AP® Physics C: Mechanics Cheat Sheet

Last Updated: September 23, 2024

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Prepare for AP Physics C: Mechanics with this detailed cheat sheet, covering kinematics, Newton’s laws, energy, momentum, rotation, oscillations, and gravitation. Ideal for quick review before exams or during problem-solving.

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Unit 1: Kinematics

Kinematics in One Dimension:
- Displacement: $\Delta x = x_f - x_i$
- Velocity: $v = \frac{dx}{dt}$
- Acceleration: $a = \frac{dv}{dt}$
- Equations of Motion:
  - $v = v_0 + at$
  - $x = x_0 + v_0t + \frac{1}{2}at^2$
  - $v^2 = v_0^2 + 2a(x - x_0)$
Kinematics in Two Dimensions:
- Position Vector: $\vec{r} = x\hat{i} + y\hat{j}$
- Velocity Vector: $\vec{v} = \frac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j}$
- Projectile Motion:
  - $x(t) = v_{0x} t$
  - $y(t) = v_{0y} t - \frac{1}{2} g t^2$
- Relative Velocity: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

Unit 2: Newton’s Laws of Motion

Newton’s First and Second Law:
- First Law (Inertia): $\vec{F}_{\text{net}} = 0 \Rightarrow \text{constant velocity}$
- Second Law: $\vec{F}_{\text{net}} = m\vec{a}$
- Weight: W = mg
Circular Motion:
- Centripetal Acceleration: $a_c = \frac{v^2}{r}$
- Centripetal Force: $F_c = m \frac{v^2}{r}$
- Uniform Circular Motion: Constant speed, changing velocity direction.
Newton’s Third Law:
- Action-Reaction: For every action, there is an equal and opposite reaction, $\vec{F}_{AB} = -\vec{F}_{BA}$ .

Unit 3: Work, Energy, and Power

Work-Energy Theorem:
- $W = \Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$
Forces and Potential Energy:
- Gravitational Potential Energy: $U_g = mgh$
- Elastic Potential Energy: $U_s = \frac{1}{2} kx^2$
- Conservative Forces: $\vec{F} = -\frac{dU}{dx}$
Conservation of Energy:
- Mechanical Energy: E = K + U
- Conservation: $E_i = E_f$ (No non-conservative forces)
Power:
- Power: $P = \frac{dW}{dt} = Fv$
- Units: Watts (W)

Unit 4: Systems of Particles and Linear Momentum

Center of Mass:
- Center of Mass (Discrete Particles): $\vec{r}_{\text{cm}} = \frac{1}{M} \sum m_i \vec{r}_i$
Impulse and Momentum:
- Momentum: $\vec{p} = m\vec{v}$
- Impulse: $\vec{J} = \Delta \vec{p} = \int \vec{F} dt$
- Impulse-Momentum Theorem: $\vec{J} = \Delta \vec{p}$
Conservation of Linear Momentum:
- Elastic Collisions: $\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}, K_{\text{initial}} = K_{\text{final}}$
- Inelastic Collisions: $\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$ , $K_{\text{initial}} > K_{\text{final}}$

Unit 5: Rotation

Torque and Rotational Statics:
- Torque: $\tau = rF \sin \theta$
- Rotational Equilibrium: $\sum \tau = 0$
Rotational Kinematics:
- Angular Displacement: $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
- Angular Velocity: $\omega = \omega_0 + \alpha t$
- Angular Acceleration: $\alpha = \frac{d\omega}{dt}$
Rotational Dynamics and Energy:
- Rotational Inertia: $I = \sum m_i r_i^2$
- Newton’s Second Law for Rotation: $\tau = I\alpha$
- Rotational Kinetic Energy: $K = \frac{1}{2} I \omega^2$
Angular Momentum and Its Conservation:
- Angular Momentum: $L = I \omega$
- Conservation of Angular Momentum: $\vec{L}_{\text{initial}} = \vec{L}_{\text{final}}$

Unit 6: Oscillations

Simple Harmonic Motion (SHM):
- Position: $x(t) = A \cos(\omega t + \phi)$
- Velocity: $v(t) = -A \omega \sin(\omega t + \phi)$
- Acceleration: $a(t) = -A \omega^2 \cos(\omega t + \phi)$
Springs:
- Hooke’s Law: $F_s = -kx$
- Spring Constant: k, $\omega = \sqrt{\frac{k}{m}}$
Pendulums:
- Simple Pendulum: $T = 2\pi \sqrt{\frac{L}{g}}$
- Angular Frequency: $\omega = \sqrt{\frac{g}{L}}$

Unit 7: Gravitation

Gravitational Forces:
- Newton’s Law of Universal Gravitation: $F_g = G \frac{m_1 m_2}{r^2}$
- $G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$
Gravitational Potential Energy:
- $U_g = -G \frac{m_1 m_2}{r}$
Orbits of Planets and Satellites:
- Orbital Speed: $v = \sqrt{\frac{GM}{r}}$
- Orbital Period: $T = 2\pi \sqrt{\frac{r^3}{GM}}$ (Kepler’s Third Law)

FRQ Tips

Start with Free-Body Diagrams: Helps visualize forces and motion.
Use Energy Methods: Especially for conservative systems.
Check Units: Ensure consistency across all calculations.
Apply Conservation Laws: Momentum, energy, and angular momentum are key.
Explain All Steps: Clear reasoning can earn partial credit.