AP Physics 1 Review

Kinematics

• $v = v_0 + at$: Final velocity equals initial velocity plus acceleration times time.
• $v^2 = v<em>0^2 + 2a(x - x</em>0)$: 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.
• $U<em>G = -\frac{Gm</em>1m_2}{r}$: 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.
• $\theta = \theta<em>0 + \omega</em>0 t + \frac{1}{2}\alpha t^2$: Angular displacement equals initial angular displacement plus initial angular velocity times time plus one-half times angular acceleration times time squared.
• $\omega^2 = \omega<em>0^2 + 2\alpha(\theta - \theta</em>0)$: 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.
• $A<em>1v</em>1 = A<em>2v</em>2$: Equation of continuity: Area times velocity is constant.
• $P<em>1 + \rho gy</em>1 + \frac{1}{2}\rho v<em>1^2 = P</em>2 + \rho gy<em>2 + \frac{1}{2}\rho v</em>2^2$: 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
• $s = r\theta$
• $\sin \theta = \frac{a}{c}$
• $\cos \theta = \frac{b}{c}$
• $\tan \theta = \frac{a}{b}$