AP Physics C: Mechanics review
AP Physics C: Mechanics is calculus-based mechanics. Here are the units, the skills that matter and how to review.
On this page
The units
- Kinematics
- Force and Translational Dynamics
- Work, Energy, and Power
- Linear Momentum
- Torque and Rotational Dynamics
- Energy and Momentum of Rotating Systems
- Oscillations
Check the current course and exam description on AP Central for weightings and exam format.
The calculus skills you need
- Derivatives and integrals of motion: v = dx/dt, x = ∫v dt.
- Work from variable forces: W = ∫F dx.
- Moment of inertia by integration: I = ∫r² dm.
- Simple differential equations, such as motion with a resistive force F = −bv.
Common free-response tasks
- Derive an expression in terms of given symbols.
- Sketch a graph of a quantity against time and justify its shape.
- Design or analyse an experiment, often by linearising data.
A review plan
- Start with the equation sheet, so you know what you are given.
- Practise one derivation a day.
- Do past free-response questions and mark them with the scoring guidelines.
- In the final weeks, take full timed practice exams.
Worked examples in the Physics C style
Example 1: motion with air resistance
An object of mass m falls with a resistive force −bv. Find its terminal velocity and describe v(t).
- At terminal velocity the net force is zero: mg = bvT, so vT = mg/b.
- Solving m dv/dt = mg − bv gives v(t) = vT(1 − e−bt/m): a rise toward vT that slows down over time.
Example 2: work from a force-position graph
F(x) = 3x² newtons acts from x = 1 m to x = 2 m. Find the work done.
- W = ∫12 3x² dx = [x³] = 8 − 1 = 7 J.
Example 3: a rolling cylinder
A solid cylinder (I = ½MR²) rolls without slipping down a height h. Find its speed at the bottom.
- Mgh = ½Mv² + ½(½MR²)(v/R)² = ¾Mv², so v = √(4gh/3).
Example 4: simple harmonic motion
A 0.4 kg mass on a spring (k = 160 N/m) oscillates with amplitude 0.05 m. Find ω and the maximum speed.
- ω = √(k/m) = √400 = 20 rad/s. vmax = Aω = 0.05 × 20 = 1 m/s.
Practice questions
Try each one before opening the answer.
Q1. x(t) = 5 sin(2t) m. Find the maximum speed and maximum acceleration.
Show answer
Q2. A thin hoop has I = MR². Find its speed after rolling down height h without slipping.
Show answer
Q3. A 2 kg object has momentum p(t) = 3t² kg·m/s. Find the net force at t = 2 s.
Show answer
Frequently asked questions
How is Physics C Mechanics different from AP Physics 1?
It covers similar topics but uses calculus, includes more derivations, and does not include fluids. See AP Physics 1 vs C.
Can I take the Mechanics exam alone?
Yes. Mechanics and E&M are separate exams with separate scores.
The key ideas in more depth
Calculus as a tool, not a topic
In Physics C, calculus is how you handle quantities that change. If the force, acceleration or velocity is constant, the familiar algebra equations still work. When something varies with time or position, switch to the calculus forms: integrate acceleration to get velocity, integrate force over distance to get work, and differentiate potential energy to get force. Before each problem, ask whether the key quantity is constant.
Differential equations you should recognise
- dv/dt = −kv: exponential decay of speed, for example a resistive force with no other forces.
- m dv/dt = mg − bv: exponential approach to terminal velocity.
- d²x/dt² = −ω²x: simple harmonic motion with angular frequency ω.
You rarely need to solve these from scratch, but you must recognise them, write them down from Newton's second law and know the shape of their solutions.
Centre of mass and systems
The centre of mass of an extended object is found by integration: xcm = (1/M)∫x dm. For a system of particles, the external forces alone determine how the centre of mass accelerates. This lets you analyse explosions and collisions by tracking a single point.
Rotation with calculus
Moments of inertia come from I = ∫r² dm, and the parallel axis theorem shifts them to a new axis. Angular momentum L = Iω is conserved when the net external torque is zero, which explains everything from spinning skaters to collapsing stars.
Exam tips for Mechanics
- Write Newton's second law as an equation before simplifying, especially for variable forces.
- Keep answers in symbols when the question uses symbols; substitute numbers only at the end.
- Check limits: does your answer make sense when a mass becomes very large or a force becomes zero?
- Linearise data in experimental questions so that the slope gives the quantity you want.
More practice
Try each one before opening the answer.
Q1. A rod of length L has linear density λ = kx. Find its centre of mass from x = 0.
Show answer
Q2. A 2 kg object moves with v(t) = 6 − 2t m/s. Find the net force on it.
Show answer
More questions students ask
How many free-response questions are there?
The format is set by the College Board and can change. Check the current course and exam description on AP Central.
Do I need to know calculus before starting?
You should at least be taking calculus at the same time. Derivatives come up within weeks, and integrals soon after.
Which topics are hardest?
Many students find rotation, variable forces and simple harmonic motion hardest. Give them extra practice time.
Extra practice
Try each one before opening the answer.
Q1. A 3 kg object attached to a spring oscillates with period 0.5 s. Find the spring constant.
Show answer
Key takeaways
- Use calculus whenever a key quantity changes with time or position.
- Recognise the standard differential equations for resistive forces and simple harmonic motion.
- Moments of inertia come from I = ∫r² dm and the parallel axis theorem.
- Angular momentum is conserved with no net external torque.
- Keep answers in symbols until the end and check limiting cases.
Quick check
A 2 kg uniform disc of radius 0.5 m (I = ½MR²) spins at 4 rad/s. Find its angular momentum.