Equation Sheet

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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 \(\tau_{\text{net}} = 0 \implies \alpha = 0\)