frqs/experimental-design) - [AP Physics C: Mechanics Qualitative/Quantitative FRQ practice](/ap-physics-c-mechanics/frqs/qualitative-quantitative-translation) - [AP Physics C: Mechanics FRQs](/ap-physics-c-mechanics/frqs) - Any question by id: the `get_frq` tool on the [Fiveable MCP server](/mcp/docs) ## About This Document Canonical Fiveable pages are available as Markdown at the same path plus `.md`. - [llms.txt](https://fiveable.me/llms.txt): index of Fiveable's sections and URL patterns - [llms-full.txt](https://fiveable.me/llms-full.txt): complete subject and unit listing - [MCP server](https://fiveable.me/mcp): call Fiveable as tools instead of fetching pages (`https://fiveable.me/api/mcp`) - [MCP server for AP teachers](https://fiveable.me/mcp/teachers): a teacher's classes, assignments and AP-rubric grading (`https://fiveable.me/api/mcp/teacher`)