# Proof 36: Gravitational potential energy

Small falling object due to weight

A flower pot falls down from a window sill due to gravity. This i because gravity does work $W_g$ on it, pulling it down with its gravitational force, called weight $w$. The distance over which this work is done is the height $h$, if we pick the reference (the level from which we measure the height) to be the ground.

\begin{align} W&=Fd\qquad\leftarrow \text{Work formula}\\ W_g& =wh \qquad\leftarrow \text{The distance moved is the height } h\\ ~& =mgh\qquad\leftarrow \text{Weight formula, } w=mg \end{align}

This derivation can be redone for any reference. The work done by gravity is what we tend to call gravitational potential energy, since gravity is a conservative force. So, symbolising it $U_g$, we have the formula:

$$U_g=mgh\qquad\blacksquare$$

## Proof 35: Kinetic energy

Linear version of kinetic energy A spaceship in outer space starts its rocket engines to propel itself forwards with a rocket force. That force does…

## Proof 37: Elastic potential energy

In a spring that is elongated/compressed the distance $\Delta L=L_\text{end}-L_\text{relaxed}$, an elastic force (due to Hooke’s law) $F_\text{elastic}=k\Delta L$ is pushing/pulling on the surroundings in…

## Proof 26: Parallel-axis theorem

When rotating about the centre-of-mass, the moment-of-inertia $I_\text{com}$ is often easier to find. When the axis-of-rotation is somewhere else, we’ll below derive a formula that…

## Proof 14: Motion equations

In motion along a path, $a$ is acceleration, $v_0$ and $v$ are initial and final (current) speed, and $t_0$ and $t$ are initial and final…

## Proof 11: Sphere and ellipsoid volume

A sphere can be cut into slices. Each slice is almost a disk, or a very flat cylinder, except for the curved edge. If we…