AP Physics Reference Sheet

Units first: convert every quantity to SI (m, kg, s, C) before you substitute, then check that the answer's units make sense.

Constant-acceleration motion

• Average velocity: \(v_{\text{avg}} = \dfrac{\Delta x}{\Delta t}\)
• Acceleration: \(a = \dfrac{\Delta v}{\Delta t}\)
• \(v = v_0 + at\)
• \(\Delta x = v_0 t + \tfrac12 a t^2\)
• \(v^2 = v_0^2 + 2a\,\Delta x\)
• \(\Delta x = \tfrac12 (v_0 + v)\,t\)

Graphs: the slope of a position–time graph is velocity; the slope of a velocity–time graph is acceleration; the area under a velocity–time graph is displacement.

Free fall & projectiles

• Drop from rest: \(t = \sqrt{\dfrac{2h}{g}}\),  \(v = \sqrt{2gh}\)
• Launch components: \(v_x = v_0\cos\theta\),  \(v_y = v_0\sin\theta\)
• Range (level ground): \(R = \dfrac{v_0^2\sin 2\theta}{g}\)
• Maximum height: \(H = \dfrac{(v_0\sin\theta)^2}{2g}\)

Independence: horizontal velocity stays constant; vertical motion has \(a = -g\). Time links the two.

Vectors

• Magnitude: \(R = \sqrt{R_x^2 + R_y^2}\)
• Direction: \(\theta = \tan^{-1}\!\left(\dfrac{R_y}{R_x}\right)\)

Forces & Newton's laws

• Second law: \(\Sigma F = ma\)
• Weight: \(F_g = mg\)
• Kinetic friction: \(f = \mu N\)
• Incline: \(N = mg\cos\theta\);  \(a = g(\sin\theta - \mu\cos\theta)\) sliding down
• Atwood machine, acceleration: \(a = \dfrac{(m_2 - m_1)\,g}{m_1 + m_2}\)
• Atwood machine, rope tension: \(T = \dfrac{2m_1 m_2\,g}{m_1 + m_2}\)

Equilibrium: \(\Sigma F_x = 0\) and \(\Sigma F_y = 0\). A block stays put on an incline when \(\mu_s \ge \tan\theta\).

Uniform circular motion

• Speed from period: \(v = \dfrac{2\pi r}{T}\)
• Centripetal acceleration: \(a_c = \dfrac{v^2}{r}\)
• Centripetal force: \(F_c = \dfrac{mv^2}{r}\)

Remember: centripetal force is not a new force — it is whichever real force (tension, gravity, friction, normal) points toward the center.

Gravitation & orbits

• Universal gravitation: \(F = \dfrac{G m_1 m_2}{r^2}\)
• Circular orbit speed: \(v = \sqrt{\dfrac{GM}{r}}\)
• Orbital period: \(T = 2\pi\sqrt{\dfrac{r^3}{GM}}\)

Set \(\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}\) to derive the orbit speed; \(r\) is measured from the center of the planet.

Rotational kinematics

• Angular velocity: \(\omega = \dfrac{\Delta\theta}{\Delta t}\)
• Angular acceleration: \(\alpha = \dfrac{\Delta\omega}{\Delta t}\)
• Linking to the rim: \(v = r\omega\),  \(a_t = r\alpha\)

Torque & angular momentum

• Torque: \(\tau = rF\sin\theta\)
• Second law for rotation: \(\Sigma\tau = I\alpha\)
• Angular momentum: \(L = I\omega\);  with no external torque \(I_i\omega_i = I_f\omega_f\)
• Balanced beam: \(\Sigma\tau = 0\), so \(F_1 d_1 = F_2 d_2\)

Work, energy & power

• Work: \(W = Fd\cos\theta\)
• Kinetic energy: \(KE = \tfrac12 mv^2\)
• Gravitational PE: \(PE_g = mgh\)
• Spring PE: \(PE_s = \tfrac12 kx^2\)
• Work–energy theorem: \(W_{\text{net}} = \Delta KE\)
• Power: \(P = \dfrac{W}{t} = Fv\)

Conservation (no friction): \(KE_i + PE_i = KE_f + PE_f\). A drop from height \(h\) gives \(v = \sqrt{2gh}\).

Momentum & collisions

• Momentum: \(p = mv\)
• Impulse: \(J = F\Delta t = \Delta p = m(v_f - v_i)\)
• Conservation: \(m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'\)
• Perfectly inelastic (stick together): \(v_f = \dfrac{m_1v_1 + m_2v_2}{m_1 + m_2}\)
• Elastic, 1-D, target at rest — first object: \(v_1' = \dfrac{m_1 - m_2}{m_1 + m_2}\,v_1\)
• Elastic, 1-D, target at rest — target: \(v_2' = \dfrac{2m_1}{m_1 + m_2}\,v_1\)

In 2-D, conserve the \(x\)- and \(y\)-components separately. Momentum is conserved in every collision; kinetic energy only in elastic ones.

Simple harmonic motion

• Hooke's law: \(F = -kx\)
• Mass on a spring: \(T = 2\pi\sqrt{\dfrac{m}{k}}\)
• Simple pendulum (small angles): \(T = 2\pi\sqrt{\dfrac{L}{g}}\)
• Total energy: \(E = \tfrac12 kA^2 = \tfrac12 m v_{\max}^2\)
• Frequency: \(f = \dfrac{1}{T}\)

Linear ↔ rotational twins

Wave basics

• Period: \(T = \dfrac{1}{f}\)
• Wave speed: \(v = f\lambda\), so \(\lambda = \dfrac{v}{f}\)
• Wave on a string: \(v = \sqrt{\dfrac{F_T}{\mu}}\)  (\(F_T\) = tension, \(\mu\) = mass per length)
• Speed of sound vs. temperature: \(v \approx 331 + 0.6\,T\) m/s (\(T\) in °C)

Standing waves & resonance

• String fixed at both ends, or a pipe open at both ends: \(f_n = \dfrac{nv}{2L}\),  \(n = 1, 2, 3, \dots\)
• Pipe closed at one end: \(f_n = \dfrac{nv}{4L}\),  \(n = 1, 3, 5, \dots\) (odd harmonics only)
• Harmonics: \(f_n = n f_1\)

Resonance happens when a driving frequency matches one of these natural frequencies.

Sound intensity & loudness

• Intensity from a point source: \(I = \dfrac{P}{4\pi r^2}\)
• Inverse-square law: \(I_2 = I_1\left(\dfrac{r_1}{r_2}\right)^2\)
• Sound level: \(\beta = 10\log\!\left(\dfrac{I}{I_0}\right)\) dB

Every \(+10\) dB is ten times the intensity.

Beats & the Doppler effect

• Beat frequency: \(f_{\text{beat}} = |f_2 - f_1|\)
• Source approaching a still listener: \(f' = f\,\dfrac{v}{v - v_s}\)
• Source moving away: \(f' = f\,\dfrac{v}{v + v_s}\)

Approaching raises the pitch; receding lowers it. \(v\) is the speed of sound, \(v_s\) the speed of the source.

Charge & electric fields

• Quantized charge: \(q = ne\)
• Coulomb's law: \(F = \dfrac{k\,q_1 q_2}{r^2}\)
• Field definition: \(E = \dfrac{F}{q}\)
• Field of a point charge: \(E = \dfrac{kq}{r^2}\)

Superposition: add the fields from each charge as vectors. Field lines point away from \(+\) charges and toward \(-\) charges.

Current, resistance & power

• Current: \(I = \dfrac{\Delta Q}{\Delta t}\)
• Ohm's law: \(V = IR\)
• Power: \(P = IV = I^2R = \dfrac{V^2}{R}\)
• Energy used: \(E = Pt\)

Resistor combinations

• Series: \(R_{\text{eq}} = R_1 + R_2 + R_3 + \cdots\)  (same current through each)
• Parallel: \(\dfrac{1}{R_{\text{eq}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots\)  (same voltage across each)
• Two in parallel: \(R_{\text{eq}} = \dfrac{R_1R_2}{R_1 + R_2}\)

Kirchhoff's rules: current into a junction equals current out; the voltage changes around any closed loop add to zero.

Capacitors

• Capacitance: \(C = \dfrac{Q}{V}\)
• Stored energy: \(U = \tfrac12 CV^2 = \tfrac12 QV\)

Capacitance is measured in farads; classroom capacitors are usually microfarads, so \(1\ \mu\text{F} = 10^{-6}\ \text{F}\).