AP Physics Formulas Cheat Sheet with Explanation

Physics can sometimes feel overwhelming, but having a well-organized cheat sheet can make a huge difference. Below, you'll find a collection of essential formulas grouped by topic to help you tackle AP Physics problems. Short explanations are included to guide you in how to use each formula.

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Kinematics Equations

$$v_x=v_{x0}+a_{x}t$$

Use this to find final velocity when you know initial velocity, acceleration, and time.

$$x=x_0+v_{x0}t+\frac{1}{2}{a}_x{t}^2$$

This equation helps find displacement, given initial velocity, acceleration, and time.

$$v_x^2=v_{x0}^2+2a_x(x-x_0)$$

Use this when you need to find velocity without knowing time but knowing displacement and acceleration.

Center of Mass

$${\overrightarrow{x}}_{\mathrm{cm}}=\frac{\sum m_i{\overrightarrow{x}}_i}{\sum m_i}$$

Calculates the center of mass of a system with multiple masses.

Newton's Second Law

$${\overrightarrow{a}}_{\mathrm{sys}}=\frac{\sum {\overrightarrow{F}}_{\mathrm{sys}}}{m_{\mathrm{sys}}}=\frac{{\overrightarrow{F}}_{\mathrm{net}}}{m_{\mathrm{sys}}}$$

Relates the net force on a system to its acceleration.

Gravitational Force

$$|{\overrightarrow{F}}_g|=G\frac{m_1{m}_2}{r^2}$$

Newton's law of universal gravitation. Use this to find the gravitational force between two masses.

Friction Force

$$|{\overrightarrow{F}}_f|\le |\mu {\overrightarrow{F}}_n|$$

This gives the maximum static or kinetic friction force. Useful for solving friction problems.

Spring Force

$${\overrightarrow{F}}_s=-k\mathrm{\Delta }\overrightarrow{x}$$

Use this to find the restoring force in a spring (Hooke's Law).

Centripetal Acceleration

$$a_c=\frac{v^2}{r}$$

This calculates the centripetal acceleration needed to keep an object moving in a circle.

Kinetic Energy

$$K=\frac{1}{2}m{v}^2$$

The kinetic energy of an object based on its mass and velocity.

Work

$$W=F_{\Vert }d=Fd\cos (\theta )$$

Work done by a force applied over a distance. Use this for forces at angles to motion.

Work-Energy Theorem

$$\mathrm{\Delta }K=\sum W_i=\sum F_{\Vert ,i}{d}_i$$

This relates the change in kinetic energy to the net work done on an object.

Spring Potential Energy

$$\mathrm{\Delta }{U}_s=\frac{1}{2}k(\mathrm{\Delta }x{)}^2$$

The potential energy stored in a compressed or stretched spring.

Gravitational Potential Energy

$$U_G=-\frac{G{m}_1{m}_2}{r}$$

Gravitational potential energy between two masses.

$$\mathrm{\Delta }{U}_g=mg\mathrm{\Delta }y$$

Gravitational potential energy relative to the Earth's surface.

Power

$$P_{\text{avg}}=\frac{W}{\mathrm{\Delta }t}=\frac{\mathrm{\Delta }E}{\mathrm{\Delta }t}$$

Average power, or the rate at which work is done or energy is transferred.

$$P_{\text{inst}}=F_{\Vert }v=Fv\cos (\theta )$$

Instantaneous power, useful when force and velocity are known.

Momentum and Impulse

$$\overrightarrow{p}=m\overrightarrow{v}$$

Momentum of an object, mass times velocity.

$${\overrightarrow{F}}_{\text{net}}=\frac{\mathrm{\Delta }\overrightarrow{p}}{\mathrm{\Delta }t}=m\frac{\mathrm{\Delta }\overrightarrow{v}}{\mathrm{\Delta }t}=m\overrightarrow{a}$$

Newton's second law in terms of momentum.

$$\overrightarrow{J}={\overrightarrow{F}}_{\text{avg}}\mathrm{\Delta }t=\mathrm{\Delta }\overrightarrow{p}$$

Impulse, or the change in momentum caused by a force over time.

$${\overrightarrow{v}}_{\text{cm}}=\frac{\sum {\overrightarrow{p}}_i}{\sum m_i}=\frac{\sum m_i{\overrightarrow{v}}_i}{\sum m_i}$$

Velocity of the center of mass of a system of particles.

Rotational Kinematics

$$\omega ={\omega }_0+\alpha t$$

Angular velocity after a time, given initial angular velocity and angular acceleration.

$$\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$$

Angular displacement given angular acceleration, time, and initial angular velocity.

$${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0)$$

Final angular velocity after an angular displacement, without using time.

$$v=r\omega$$

Linear velocity from angular velocity in a rotating system.

$$a_T=r\alpha$$

Tangential acceleration from angular acceleration.

Torque

$$\tau =r_{\perp }F=rF\sin (\theta )$$

Torque, or the rotational force applied at a radius.

Moment of Inertia

$$I=\sum m_i{r}_i^2$$

Moment of inertia for point masses in a system.

$$I^{\prime }=I_{\text{cm}}+M{d}^2$$

Parallel axis theorem, relating moment of inertia about a center of mass to another axis.

Rotational Acceleration (System-Wide)

$${\alpha }_{\text{sys}}=\frac{\mathrm{\Sigma }\tau }{I_{\text{sys}}}=\frac{{\tau }_{\text{net}}}{I_{\text{sys}}}$$

System-wide rotational acceleration, from net torque and moment of inertia.

Rotational Power and Work

$$K=\frac{1}{2}I{\omega }^2$$

Kinetic energy of a rotating object.

$$W=\tau \mathrm{\Delta }\theta$$

Work done in rotational motion, using torque and angular displacement.

Angular Momentum

$$L=I\omega$$

Angular momentum from moment of inertia and angular velocity.

$$L=rmv\sin \theta$$

Angular momentum of a point mass moving in a straight line.

$$\mathrm{\Delta }L=\tau \mathrm{\Delta }t$$

Change in angular momentum due to torque over time (impulse).

$$\mathrm{\Delta }{x}_{\text{cm}}=r\mathrm{\Delta }\theta$$

Linear displacement of the center of mass in rotational motion.

Oscillations

$$T=\frac{1}{f}$$

Period of oscillation, inverse of frequency.

$$T_s=2\pi \sqrt{\frac{m}{k}}$$

Period of a mass-spring system.

$$T_p=2\pi \sqrt{\frac{\ell }{g}}$$

Period of a simple pendulum.

$$x=A\cos (2\pi ft)$$

Position of an oscillating object at time t, using cosine.

$$x=A\sin (2\pi ft)$$

Position of an oscillating object at time t, using sine.

Fluid Mechanics

$$\rho =\frac{m}{V}$$

Density, mass per unit volume.

$$P=\frac{F_{\perp }}{A}$$

Pressure, force applied perpendicularly per unit area.

$$P=P_0+\rho gh$$

Absolute pressure at a depth in a fluid.

$$P_{\text{gauge}}=\rho gh$$

Gauge pressure, pressure above atmospheric pressure.

$$F_b=\rho Vg$$

Buoyant force on an object submerged in a fluid.

Continuity Equation

$$A_1{v}_1=A_2{v}_2$$

Flow rate of an incompressible fluid through two sections of a pipe.

Bernoulli's Equation

$$P_1+\rho g{y}_1+\frac{1}{2}\rho v_1^2=P_2+\rho g{y}_2+\frac{1}{2}\rho v_2^2$$

Bernoulli's equation, relating pressure, velocity, and height in a moving fluid.

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