Physics C: Mech
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AP Physics C: Mechanics — Equations & Constants

7 sections · 43 entries · print it and keep it beside your practice sets

The calculus-based mechanics sheet. Notice how many entries are derivatives or integrals: the exam expects you to reach for the general definition (a = dv/dt, W = ∫F·dr) whenever a quantity is not constant, and to fall back on the constant-acceleration shortcuts only when they apply.

Practise with the sheet, not from memory. The College Board hands out its own version of this page on exam day, so nothing here is worth memorising for its own sake. What earns points is speed: knowing which section a quantity lives in, and reading off the right line without breaking your train of thought. Keep this open (or printed) for every practice set you do.

Kinematics

v, a
v = dx/dt, a = dv/dt = d²x/dt²

General definitions — always valid

Δx, Δv
Δx = ∫ v dt, Δv = ∫ a dt

Integrating back up

v(t)
v = v₀ + a t

Constant acceleration only

x(t)
x = x₀ + v₀ t + ½ a t²

Constant acceleration only

v²
v² = v₀² + 2a(x − x₀)

Constant acceleration, time eliminated

Newton’s laws, momentum, impulse

F_net
F_net = dp/dt = m a (constant m)

Second law in its general form

p
p = m v

Linear momentum

J
J = ∫ F dt = Δp

Impulse

F_f
|F_f| ≤ μ |F_N|

Friction

x_cm
x_cm = (Σ mᵢ xᵢ)/(Σ mᵢ) = (1/M) ∫ x dm

Centre of mass of a system

v_cm
p_total = M v_cm

Momentum of the system

Work, energy, power

W
W = ∫ F · dr

Work done by a variable force along a path

K
K = ½ m v², W_net = ΔK

Kinetic energy; work–energy theorem

F(x)
F_x = −dU/dx

Conservative force from its potential energy

U_s
U_s = ½ k x²

Spring potential energy (F_s = −k x)

ΔU_g
ΔU_g = m g Δh

Gravitational PE near a surface

P
P = dE/dt = F · v

Power, instantaneous and average

Circular motion & rotation

a_c
a_c = v²/r = ω² r

Centripetal acceleration

ω, α
ω = dθ/dt, α = dω/dt

Angular velocity and acceleration

θ(t), ω(t)
θ = θ₀ + ω₀t + ½αt², ω = ω₀ + αt

Constant α only

τ
τ = r × F, |τ| = r F sin θ

Torque as a cross product

τ_net
τ_net = dL/dt = I α

Rotational second law

I
I = Σ mᵢ rᵢ² = ∫ r² dm

Rotational inertia

Parallel axis
I = I_cm + M d²

Shifting the axis a distance d from the centre of mass

L
L = r × p, L = I ω

Angular momentum

K_rot
K = ½ I ω²

Rotational kinetic energy

Rolling
v_cm = ω R, a_cm = α R

Rolling without slipping

Oscillations

x(t)
x = A cos(ω t + φ)

SHM solution of d²x/dt² = −ω²x

T
T = 2π/ω = 1/f

Period and frequency

T_s
T_s = 2π √(m/k)

Mass on a spring

T_p
T_p = 2π √(ℓ/g)

Simple pendulum, small amplitude

T_phys
T = 2π √(I/(m g d))

Physical pendulum about a pivot a distance d from the cm

Gravitation

F_G
F_G = −(G m₁ m₂ / r²) r̂

Newton’s law of gravitation (attractive, along the line of centres)

U_G
U_G = − G m₁ m₂ / r

Gravitational potential energy, zero at infinity

Sign matters. Bound orbits have total energy E = K + U < 0.

v_orbit
v = √(G M / r)

Speed of a circular orbit of radius r

v_esc
v_esc = √(2 G M / r)

Escape speed from radius r

Kepler III
T² = (4π²/(G M)) r³

Period of a circular orbit

Calculus tools you will need

Separable ODE
dv/dt = −(b/m) v ⇒ v = v₀ e^(−bt/m)

Resistive-force problems (e.g. F = −b v)

Terminal speed
m g = b v_t or m g = c v_t²

When drag balances gravity

# Constants & conversions

g
g = 9.8 m/s²

Free-fall acceleration near Earth’s surface

The exam sheet rounds to 9.8; 10 m/s² is fine for estimates but show the value you used.

G
G = 6.67 × 10⁻¹¹ N·m²/kg²

Universal gravitational constant

M_E
M_E = 5.97 × 10²⁴ kg

Mass of Earth

R_E
R_E = 6.37 × 10⁶ m

Radius of Earth

Practise with the sheet openCheatsheet