Work-energy principle and conservation of mechanical energy

Quick recap of the three ideas this lesson joins together (all measured in joules, J):

• Work done by a force $= Fd\cos\theta$, where $F$ is the force in newtons, $d$ is the distance moved in metres, and $\theta$ is the angle between the force and the direction of motion. If the force is along the motion, $\cos 0 = 1$ and the work is just $Fd$.
• Kinetic energy (KE), the energy of movement: $\tfrac12 mv^2$, where $m$ is the mass in kg and $v$ the speed in m s⁻¹.
• Gravitational potential energy (GPE), the energy of height: $mgh$, where $h$ is the height above a level you choose as zero. Take $g = 10\text{ m s}^{-2}$.

The symbol $\Delta$ ("delta") means "change in", so $\Delta\text{KE}$ is the final KE minus the initial KE.

Think of an object's energy (its KE plus its GPE) as a bank balance.

• A force that pushes with the motion (an engine, a pull, a push) does positive work: it pays in.
• A force that acts against the motion (friction, air resistance) takes money out, turning energy into heat and sound. We write this as a positive amount, the work done against the resistance (force × distance), and subtract it.
• Gravity is already counted in the GPE, so don't count it again as work. (Equivalently, the work done by gravity equals the GPE lost; use one or the other, never both.) Normal reactions are at right angles to the motion, so they do no work at all ($\cos 90^\circ = 0$).

That gives the one equation this lesson is about, the work–energy principle:

$\text{work done by driving forces} - \text{work done against resistances} = \Delta\text{KE} + \Delta\text{GPE}.$

Here "driving forces" means anything except gravity that helps the motion: an engine, a push or a pull. In words: what you pay in, minus what's taken out, equals how much the balance changed.

If no forces other than gravity do any work (the surface is smooth, there's no air resistance and no engine), nothing is paid in or taken out. The balance stays the same:

$\text{KE}_{\text{start}} + \text{GPE}_{\text{start}} = \text{KE}_{\text{end}} + \text{GPE}_{\text{end}}.$

This is the principle of conservation of energy. Energy just swaps between height and speed.

A big bonus: it only cares about the start and the end, not the path. So it works for curved slides and ramps, where $F = ma$ would be very hard because the forces keep changing direction.