AP Physics 1 Review

Kinematics

• v = v_0 + at: Final velocity equals initial velocity plus acceleration times time.
• v^2 = v0^2 + 2a(x - x0): Final velocity squared equals initial velocity squared plus two times acceleration times the change in position.

Mechanics and Fluids

Variables and Symbols

• a = acceleration
• A = amplitude or area
• d = distance
• E = energy
• f = frequency
• F = force
• h = height
• I = rotational inertia
• J = impulse
• k = spring constant
• K = kinetic energy
• l = length
• L = angular momentum
• m = mass
• M = mass
• p = momentum
• P = power
• r = radius, distance, or position
• t = time
• T = period
• U = potential energy
• v = velocity or speed
• V = volume
• W = work
• x = position
• y = height
• $\theta$ = angle
• $\alpha$ = angular acceleration
• $\rho$ = density
• $\tau$ = torque
• $\omega$ = angular speed
• $\mu$ = coefficient of friction

Equations

• K = \frac{1}{2}mv^2: Kinetic energy equals one-half times mass times velocity squared.
• W = Fd = Fd \cos\theta: Work equals force times distance, also expressed as force times distance times the cosine of the angle between them.
• \Delta K = \Sigma W = \Sigma F_id: Change in kinetic energy equals the sum of work done, which equals the sum of the force times distance.
• U_s = \frac{1}{2}k(\Delta x)^2: Potential energy of a spring equals one-half times the spring constant times the change in displacement squared.
• UG = -\frac{Gm1m_2}{r}: Gravitational potential energy equals the negative of the gravitational constant times the product of two masses divided by the distance between them.
• \Delta U = mg\Delta y: Change in gravitational potential energy equals mass times the acceleration due to gravity times the change in height.
• P_{avg} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}: Average power equals work divided by change in time, which equals the change in energy divided by change in time.
• P_{inst} = Fv = Fv \cos\theta: Instantaneous power equals force times velocity, also expressed as force times velocity times the cosine of the angle between them.
• p = mv: Momentum equals mass times velocity.

Rotational Motion

• \omega = \omega_0 + \alpha t: Final angular velocity equals initial angular velocity plus angular acceleration times time.
• \theta = \theta0 + \omega0 t + \frac{1}{2}\alpha t^2: Angular displacement equals initial angular displacement plus initial angular velocity times time plus one-half times angular acceleration times time squared.
• \omega^2 = \omega0^2 + 2\alpha(\theta - \theta0): Final angular velocity squared equals initial angular velocity squared plus two times angular acceleration times the change in angular displacement.
• v = r\omega: Tangential velocity equals radius times angular velocity.
• a = r\alpha: Tangential acceleration equals radius times angular acceleration.
• \tau = rF = rF \sin\theta: Torque equals radius times force, also expressed as radius times force times the sine of the angle between them.
• I = \Sigma mr^2: Rotational inertia equals the sum of mass times radius squared.
• I' = I_{cm} + Md^2: Parallel axis theorem: Rotational inertia about a new axis equals the rotational inertia about the center of mass plus mass times the distance between the axes squared.
• \Sigma \tau = I\alpha: Net torque equals rotational inertia times angular acceleration.
• K = \frac{1}{2}I\omega^2: Rotational kinetic energy equals one-half times rotational inertia times angular velocity squared.
• W = \tau \Delta \theta: Work equals torque times the change in angular displacement.
• L = I\omega: Angular momentum equals rotational inertia times angular velocity.
• L = rmv \sin\theta: Angular momentum equals radius times mass times velocity times the sine of the angle between them.
• \Delta L = \tau \Delta t: Change in angular momentum equals torque times the change in time.
• \Delta x = r \Delta \theta: Arc length equals radius times the change in angle.

Simple Harmonic Motion

• T = \frac{1}{f}: Period equals one divided by frequency.
• T = 2\pi\sqrt{\frac{m}{k}}: Period of a spring-mass system equals two pi times the square root of mass divided by spring constant.
• T = 2\pi\sqrt{\frac{l}{g}}: Period of a pendulum equals two pi times the square root of length divided by the acceleration due to gravity.
• x = A \cos(2\pi ft): Position as a function of time for cosine.
• x = A \sin(2\pi ft): Position as a function of time for sine.

Fluids

• \rho = \frac{m}{V}: Density equals mass divided by volume.
• P = \frac{F}{A}: Pressure equals force divided by area.
• \Delta P = -B\frac{\Delta V}{V}: Change in pressure, bulk modulus times the fractional change in volume.
• J = F \Delta t = \Delta p: Impulse equals force times the change in time equals the change in momentum.
• P = P_0 + \rho gh: Pressure at a depth h equals the surface pressure plus density times the acceleration due to gravity times depth.
• P_{gauge} = \rho gh: Gauge pressure equals density times the acceleration due to gravity times depth.
• F_b = \rho Vg: Buoyant force equals density of the fluid times the volume of the displaced fluid times the acceleration due to gravity.
• A1v1 = A2v2: Equation of continuity: Area times velocity is constant.
• P1 + \rho gy1 + \frac{1}{2}\rho v1^2 = P2 + \rho gy2 + \frac{1}{2}\rho v2^2: Bernoulli's equation: Pressure plus density times the acceleration due to gravity times height plus one-half times density times velocity squared is constant.

Constants and Conversion Factors

• Universal gravitational constant: G = 6.67 \times 10^{-11} \frac{m^3}{kg \cdot s^2} = 6.67 \times 10^{-11} \frac{N \cdot m^2}{kg^2}
• 1 atmosphere of pressure: 1 \text{ atm} = 1.0 \times 10^5 \frac{N}{m^2} = 1.0 \times 10^5 \text{ Pa}
• Acceleration due to gravity at Earth's surface: g = 9.8 \frac{m}{s^2}
• Magnitude of the gravitational field strength at the Earth's surface: g = 9.8 \frac{N}{kg}

Prefixes

• tera: T, 10^{12}
• giga: G, 10^9
• mega: M, 10^6
• kilo: k, 10^3
• centi: c, 10^{-2}
• milli: m, 10^{-3}
• micro: $\mu$, 10^{-6}
• nano: n, 10^{-9}
• pico: p, 10^{-12}

Unit Symbols

• hertz: Hz
• newton: N
• joule: J
• pascal: Pa
• kilogram: kg
• second: s
• meter: m
• watt: W

Common Angles

Conventions

• The frame of reference of any problem is assumed to be inertial unless otherwise stated.
• Air resistance is assumed to be negligible unless otherwise stated.
• Springs and strings are assumed to be ideal unless otherwise stated.
• Fluids are assumed to be ideal, and pipes are assumed to be completely filled by fluid, unless otherwise stated.

Geometry and Trigonometry

Rectangle

• A = area
• A = bh (area equals base times height)

Rectangular Solid

• V = volume
• V = lwh (volume equals length times width times height)

Right Triangle

• a^2 + b^2 = c^2 (Pythagorean theorem)

Triangle

• A = area
• A = \frac{1}{2}bh

Cylinder

• V = volume
• V = \pi r^2 l
• S = surface area
• S = 2\pi rl + 2\pi r^2

Circle

• A = area
• C = circumference
• A = \pi r^2
• C = 2\pi r

Sphere

• V = volume
• S = surface area
• V = \frac{4}{3}\pi r^3
• S = 4\pi r^2

Trigonometry

• s = arc length
• s = r\theta
• \sin \theta = \frac{a}{c}
• \cos \theta = \frac{b}{c}
• \tan \theta = \frac{a}{b}