Work

$W = F\,d\cos\theta \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

$W = \vec{F} \cdot \vec{d} \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

$W = \int_{x_1}^{x_2} F(x)\, dx \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

Kinetic Energy

$KE = \frac{1}{2}mv^2 \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

$KE = \frac{p^2}{2m} \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

Potential Energy

$PE_g = mgh \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

$PE_s = \frac{1}{2}kx^2 \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

Work-Energy Theorem

$W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

Energy Conservation

$KE + PE = E = \text{constant (no non-conservative forces)} \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

$\Delta E_{\text{mech}} = W_{\text{friction}} = -f_k d \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

Power

$P = \frac{W}{t} \quad [\text{M}^1\text{L}^2\text{T}^{-3}]\ [\text{SI: W}]$

$P = Fv\cos\theta \quad [\text{M}^1\text{L}^2\text{T}^{-3}]\ [\text{SI: W}]$

$P = \vec{F} \cdot \vec{v} \quad [\text{M}^1\text{L}^2\text{T}^{-3}]\ [\text{SI: W}]$

$1\ \text{hp} = 746\ \text{W}$

$1\ \text{kWh} = 3.6 \times 10^6\ \text{J}\ [\text{SI: J}]$

Vertical Circular Motion (String)

$v_{\text{top,min}} = \sqrt{gR} \quad [\text{L}^1\text{T}^{-1}]\ [\text{SI: m/s}]$

$v_{\text{bottom,min}} = \sqrt{5gR} \quad [\text{L}^1\text{T}^{-1}]\ [\text{SI: m/s}]$

$v_{\text{side,min}} = \sqrt{3gR} \quad [\text{L}^1\text{T}^{-1}]\ [\text{SI: m/s}]$

$T_{\text{bottom}} = \frac{mv^2}{R} + mg \quad [\text{M}^1\text{L}^1\text{T}^{-2}]\ [\text{SI: N}]$

$T_{\text{top}} = \frac{mv^2}{R} - mg \quad [\text{M}^1\text{L}^1\text{T}^{-2}]\ [\text{SI: N}]$

$T_{\text{bottom}} - T_{\text{top}} = 6mg \quad [\text{M}^1\text{L}^1\text{T}^{-2}]\ [\text{SI: N}]$

$v_b^2 - v_t^2 = 4gR \quad [\text{L}^2\text{T}^{-2}]\ [\text{SI: m}^2/\text{s}^2]$

Vertical Circular Motion (Rod)

$v_{\text{top,min}} = 0 \quad [\text{SI: m/s}]$

$v_{\text{bottom,min}} = \sqrt{4gR} = 2\sqrt{gR} \quad [\text{L}^1\text{T}^{-1}]\ [\text{SI: m/s}]$

Elastic Collision

$v_1 = \frac{(m_1 - m_2)u_1 + 2m_2 u_2}{m_1 + m_2} \quad [\text{L}^1\text{T}^{-1}]\ [\text{SI: m/s}]$

$v_2 = \frac{(m_2 - m_1)u_2 + 2m_1 u_1}{m_1 + m_2} \quad [\text{L}^1\text{T}^{-1}]\ [\text{SI: m/s}]$

$e = \frac{v_2 - v_1}{u_1 - u_2} = 1 \quad \text{(elastic)}$

Perfectly Inelastic Collision

$v_f = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2} \quad [\text{L}^1\text{T}^{-1}]\ [\text{SI: m/s}]$

$\Delta KE_{\text{loss}} = \frac{m_1 m_2 (u_1 - u_2)^2}{2(m_1 + m_2)} = \frac{1}{2}\mu\, u_{\text{rel}}^2 \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ [\text{SI: J}]$

$e = 0 \quad \text{(perfectly inelastic)}$

Bouncing Ball

$e = \sqrt{\frac{h}{H}} \quad \text{(dimensionless)}$

where H = drop height, h = bounce height.

Spring Constant

$k = \frac{F}{x} \quad [\text{M}^1\text{L}^0\text{T}^{-2}]\ [\text{SI: N/m}]$