Kinetic Energy and Gravitational Potential Energy

Kinetic energy — the energy a body has because of its motion:

Eₖ = ½mv²

Where the formula comes from — a resultant force F accelerating a mass from rest through a distance s does work Fs. Using v² = u² + 2as with u = 0 gives s = v²/2a, and F = ma, so the work done is ma × v²/2a = ½mv². The acceleration cancels, which is why the result holds regardless of how the body was accelerated.

The square matters — doubling the speed quadruples the kinetic energy. This is the same relationship that makes braking distance proportional to v², and it is why the difference between 30 and 40 mph matters so much more than the numbers suggest.

Gravitational potential energy — the energy a body has because of its position in a gravitational field. For changes near the Earth's surface:

ΔEₚ = mgΔh

This uses a constant g, so it is valid only over heights small enough that the field strength does not change appreciably — fine for a building, not for a satellite.

Height is measured vertically — Δh is the vertical height gained, not the distance travelled. Dragging a box 5 m up a ramp that rises 2 m gives a gain of mg × 2, not mg × 5.

Example: a 2.5 kg book lifted 1.8 m onto a shelf gains 2.5 × 9.81 × 1.8 = 44 J. Lifted at an angle along a 3 m path to the same shelf, the gain is identical, because only the vertical rise counts.

Exchanging one for the other — for an object falling freely, the gravitational potential energy lost equals the kinetic energy gained:

mgΔh = ½mv²

The mass cancels, giving v = √(2gΔh). The speed reached in a fall does not depend on the mass, which is the same result as free fall seen through energy rather than forces.