--- title: "AP Physics 1 Mathematical Routines FRQ Practice" description: "68 Mathematical Routines questions written in the AP format, each scored part by part." canonical: "https://fiveable.me/ap-physics-1-revised/frqs/mathematical-routines" type: "frq-type-practice" subject: "AP Physics 1" --- # AP Physics 1 Mathematical Routines FRQ Practice ## 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 | What it asks | Points | | --- | --- | --- | | A: Analysis | Sketch a graph, then derive or calculate in two more subparts | 7 | | B: Claim | Indicate an answer, such as whether a claim is correct, and justify it | 3 | ## How Mathematical Routines FRQ practice works 1. **Read the full question.** The scenario, its figures, and parts A and B, 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 68 questions ### Unit 1: Kinematics (14) - [Horizontal velocity magnitude across different reference frames](/ap-physics-1-revised/frq-practice?id=6977e1c7c85fd165a6d493b6&type=frq-1-mathematical-routines) - [Ball velocity in reference frames](/ap-physics-1-revised/frq-practice?id=698a8d3e7c0d49fedeed5644&type=frq-1-mathematical-routines) - [Relative velocity of projectile in moving reference frame](/ap-physics-1-revised/frq-practice?id=698a8e057c0d49fedeed564c&type=frq-1-mathematical-routines) - [Drone displacement components under wind influence](/ap-physics-1-revised/frq-practice?id=698a8e997c0d49fedeed564f&type=frq-1-mathematical-routines) - [Relative velocity in reference frames](/ap-physics-1-revised/frq-practice?id=698a8f3b7c0d49fedeed5655&type=frq-1-mathematical-routines) - [Relative velocity transformations between inertial reference frames](/ap-physics-1-revised/frq-practice?id=698a90617c0d49fedeed565c&type=frq-1-mathematical-routines) - [Boat velocity relative to water and riverbank frames](/ap-physics-1-revised/frq-practice?id=698a91897c0d49fedeed5667&type=frq-1-mathematical-routines) - [Relative velocity in different reference frames](/ap-physics-1-revised/frq-practice?id=698a92167c0d49fedeed566d&type=frq-1-mathematical-routines) - [Relative velocity of puck in different reference frames](/ap-physics-1-revised/frq-practice?id=698a92a47c0d49fedeed5675&type=frq-1-mathematical-routines) - [Velocity transformations between inertial reference frames](/ap-physics-1-revised/frq-practice?id=698a936d7c0d49fedeed567a&type=frq-1-mathematical-routines) - [Relative velocity across inertial reference frames](/ap-physics-1-revised/frq-practice?id=698a949c7c0d49fedeed5685&type=frq-1-mathematical-routines) - [Relative velocity observations from different reference frames](/ap-physics-1-revised/frq-practice?id=698a97157c0d49fedeed569d&type=frq-1-mathematical-routines) - [Motion of cart in moving and stationary reference frames](/ap-physics-1-revised/frq-practice?id=698a97ae7c0d49fedeed56a4&type=frq-1-mathematical-routines) - [Relative velocity in moving reference frames](/ap-physics-1-revised/frq-practice?id=698a98467c0d49fedeed56aa&type=frq-1-mathematical-routines) ### Unit 2: Force and Translational Dynamics (6) - [Connected blocks, friction, tension forces](/ap-physics-1-revised/frq-practice?id=698a8de47c0d49fedeed5649&type=frq-1-mathematical-routines) - [Connected blocks, tension, friction forces](/ap-physics-1-revised/frq-practice?id=698a92547c0d49fedeed5670&type=frq-1-mathematical-routines) - [Circular motion friction on rotating platform](/ap-physics-1-revised/frq-practice?id=698a93167c0d49fedeed5677&type=frq-1-mathematical-routines) - [Two-block pulley system motion and forces](/ap-physics-1-revised/frq-practice?id=698a96957c0d49fedeed5699&type=frq-1-mathematical-routines) - [Circular motion, friction, and tension forces](/ap-physics-1-revised/frq-practice?id=698a9af27c0d49fedeed56c6&type=frq-1-mathematical-routines) - [Applied force, friction, and crate acceleration](/ap-physics-1-revised/frq-practice?id=6ab29cfbd9cf3047b278bffc&type=frq-1-mathematical-routines) ### Unit 3: Work, Energy, and Power (12) - [Block-spring system energy with friction losses](/ap-physics-1-revised/frq-practice?id=698a8e057c0d49fedeed564d&type=frq-1-mathematical-routines) - [Energy dissipation during spring compression with friction](/ap-physics-1-revised/frq-practice?id=698a8ffa7c0d49fedeed565a&type=frq-1-mathematical-routines) - [Spring potential energy, friction, and ramp motion](/ap-physics-1-revised/frq-practice?id=698a91f17c0d49fedeed566c&type=frq-1-mathematical-routines) - [Block motion, friction, mechanical energy conversion](/ap-physics-1-revised/frq-practice?id=698a92827c0d49fedeed5671&type=frq-1-mathematical-routines) - [Spring-launched block on inclined ramp with friction](/ap-physics-1-revised/frq-practice?id=698a931c7c0d49fedeed5679&type=frq-1-mathematical-routines) - [Energy dissipation through friction on surfaces](/ap-physics-1-revised/frq-practice?id=698a94c87c0d49fedeed5686&type=frq-1-mathematical-routines) - [Block sliding down ramp, compressing spring with friction](/ap-physics-1-revised/frq-practice?id=698a95867c0d49fedeed568f&type=frq-1-mathematical-routines) - [Block sliding down ramp with friction and spring compression](/ap-physics-1-revised/frq-practice?id=698a96677c0d49fedeed5697&type=frq-1-mathematical-routines) - [Block pulled up incline by motor](/ap-physics-1-revised/frq-practice?id=698a97027c0d49fedeed569b&type=frq-1-mathematical-routines) - [Energy dissipation through friction on inclined plane](/ap-physics-1-revised/frq-practice?id=698a979c7c0d49fedeed56a3&type=frq-1-mathematical-routines) - [Block sliding down ramp, spring compression power comparison](/ap-physics-1-revised/frq-practice?id=698a99ca7c0d49fedeed56ba&type=frq-1-mathematical-routines) - [Block energy conversion and system work analysis](/ap-physics-1-revised/frq-practice?id=698a9ab47c0d49fedeed56c3&type=frq-1-mathematical-routines) ### Unit 4: Linear Momentum (5) - [Connected blocks with applied force and friction](/ap-physics-1-revised/frq-practice?id=698a95377c0d49fedeed568a&type=frq-1-mathematical-routines) - [Momentum and kinetic energy in cart collision](/ap-physics-1-revised/frq-practice?id=698a95ff7c0d49fedeed5692&type=frq-1-mathematical-routines) - [Collision elasticity and momentum conservation](/ap-physics-1-revised/frq-practice?id=698a97277c0d49fedeed56a0&type=frq-1-mathematical-routines) - [Collision type determination using momentum and energy](/ap-physics-1-revised/frq-practice?id=698a98427c0d49fedeed56a9&type=frq-1-mathematical-routines) - [Collision elasticity and kinetic energy conservation](/ap-physics-1-revised/frq-practice?id=698a99667c0d49fedeed56b4&type=frq-1-mathematical-routines) ### Unit 5: Torque and Rotational Dynamics (3) - [Rotational dynamics of disk-block system](/ap-physics-1-revised/frq-practice?id=698a971c7c0d49fedeed569e&type=frq-1-mathematical-routines) - [Rotational inertia and angular acceleration of composite systems](/ap-physics-1-revised/frq-practice?id=6a8f9a45964e4d33409cf123&type=frq-1-mathematical-routines) - [Rotational motion of rigid system with friction](/ap-physics-1-revised/frq-practice?id=6a8f9aa7964e4d33409cf135&type=frq-1-mathematical-routines) ### Unit 6: Energy and Momentum of Rotating Systems (6) - [Rotational motion and angular momentum of disk](/ap-physics-1-revised/frq-practice?id=698a98ea7c0d49fedeed56b1&type=frq-1-mathematical-routines) - [Angular momentum conservation in coupled rotation](/ap-physics-1-revised/frq-practice?id=698a9d287c0d49fedeed56d9&type=frq-1-mathematical-routines) - [Rotating disk-ring system angular momentum behavior](/ap-physics-1-revised/frq-practice?id=698aa3a27c0d49fedeed5710&type=frq-1-mathematical-routines) - [Rotational impulse and angular momentum change equivalence](/ap-physics-1-revised/frq-practice?id=698aa6f97c0d49fedeed572b&type=frq-1-mathematical-routines) - [Rotational motion of pivoted rod under gravity](/ap-physics-1-revised/frq-practice?id=6a8fa5c2fae433b4a83179a1&type=frq-1-mathematical-routines) - [Rotational deceleration from applied tangential force](/ap-physics-1-revised/frq-practice?id=6a8fa9a5e4a89296bc0b3939&type=frq-1-mathematical-routines) ### Unit 7: Oscillations (9) - [Period changes in two-block oscillating system](/ap-physics-1-revised/frq-practice?id=6977e6bcc5bfc64c1a71b07e&type=frq-1-mathematical-routines) - [Spring-mass system energy conservation](/ap-physics-1-revised/frq-practice?id=698a9d7e7c0d49fedeed56dc&type=frq-1-mathematical-routines) - [Horizontal spring-mass system, oscillatory motion](/ap-physics-1-revised/frq-practice?id=698a9f2a7c0d49fedeed56ea&type=frq-1-mathematical-routines) - [Simple harmonic motion, spring forces, acceleration](/ap-physics-1-revised/frq-practice?id=698aa0487c0d49fedeed56f5&type=frq-1-mathematical-routines) - [Simple harmonic motion energy comparison](/ap-physics-1-revised/frq-practice?id=698aa0d07c0d49fedeed56f7&type=frq-1-mathematical-routines) - [Block-spring system oscillation period changes](/ap-physics-1-revised/frq-practice?id=698aa1557c0d49fedeed56fb&type=frq-1-mathematical-routines) - [Simple harmonic motion acceleration at different displacements](/ap-physics-1-revised/frq-practice?id=698aa2a97c0d49fedeed5706&type=frq-1-mathematical-routines) - [Simple harmonic motion: spring force and acceleration](/ap-physics-1-revised/frq-practice?id=698aa3ce7c0d49fedeed5711&type=frq-1-mathematical-routines) - [Spring-mass oscillation period comparison](/ap-physics-1-revised/frq-practice?id=698aa5857c0d49fedeed5721&type=frq-1-mathematical-routines) ### Unit 8: Fluids (13) - [Buoyant force on rising bubble in water column](/ap-physics-1-revised/frq-practice?id=6977e9402fb89df59930c842&type=frq-1-mathematical-routines) - [Water pressure and flow through sealed tank nozzle](/ap-physics-1-revised/frq-practice?id=698a9c437c0d49fedeed56d1&type=frq-1-mathematical-routines) - [Fluid flow through narrowing pipe sections](/ap-physics-1-revised/frq-practice?id=698a9d4e7c0d49fedeed56db&type=frq-1-mathematical-routines) - [Fluid flow through pipe with pressure variations](/ap-physics-1-revised/frq-practice?id=698a9e8f7c0d49fedeed56e4&type=frq-1-mathematical-routines) - [Water discharge velocity from tank nozzle](/ap-physics-1-revised/frq-practice?id=698aa01c7c0d49fedeed56f4&type=frq-1-mathematical-routines) - [Water pressure and flow through narrowing pipe](/ap-physics-1-revised/frq-practice?id=698aa2b67c0d49fedeed5707&type=frq-1-mathematical-routines) - [Fluid flow through narrowing pipe sections](/ap-physics-1-revised/frq-practice?id=698aa35b7c0d49fedeed570d&type=frq-1-mathematical-routines) - [Water flow through tapered pipe nozzle](/ap-physics-1-revised/frq-practice?id=698aa4c07c0d49fedeed571d&type=frq-1-mathematical-routines) - [Fluid pressure and buoyant force in pipe flow](/ap-physics-1-revised/frq-practice?id=698aa5c57c0d49fedeed5724&type=frq-1-mathematical-routines) - [Fluid pressure and net force in narrowing pipe](/ap-physics-1-revised/frq-practice?id=698aa71b7c0d49fedeed572f&type=frq-1-mathematical-routines) - [Pressurized water flow through tapered pipe sections](/ap-physics-1-revised/frq-practice?id=698aa8457c0d49fedeed5735&type=frq-1-mathematical-routines) - [Fluid flow through pipe constriction, pressure differences](/ap-physics-1-revised/frq-practice?id=698aa8fa7c0d49fedeed5737&type=frq-1-mathematical-routines) - [Fluid flow through narrowing pipe sections](/ap-physics-1-revised/frq-practice?id=698aa9947c0d49fedeed573b&type=frq-1-mathematical-routines) ## Sample question **Horizontal velocity magnitude across different reference frames** 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. Figures: Type: setup_diagram Caption: Figure 1. Student throwing ball on platform URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-1-revised/593f2eb3-f7ed-474c-a228-306e13a63ffd_fig1.jpg Placement: after-background Figure 2: Type: graph_axes Caption: Figure 2. Ball vertical velocity vs. time URL: https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-1-revised/593f2eb3-f7ed-474c-a228-306e13a63ffd_fig2.jpg Placement: before-part Part Letter: A Part A (7 points): Context: Consider the motion of the ball as observed by a stationary observer standing on the ground. i. 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. (2 pt) Verb: sketch ii. 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. (2 pt) Verb: derive iii. 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. (3 pt) Verb: calculate Part B (3 points): Prompt: 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. ______ Greater than ______ Less than ______ Equal to Justify your response using principles of reference frames and vector addition. Context: 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. Verb: indicate and justify Image URLs (2): - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-1-revised/593f2eb3-f7ed-474c-a228-306e13a63ffd_fig1.jpg - https://kcbwzropdukwhmmvdylp.supabase.co/storage/v1/object/public/frq_stimulus/ap-physics-1-revised/593f2eb3-f7ed-474c-a228-306e13a63ffd_fig2.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 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. ## Related - [Mathematical Routines FRQ exam guide](/ap-physics-1-revised/ap-physics-1-exam/ap-physics-1-ap-phys-1-frq-mathematical-routines/study-guide/ap-physics-1-ap-phys-1-frq-mathematical-routines) - [AP Physics 1 Representations FRQ practice](/ap-physics-1-revised/frqs/translation-between-representations) - [AP Physics 1 Experimental Design FRQ practice](/ap-physics-1-revised/frqs/experimental-design) - [AP Physics 1 Qualitative/Quantitative FRQ practice](/ap-physics-1-revised/frqs/qualitative-quantitative-translation) - [AP Physics 1 FRQs](/ap-physics-1-revised/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`)