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Kinematics

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Velocity \(v = v_0 + at\)
Position \(x = x_0 + v_0 t + \tfrac{1}{2}at^2\)
No-time \(v^2 = v_0^2 + 2a\Delta x\)
Average \(\Delta x = \tfrac{1}{2}(v_0 + v)t\)
Horizontal component \(v_x = v_0\cos\theta\)
Vertical component \(v_y = v_0\sin\theta - gt\)
Projectile range \(R = \dfrac{v_0^2\sin 2\theta}{g}\)
Gravity \(g = 9.8\text{ m/s}^2\)

Force & Dynamics

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Newton's 2nd law \(\sum \vec{F} = m\vec{a}\)
Newton's 3rd law \(\vec{F}_{A \text{ on } B} = -\vec{F}_{B \text{ on } A}\)
Weight \(F_g = mg\)
Elevator scale \(F_n = m(g + a)\)
Center of mass \(x_{\text{cm}} = \dfrac{\sum m_i x_i}{\sum m_i}\)

Contact Forces & Gravity

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Gravity parallel (incline) \(F_{g\parallel} = mg\sin\theta\)
Gravity perpendicular \(F_{g\perp} = mg\cos\theta\)
Incline acceleration \(a = g(\sin\theta - \mu_k\cos\theta)\)
Static friction (max) \(f_{s,\max} = \mu_s F_n\)
Kinetic friction \(f_k = \mu_k F_n\)
Hooke's law \(F_s = -k\Delta x\)
Spring equilibrium \(\Delta x = \dfrac{mg}{k}\)
Universal gravitation \(F_g = G\dfrac{m_1 m_2}{r^2}\)
Gravitational field \(g = \dfrac{GM}{r^2}\)

Circular Motion & Orbits

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Centripetal acceleration \(a_c = \dfrac{v^2}{r}\)
Centripetal force \(\sum F_{\text{rad}} = \dfrac{mv^2}{r}\)
Min speed (top of loop) \(v_{\min} = \sqrt{gr}\)
Normal force (top of loop) \(F_n = m\!\left(\dfrac{v^2}{r} - g\right)\)
Weightless speed (hilltop) \(v = \sqrt{gr}\)
Period \(T = \dfrac{2\pi r}{v}\)
Frequency \(f = \dfrac{1}{T}\)
Orbital speed \(v = \sqrt{\dfrac{GM}{r}}\)
Kepler's 3rd law \(T^2 = \dfrac{4\pi^2}{GM}\,r^3\)

Work & Energy

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Work \(W = Fd\cos\theta\)
Kinetic energy \(K = \tfrac{1}{2}mv^2\)
Work-energy theorem \(W_{\text{net}} = \Delta K\)
Gravitational PE (near Earth) \(\Delta U_g = mg\Delta y\)
Gravitational PE (general) \(U_g = -G\dfrac{m_1 m_2}{r}\)
Elastic PE \(U_s = \tfrac{1}{2}k(\Delta x)^2\)
Mechanical energy \(E_{\text{mech}} = K + U\)
Energy conservation \(W_{\text{ext}} = \Delta E_{\text{mech}} + \Delta E_{\text{th}}\)
Average power \(P_{\text{avg}} = \dfrac{W}{\Delta t}\)
Instantaneous power \(P = Fv\cos\theta\)

Impulse & Momentum

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Momentum \(\vec{p} = m\vec{v}\)
Newton's 2nd law (momentum form) \(\vec{F}_{\text{net}} = \dfrac{\Delta \vec{p}}{\Delta t}\)
Impulse \(\vec{J} = \vec{F}\,\Delta t\)
Impulse-momentum theorem \(\vec{J}_{\text{net}} = \Delta \vec{p}\)
Impulse from \(F\) vs. \(t\) graph \(J = \text{area under the curve} = \Delta p\)
Average force \(F_{\text{avg}} = \dfrac{\Delta p}{\Delta t}\)
Conservation of momentum \(\sum \vec{p}_i = \sum \vec{p}_f\)
Open system (momentum transfer) \(\vec{F}_{\text{ext,net}}\,\Delta t = \Delta \vec{p}_{\text{sys}}\)
Center of mass velocity \(\vec{v}_{\text{cm}} = \dfrac{\vec{p}_{\text{tot}}}{M}\)

Collisions

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Perfectly inelastic collision \(\vec{v}_f = \dfrac{m_1\vec{v}_{1i} + m_2\vec{v}_{2i}}{m_1 + m_2}\)
Elastic 1D (target at rest) \(v_{1f} = \dfrac{m_1 - m_2}{m_1 + m_2}v_{1i}\), \(\ v_{2f} = \dfrac{2m_1}{m_1 + m_2}v_{1i}\)
Elastic relative-speed test \(v_{1i} - v_{2i} = -(v_{1f} - v_{2f})\)
Lorentz factor \(\gamma = \dfrac{1}{\sqrt{1 - v^2/c^2}}\)
Relativistic momentum \(\vec{p} = \gamma m\vec{v}\)
Relativistic energy \(E^2 = p^2c^2 + m^2c^4\)
Mass energy (at rest) \(E = mc^2\)
Massless particle (photon) \(E = pc \implies p = E/c\)

Rotational Kinematics

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Average angular velocity \(\omega_{\text{avg}} = \dfrac{\Delta\theta}{\Delta t}\)
Average angular acceleration \(\alpha_{\text{avg}} = \dfrac{\Delta\omega}{\Delta t}\)
Radian measure \(\theta = \dfrac{s}{r}\), \(\ 1\text{ rev} = 360^\circ = 2\pi\text{ rad}\)
Arc length \(\Delta s = r\,\Delta\theta\)
Linear speed \(v = r\omega\)
Tangential acceleration \(a_{\text{tan}} = r\alpha\)
Kinematic eq. (no \(\theta\)) \(\omega = \omega_0 + \alpha t\)
Kinematic eq. (no \(\omega\)) \(\Delta\theta = \omega_0 t + \tfrac{1}{2}\alpha t^2\)
Kinematic eq. (no \(t\)) \(\omega^2 = \omega_0^2 + 2\alpha\,\Delta\theta\)

Torque & Rotational Dynamics

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Torque \(\tau = rF\sin\theta\)
Lever arm \(r_\perp = r\sin\theta\), so \(\tau = r_\perp F\)
Rotational inertia (definition) \(I = \sum mr^2\)
Hoop / point mass \(I = MR^2\)
Solid disk or cylinder \(I = \tfrac{1}{2}MR^2\)
Solid sphere \(I = \tfrac{2}{5}MR^2\)
Thin rod (center / end) \(I = \tfrac{1}{12}ML^2\ /\ \tfrac{1}{3}ML^2\)
Newton's 2nd law (rotation) \(\tau_{\text{net}} = I\alpha\)
Rotational equilibrium \(\tau_{\text{net}} = 0 \implies \alpha = 0\)
Angular momentum (rigid system) \(L = I\omega\)
Angular impulse-momentum theorem \(\Delta\vec{L} = \sum\vec{\tau}\,\Delta t\)

Rotational Energy

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Rotational kinetic energy \(K_{\text{rot}} = \tfrac{1}{2}I\omega^2\)
Total kinetic energy \(K = \tfrac{1}{2}Mv_{\text{cm}}^2 + \tfrac{1}{2}I_{\text{cm}}\omega^2\)
Rolling without slipping \(v_{\text{cm}} = R\omega\), \(a_{\text{cm}} = R\alpha\)
Top / contact point of a rolling wheel \(v_{\text{top}} = 2v_{\text{cm}}\ /\ v_{\text{contact}} = 0\)
Rolling down a ramp (energy) \(Mgh = \tfrac{1}{2}Mv_{\text{cm}}^2 + \tfrac{1}{2}I_{\text{cm}}\omega^2\)
Ramp speed for \(I = cMR^2\) \(v_{\text{cm}} = \sqrt{2gh/(1+c)}\)

Angular Momentum & Orbital Energy

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Angular momentum (point object) \(L = rmv\sin\theta\)
Perpendicular-distance form \(L = r_\perp mv\), \(r_\perp = r\sin\theta\)
Straight-line motion past a point \(L = mvd\), constant
Orbital \(L\) at closest / farthest point \(r_1v_1 = r_2v_2\)
Total mechanical energy (orbit) \(E = \tfrac{1}{2}mv^2 - G\dfrac{mM}{r}\)
Circular orbit energy \(E = -\dfrac{GmM}{2r} = -K = \tfrac{1}{2}U_g\)
Circular orbit kinetic energy \(K = \dfrac{GmM}{2r}\)
Escape velocity \(v_{\text{esc}} = \sqrt{\dfrac{2GM}{r}} = \sqrt{2}\,v_{\text{orbit}}\)

Oscillations

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Condition for SHM \(F_{\text{net}} = -kx\), so \(a = -\omega^2 x\)
Period, frequency, angular frequency \(f = 1/T\), \(\omega = 2\pi f = 2\pi/T\)
Spring-mass oscillator \(T_s = 2\pi\sqrt{\dfrac{m}{k}}\), \(\omega = \sqrt{k/m}\)
Position in SHM \(x(t) = A\cos(\omega t + \varphi)\)
Velocity and acceleration in SHM \(v = -A\omega\sin(\omega t + \varphi)\), \(a = -A\omega^2\cos(\omega t + \varphi)\)
Elastic potential energy \(U_s = \tfrac{1}{2}kx^2\)
Total oscillator energy \(E = \tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2\)
Maximum speed \(v_{\text{max}} = A\omega = A\sqrt{k/m}\)
Pendulum restoring torque \(\tau = -mgL\sin\theta \approx -mgL\theta\)
Simple pendulum period \(T_p = 2\pi\sqrt{\dfrac{L}{g}}\)

Density, Pressure & Buoyancy

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Density \(\rho = \dfrac{m}{V}\)
Pressure \(P = \dfrac{F_\perp}{A}\), in pascals
Atmospheric pressure \(1\text{ atm} = 1.013\times10^{5}\text{ Pa}\)
Pressure with depth \(P = P_0 + \rho g h\)
Gauge pressure \(P_{\text{gauge}} = P - P_0 = \rho g h\)
Buoyant force \(F_b = \rho_{\text{fluid}}V_{\text{disp}}\,g\)
Floating condition \(\dfrac{V_{\text{disp}}}{V_{\text{obj}}} = \dfrac{\rho_{\text{obj}}}{\rho_{\text{fluid}}}\)
Apparent weight \(W_{\text{app}} = W - F_b\)

Fluid Flow & Bernoulli's Principle

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What drives flow \(F_{\text{net}} = (P_1 - P_2)A\)
Volume flow rate \(Q = Av\), in m³/s
Continuity equation \(A_1v_1 = A_2v_2\)
Round pipe \(A = \pi r^2\), so \(v \propto 1/r^2\)
Bernoulli's equation \(P + \rho g y + \tfrac{1}{2}\rho v^2 = \text{constant}\)
Level pipe \(P_1 - P_2 = \tfrac{1}{2}\rho(v_2^2 - v_1^2)\)
Tank with a hole (Torricelli) \(v = \sqrt{2gh}\)