Gravitational Potential Energy — GPE = mgh

Gravitational potential energy (GPE) is the energy stored by an object due to its height above the ground. The greater the height and mass, the more GPE it has. As an object falls, its GPE converts to kinetic energy.

What is Gravitational Potential Energy?

Gravitational potential energy (GPE) is the energy stored in an object due to its position above the ground (or any reference level).

$GPE = mgh$

Symbol	Quantity	Unit
$GPE$	gravitational potential energy	Joules (J)
$m$	mass	kg
$g$	gravitational field strength	N/kg (on Earth ≈ 10 N/kg)
$h$	height above reference level	metres (m)

Calculating GPE

A 5 kg book is placed on a shelf 1.2 m above the floor. Calculate its GPE. (g = 10 N/kg)

$GPE = mgh = 5 \times 10 \times 1.2 = 60 \text{ J}$

GPE and Kinetic Energy

When an object falls from a height (ignoring air resistance), GPE converts entirely to KE:

$GPE_{\text{lost}} = KE_{\text{gained}}$

$mgh = \frac{1}{2}mv^2$

Falling Object

A 2 kg ball is dropped from a height of 5 m. What is its speed just before hitting the ground? (g = 10 N/kg)

$mgh = \frac{1}{2}mv^2$ $10 \times 5 = \frac{1}{2}v^2$ $v^2 = 100 \Rightarrow v = 10 \text{ m/s}$

Note

GPE is always measured relative to a chosen reference level (usually the ground). An object 2 m above the ground has less GPE than the same object 5 m above the ground.

Real world: Roller coaster drop

A roller coaster car with passengers has a total mass of 600 kg. It is lifted to the top of a 40 m drop.

GPE stored at the top: $GPE = 600 \times 10 \times 40 = 240{,}000 \text{ J} = 240 \text{ kJ}$

At the bottom (ignoring friction), all this converts to KE: $v = \sqrt{2gh} = \sqrt{2 \times 10 \times 40} = \sqrt{800} \approx 28 \text{ m/s}$ (about 100 km/h).

Real world: Hydroelectric dam

Water in a reservoir sits 80 m above the turbines. Every second, 500 kg of water flows through.

GPE converted per second: $GPE = 500 \times 10 \times 80 = 400{,}000 \text{ J} = 400 \text{ kJ}$

This energy drives the turbines to generate electricity. The taller the dam, the more energy is available per kilogram of water.

Real world: Lift (elevator) rising

A 750 kg lift cabin (with passengers) rises 30 m to the top floor of a building.

Minimum energy needed: $GPE = 750 \times 10 \times 30 = 225{,}000 \text{ J} = 225 \text{ kJ}$

The electric motor must supply at least this much energy, plus extra to overcome friction and cable weight.

Tip

Where do you see this in real life? Roller coasters, hydroelectric power stations, ski lifts, and even water towers all store gravitational potential energy — height is a natural battery.

Practice Problems

Calculate the GPE of a 3 kg stone at a height of 4 m. (g = 10 N/kg)

Show answer

GPE = mgh = 3 × 10 × 4 = 120 J

A 10 kg rock rolls off a cliff 20 m high and falls. What is the maximum KE it can have at the bottom? (g = 10 N/kg)

Show answer

KE = GPE = mgh = 10 × 10 × 20 = 2,000 J (assuming no air resistance)

A 1 kg ball is released from rest at 8 m height. What is its speed just before landing? (g = 10 N/kg)

Show answer

mgh = ½mv² → v² = 2gh = 2 × 10 × 8 = 160 → v = √160 ≈ 12.6 m/s