Formula of Potential Energy: PE = mgh, Derivation and Unit

The formula of potential energy for an object near the Earth's surface is $PE = mgh$ , where m is the mass, g is the acceleration due to gravity and h is the height above a level you choose as zero. Lift a 2 kg bag onto a 5 m high shelf and you store 2 × 9.8 × 5 = 98 J in it.

Potential energy is energy of position or shape. Height in a gravitational field is the most common case, a stretched or squashed spring is the second, and charges in an electric field give a third. This page focuses on the gravitational form, then shows the others and where each one fits in the work, energy and power chapter.

Quick facts

Formula
$PE = mgh$
SI unit
joule (J)
Dimensional formula
[M¹ L² T⁻²]
Scalar or vector
Scalar
General gravity form
$U = -\frac{GMm}{r}$
Spring form
$U = \frac{1}{2}kx^2$
Value of g
9.8 m/s² (9.81 is also used)

What is the formula of potential energy telling you

Symbols in PE = mgh
Symbol	Stands for	Unit
PE or U	stored gravitational energy	J
m	mass of the raised object	kg
g	gravitational field strength, 9.8 near the ground	m/s² or N/kg
h	vertical height above your zero level	m

Only the vertical height counts. A trolley pushed 10 m along a ramp that rises 3 m gains the potential energy of a 3 m lift. The zero level is your choice: the floor, the table top or the ground outside. What matters in every real problem is the change, $\Delta PE = mg\,\Delta h$ , and that change is the same whichever zero you pick.

Derive the formula of potential energy

Lift an object of mass m slowly, at constant speed, from the ground to a height h.

To move it without speeding up, you push upward with a force equal to its weight, $F = mg$ .
The force and the displacement h point the same way, so the work you do is $W = Fh = mgh$ .
This work is not lost. It is stored in the object and Earth system as potential energy, so $PE = mgh$ .

Let go and gravity does the same amount of positive work on the way down, turning the stored energy into kinetic energy. This derivation assumes g stays constant, which is true to a very good approximation over heights much smaller than the radius of the Earth (6.37 × 10⁶ m).

Solved examples

Example 1: a bag on a shelf

Find the potential energy gained by a 2.0 kg bag lifted 5.0 m.

Given: m = 2.0 kg, g = 9.8 m/s², h = 5.0 m.

$PE = (2.0)(9.8)(5.0) = 98\ \text{J}$

Answer: 98 J.

Example 2: grams and centimetres

A 500 g book is raised from the floor to a table 80 cm high. How much potential energy does it gain? If it then falls back, at what speed does it hit the floor, ignoring air resistance?

Convert: m = 500 g = 0.500 kg and h = 80 cm = 0.80 m.

$\Delta PE = (0.500)(9.8)(0.80) = 3.92\ \text{J} \approx 3.9\ \text{J}$

During the fall, the stored energy turns into motion: $\frac{1}{2}(0.500)v^2 = 3.92$ , so $v^2 = 15.68$ and $v \approx 3.96\ \text{m/s}$ .

Answer: about 3.9 J gained, and the book lands at about 4.0 m/s.

Notice that the mass cancels in the speed step. That is why, without air resistance, a heavy and a light object dropped together land together.

Other forms of potential energy

Potential energy in different situations
Situation	Formula	Notes
Near Earth's surface	$U = mgh$	g treated as constant
Any distance from a planet	$U = -\frac{GMm}{r}$	zero at infinity, always negative
Stretched or compressed spring	$U = \frac{1}{2}kx^2$	see spring potential energy
Two point charges	$U = \frac{k\,q_1 q_2}{r}$	positive for like charges

The general gravity form uses the gravitational constant, 6.674 × 10⁻¹¹ in SI units. For a small lift Δh close to the surface, the change in $-GMm/r$ reduces to $mg\,\Delta h$ , so the two forms agree where both apply. Satellites and escape speed questions need the general form; everyday heights use mgh.

Units, dimensions and common mistakes

Units: multiplying kilograms by metres per second squared and then by metres gives the joule. In bracket form, mass times acceleration times length gives [M¹ L² T⁻²], the same as kinetic energy and work.

Using the slope length instead of the vertical height. Only the rise counts.
Mass in grams or height in centimetres. Convert to kg and m first.
Using mgh for a satellite. At large heights g changes, so use the $-GMm/r$ form.
Thinking a negative value is wrong. Below your zero level, h is negative and so is PE. That is fine.

Potential and kinetic energy together make mechanical energy; see the conservation of energy formula and the energy formula overview.

Frequently asked questions

What is the formula of potential energy?

Near the Earth's surface it is PE = mgh: mass times g times height. For large distances from a planet, use U = −GMm/r instead.

What is the SI unit of potential energy?

The joule (J), the same unit as work and kinetic energy. One joule is the energy needed to lift about 102 g through 1 m.

Is potential energy a scalar or a vector?

Scalar. PE carries a magnitude, which may be positive or negative compared with the zero level, yet it points nowhere.

Does the path matter when you lift an object?

No. Gravity is a conservative force, so the change in potential energy depends only on the starting and ending heights, not on the route between them.

What is the difference between potential energy and kinetic energy?

Potential energy is stored because of position or shape, such as height or a stretched spring. Kinetic energy is the energy an object has because it is moving, ½mv².