Work, Energy and Power

Work is done when a force moves its point of application. It is the energy transferred, measured in joules, and one joule is one newton-metre.

Only the component of force $\text{along}$ the displacement does work, which is where the $\cos \theta$ comes from. A force perpendicular to the motion does no work at all — which is why carrying a bag horizontally transfers no energy to the bag, however tiring it feels, and why the tension in a string does no work on an object moving in a circle.

This projection is also why energy is a scalar: taking the component along the displacement removes the direction, leaving a plain number.

$\text{Kinetic energy}$ is the energy of motion, $E_k=\frac{1}{2}m{v}^2$. Note the square: doubling the speed quadruples the kinetic energy, which is why stopping distances grow so sharply with speed.

$\text{Gravitational potential energy}$ is the energy of position in a gravitational field, $\Delta E_p=mg\Delta h$. Only the $\text{change}$ in height matters, and only the vertical component of any movement counts — moving horizontally changes nothing.

The $mg\Delta h$ form assumes $g$ is constant, which is fine near the Earth’s surface. Over astronomical distances it fails, and Topic 12 replaces it.

Energy cannot be created or destroyed, only transferred. With no resistive forces, the total mechanical energy stays constant, so potential energy lost equals kinetic energy gained.

That gives a route to final speed that avoids the path entirely: $mgh=\frac{1}{2}m{v}^2$, and the mass cancels, leaving $v=\sqrt{2gh}$. The result is independent of the shape of the slope, which no force-based method would show so quickly.

When friction or drag acts, energy is transferred to thermal energy instead of disappearing. The bookkeeping becomes: initial energy = final energy + work done against resistance.

$\text{Power}$ is the rate of energy transfer, in watts. One watt is one joule per second.

For an object moving at steady speed against resistance, combining $P=W/t$ with $W=Fs$ gives the useful form $P=Fv$ — which is why a car needs far more power at high speed even to maintain it.

$\text{Efficiency}$ is the fraction of input energy that becomes useful output. It can never exceed 1 (or 100%), because the remainder is transferred to the surroundings, usually as thermal energy.