Derivations for Formulas of Energy

Gravitational potential energy

\begin{align*} W &= Fs \qquad \scriptsize W = \Delta E_p, \enspace F = W = mg, \enspace s = \Delta h \\ \\ \Delta E_p &= m g \Delta h \\ \therefore E_p &= mgh \end{align*}

\text{Area under force-extension graph that obeys Hooke's Law} \\ \begin{align*} E_p &= \frac{1}{2} Fx \qquad \scriptsize F=kx \\ \therefore E_p &= \frac{1}{2} kx^2 \end{align*}

\begin{align*} v^2 &= u^2 + 2as \qquad \scriptsize \text{multiply everything by } \frac{1}{2}m \\ \frac{1}{2} m v^2 &= \frac{1}{2} m u^2 + mas \\ \frac{1}{2} m v^2 &= \frac{1}{2} m u^2 + Fs \\ Fs &= \frac{1}{2} m v^2 - \frac{1}{2} m u^2 \qquad \scriptsize Fs = \Delta E_k \\ \\ \underset{\text{change in KE}}{\Delta E_k} &= \underset{\text{final KE}}{\frac{1}{2} m v^2} - \underset{\text{initial KE}}{\frac{1}{2} m u^2} \\ \therefore E_k &= \frac{1}{2} m v^2 \end{align*}