Kinetic and Potential Energy

Kinetic energy is the energy of motion, $\tfrac12mv^2$ ; potential energy is the energy of height, $mgh$ . With no friction or air resistance, one simply turns into the other and the total never changes.

Cambridge 9709 Paper 4Edexcel YMA01 WME02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Calculate kinetic energy $\tfrac12mv^2$ and changes in potential energy $mgh$ .
Use conservation of energy when only gravity does work.
Solve problems on smooth curved paths, where suvat cannot be used.
Recognise that the speed gained depends only on the height dropped, not the mass or path.

Pause and think

Two children slide from the same height: one down a short steep slide, the other down a long twisting one. Ignoring friction, who is faster at the bottom?

Section 2 of 14

Core idea

Start hereA rollercoaster needs no engine

Once hauled to the top, a rollercoaster runs on stored energy alone. Height becomes speed, speed becomes height. Ignoring friction, the total of kinetic and potential energy never changes.

The big idea

$E_k=\tfrac12mv^2$ ; $E_p=mgh$ . Without friction, $E_k+E_p$ is constant.

1. Where ½mv² comes from

Work done accelerating from rest

$W=Fs=mas$
$F=ma$ .
$v^2=0+2as\Rightarrow as=\tfrac12v^2$
suvat from rest.
$W=\tfrac12mv^2$
Substitute: this is the kinetic energy gained.

A 2 kg ball dropped from 20 m: PE and KE as it falls

height h (m) = 20.0

The PE bar shrinks exactly as fast as the KE bar grows. The total stays at

mgH=2(10)(20)=400

Worked example2 kg ball dropped from 20 m

Speed at the bottom, and at half way.

Goal: Speeds.
Strategy: PE lost = KE gained.

$mgh=2(10)(20)=400\text{ J}$
Total energy.
$\tfrac12(2)v^2=400\Rightarrow v=20$
Bottom.
$\tfrac12(2)v^2=200\Rightarrow v=14.1$
Half way: half the PE converted.

Answer

20

m/s;

14.1

m/s.

✓ Checksuvat:

v^2=2(10)(20)=400

✓.

Worked exampleSkier on a smooth slope

At 5 m/s, drops 15 m. Final speed?

Goal: $v$ .
Strategy: Energy; mass cancels.

$\tfrac12v^2=\tfrac12(25)+10(15)$
Divide through by $m$ .
$v^2=325\Rightarrow v=18.0$
Solve.

Answer

18.0

m/s.

✓ CheckIndependent of the slope's shape ✓ (only height matters).

Pause and think

Why does mass cancel?

Summary

$\frac12mv^2$ , $mgh$ .
Smooth: total constant.

Kinetic energy: Energy of motion.
Potential energy: Energy of position.

A 2 kg ball dropped from 10 m (g = 10 m s⁻²)

Ignoring air resistance, every joule of potential energy lost becomes kinetic energy, so the total stays at

mgh=2\times10\times10=200\text{ J}

HistoryJoule's paddle wheel. In 1845 James Prescott Joule let a falling weight turn a paddle wheel in a tank of water, and measured the tiny rise in temperature. He showed that a fixed amount of mechanical work always produces the same amount of heat, which was the key evidence that energy is conserved, just changing form. The unit of energy is named after him.

Fun factWhy the first hill is the tallest. A roller coaster is hauled up its first hill and then runs on energy alone. Every later hill must be lower, because some energy is lost to friction and air resistance along the way. If a later hill were as tall as the first, the train would stall near the top.

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