AP Physics C: Mechanics cheat sheet

Average velocity is Δx/Δt; instantaneous velocity is dx/dt, the slope of the position–time graph; instantaneous acceleration is dv/dt, the slope of the velocity–time graph.

For constant acceleration: v = v0 + at, x = x0 + v0t + ½at², and v² = v0² + 2a(x − x0); these apply only when a is constant.

The area under a velocity–time graph equals displacement, and the area under an acceleration–time graph equals change in velocity; signed areas count.

When acceleration depends on position, use a = v dv/dx to relate speed and position; when it depends on time, integrate a(t) to get velocity.

Projectile motion separates into constant-velocity horizontal motion and constant-acceleration vertical motion with a_y = −g; the components share only the time.

Relative velocity in one dimension is v_AB = v_A − v_B when both are measured in a common inertial frame; accelerations are equal in all inertial frames.

Uniform circular motion has constant speed but changing direction; a_c = v²/r points toward the center and the period is T = 2πr/v.

For motion along a curved path, acceleration has a tangential component dv/dt and a centripetal component v²/r; the velocity vector is tangent to the path.

Newton's second law: ΣF = ma = dp/dt; the first law states that zero net force means zero acceleration, and the third law pairs act on different bodies.

Static friction satisfies f_s ≤ μ_sN and takes the value needed to prevent relative slipping; kinetic friction is f_k = μ_kN and opposes relative sliding.

On a frictionless incline of angle θ, the acceleration down the slope is g sinθ and the normal force is mg cosθ.

An ideal massless string has the same tension throughout; a massless, frictionless pulley changes the direction of tension but not its magnitude.

Hooke's law gives spring force F = −kx, where x is displacement from equilibrium and the minus sign shows the force is restoring.

For an object in circular motion, the radial component of the net force equals mv²/r; centripetal force is a role played by real forces, not an extra force.

A system's center of mass obeys ΣF_ext = M a_cm; internal forces cancel in pairs and cannot change the motion of the center of mass.

Equilibrium requires ΣF = 0; if the net force is constant and nonzero, velocity changes linearly with time.

Work is W = ∫F·dr; for a constant force, W = Fd cosθ, where θ is the angle between force and displacement.

The work–energy theorem states W_net = ΔK = ½mv_f² − ½mv_i² for a particle of constant mass.

Gravity near Earth gives ΔU = mgΔh; an ideal spring gives U = ½kx², with x measured from the spring's equilibrium length.

A force is conservative if the work it does between two points is path-independent; friction is nonconservative and its work depends on path length.

For conservative forces, F_x = −dU/dx; equilibrium occurs where dU/dx = 0, stable if d²U/dx² > 0 and unstable if d²U/dx² < 0.

Mechanical energy K + U is conserved when only conservative forces do work; if nonconservative forces act, W_nc = Δ(K + U).

Power is the rate of doing work, P = dW/dt; for a constant force, P = F·v = Fv cosθ.

The work done by kinetic friction is negative and converts mechanical energy into thermal energy; the amount depends on the path.

Momentum is p = mv; impulse is J = ∫F dt and equals Δp, so a given impulse produces a larger velocity change for a smaller mass.

Total momentum is conserved when the net external impulse is zero; internal forces cancel in pairs and do not change total momentum.

In collisions, momentum is conserved in every direction separately; kinetic energy is conserved only in elastic collisions.

Perfectly inelastic collisions make the bodies stick together; use m1v1 + m2v2 = (m1 + m2)v_f and expect kinetic energy loss.

For a one-dimensional elastic collision, the relative velocity reverses: v1' − v2' = −(v1 − v2), in addition to momentum conservation.

The center of mass is x_cm = Σm_i x_i / Σm_i; its velocity is v_cm = p_total/M, and F_ext = M a_cm.

In an explosion, total momentum stays constant while internal energy can increase the total kinetic energy.

For a rocket, thrust equals v_rel(dm/dt) in the ideal case, where v_rel is the exhaust speed relative to the rocket.

Torque about a point is τ = r × F, with magnitude rF sinθ; only the force component perpendicular to r produces torque.

Rotational inertia is I = Σm_i r_i²; for a uniform rod about its center I = ML²/12, a disk about its center I = ½MR², and a solid sphere I = ⅖MR².

Newton's second law for rotation about a fixed axis is Στ = Iα, where τ and α are signed according to a chosen rotation direction.

The parallel-axis theorem is I = I_cm + Md², where d is the distance between the parallel axes.

Angular kinematics for constant α mirror linear kinematics: ω = ω0 + αt, θ = θ0 + ω0t + ½αt², and ω² = ω0² + 2α(θ − θ0).

A rigid body in static equilibrium requires both ΣF = 0 and Στ = 0 about any chosen axis.

Rolling without slipping requires v = Rω and a = Rα; static friction enforces this constraint and does no work in ideal rolling.

Angular momentum about a fixed axis is L = Iω, and Στ = dL/dt; a net external torque is required to change L.

A rigid body rotating about a fixed axis has rotational kinetic energy K = ½Iω².

Total kinetic energy of a rolling body is K = ½Mv_cm² + ½I_cmω²; with rolling without slipping, substitute ω = v_cm/R.

Work done by a torque is W = ∫τ dθ, and instantaneous rotational power is P = τω.

Angular impulse equals change in angular momentum: ∫τ dt = ΔL.

If the net external torque is zero, total angular momentum is conserved; for a rigid body changing shape, I1ω1 = I2ω2.

For a particle, angular momentum is L = r × p, with magnitude mvr sinθ; it depends on the chosen origin.

A body rolling from rest down an incline has acceleration a = g sinθ / (1 + I_cm/(MR²)), so smaller I_cm/(MR²) gives greater acceleration.

In a rotational collision with zero external torque, angular momentum is conserved, but kinetic energy may be lost to internal friction or deformation.

Simple harmonic motion requires a restoring acceleration proportional to displacement and opposite it: a = −ω²x.

Position, velocity, and acceleration in SHM are x = A cos(ωt+φ), v = −Aω sin(ωt+φ), and a = −Aω² cos(ωt+φ).

For SHM, ω = 2πf = 2π/T; maximum speed is Aω at equilibrium and maximum acceleration magnitude is Aω² at the endpoints.

A mass on an ideal spring has ω = √(k/m) and T = 2π√(m/k); the period does not depend on amplitude.

A simple pendulum has T = 2π√(L/g) for small angular amplitudes; the period is independent of the bob mass.

A physical pendulum has T = 2π√(I/(mgd)), where I is rotational inertia about the pivot and d is the pivot-to-center-of-mass distance.

Total energy of an undamped SHM oscillator is E = ½kA² = ½mv² + ½kx², so speed at displacement x is v = ±√(k/m)√(A² − x²).

With damping, underdamped motion oscillates with decreasing amplitude, critical damping returns to equilibrium fastest without oscillating, and overdamping returns slowly without oscillating.