Potential energy: what it is, gravitational potential energy (E p = mgh) and elastic potential energy (E e = ½kx²), chemical, electrical and nuclear potential energy, the energy-store model used in GCSE, ten worked examples (a lifted book, a roller coaster, a stretched spring, a pendulum), potential against kinetic energy and conservation, the mistakes, and a printable formula sheet with practice

"Potential energy" is searched about 1,900 times a month in the UK and 2,300 in Australia, and its two formulas appear on every physics paper from Year 9 to GCSE and Year 10 to 12. The idea is simple (energy stored by position or shape); the marks are lost on units, on where the height is measured from, and on the square in ½kx². This page explains both formulas, lists the other kinds of potential energy, works ten examples with the units carried through, shows the trade with kinetic energy on a roller coaster and a pendulum, translates into the GCSE energy-store language, lists the mistakes and gives a printable formula sheet with practice.

Download the potential energy formula sheet with practice (PDF) (A4, the formulas, the stores table and the worked examples on page 1, the conservation examples and the practice set with answers on page 2, free).

What potential energy is

Energy can be stored or in motion. Kinetic energy is the energy of a moving object. Potential energy is energy stored in an object because of its position (a raised weight), its shape (a stretched spring) or its state (the arrangement of atoms in a fuel, of charges in a battery, of nucleons in a nucleus). It is "potential" because it can become something else: the raised weight can fall and gain kinetic energy, the spring can snap back, the fuel can burn and heat a room.

Potential energy is measured in joules (J), like every form of energy. One joule is the energy needed to lift a 100 g apple about one metre.

Gravitational potential energy: E p = mgh

An object raised against gravity stores energy. The amount is the work done lifting it: force × distance, where the force is the weight (mg) and the distance is the height gained (h).

E p = m × g × h

• m: mass in kilograms (kg)
• g: gravitational field strength, 9.8 N/kg on Earth's surface (many school papers use 10; the Moon is 1.6, Mars 3.7)
• h: height gained, in metres, measured from whatever level you choose as zero

Three things to notice. First, only the change in height matters, so you can set zero wherever is convenient (the floor, the table, sea level); the energy difference between two heights is the same whatever zero you choose. Second, the path does not matter: a ramp, stairs or a straight lift all give mgh for the same height. Third, the answer is in joules when m, g and h are in kg, N/kg and m; a mass given in grams or a height in centimetres must be converted first.

Worked: a 0.5 kg ball lifted 2 m: E p = 0.5 × 9.8 × 2 = 9.8 J. A 60 kg climber 200 m up a cliff: 60 × 9.8 × 200 = 117,600 J = 117.6 kJ. A 1,000 kg car driven up a 50 m hill: 1,000 × 9.8 × 50 = 490,000 J = 490 kJ.

Elastic potential energy: E e = ½kx²

A spring, elastic band, bow or trampoline stores energy when stretched or compressed. Hooke's law says the force needed is proportional to the extension (F = kx); the energy stored is the area under that force-extension line, a triangle, which gives the half:

E e = ½ × k × x²

• k: the spring constant in newtons per metre (N/m), a measure of stiffness
• x: the extension (or compression) in metres, measured from the spring's natural, unstretched length

The square matters: double the stretch and you store four times the energy. The formula holds up to the limit of proportionality; beyond it the spring deforms permanently and the line bends.

Worked: a spring with k = 200 N/m stretched 0.1 m: E e = ½ × 200 × 0.1² = ½ × 200 × 0.01 = 1 J. A bow with an effective k of 500 N/m drawn 0.6 m: ½ × 500 × 0.36 = 90 J, which becomes the arrow's kinetic energy. A car's suspension spring, k = 50,000 N/m, compressed 0.05 m: ½ × 50,000 × 0.0025 = 62.5 J.

The other kinds of potential energy

GCSE Physics (England) and many other syllabuses now use energy stores instead of "types of energy": the gravitational store, elastic store, chemical store, kinetic store, thermal store, magnetic, electrostatic and nuclear stores, with energy transferred between stores mechanically (by a force), electrically (by a current), by heating or by radiation (light, sound). In that language, lifting a book "transfers energy mechanically from the chemical store of your muscles to the gravitational store of the book". The physics is identical; the wording is what the mark scheme wants.

Potential against kinetic energy: conservation

With no friction or air resistance, the total of potential and kinetic energy stays constant: what one loses the other gains.

The roller coaster. A 500 kg car at rest at the top of a 40 m drop has E p = 500 × 9.8 × 40 = 196,000 J and E k = 0. At the bottom, E p = 0 and E k = 196,000 J, so ½ × 500 × v² = 196,000, v² = 784, v = 28 m/s (about 100 km/h). Halfway down, 98,000 J of each. Real coasters lose some to friction and air, so the next hill must be lower than the first.

The pendulum. Pulled aside so the bob rises 0.2 m, a 0.3 kg bob has E p = 0.3 × 9.8 × 0.2 = 0.588 J at each end of its swing and E k = 0.588 J at the bottom: v = √(2 × 0.588 ÷ 0.3) = 1.98 m/s. The height, not the length of the string or the mass, fixes the speed at the bottom.

The dropped ball. Height h, speed at the ground v: mgh = ½mv², so v = √(2gh) for any mass. From 5 m, v = √(2 × 9.8 × 5) = 9.9 m/s. The mass cancels, which is why a heavy and a light ball hit the ground together (without air resistance).

The bouncing ball does not return to its starting height because some energy goes to the thermal store (the ball and the floor warm slightly) and to sound. The missing height is the energy "lost", which is never destroyed, only spread out.

Ten worked examples in one table

The mistakes that cost marks

• Using the mass in grams or the height in centimetres without converting (a 200 g, 50 cm problem is 0.2 kg and 0.5 m).
• Confusing mass (kg) with weight (N). E p = mgh uses the mass; if the weight is given, E p = weight × height.
• Measuring h from the wrong place. Only the height change between the two positions matters.
• Forgetting the ½ or the square in ½kx², or squaring k instead of x.
• Using the spring's total length for x. It is the extension from the natural length.
• Saying energy is "lost" in a bounce without saying where it went (thermal store, sound).
• Writing "potential energy becomes kinetic energy" for a ball thrown upward. On the way up it is the other way round.
• Leaving out the unit: joules, always.

Where it sits in each curriculum

Practice (answers in the PDF)

1. A 3 kg bag is lifted 1.2 m onto a table. How much gravitational potential energy does it gain? (g = 9.8 N/kg)
2. A 45 kg gymnast on a 4 m high beam. What is her potential energy relative to the floor?
3. A spring with k = 150 N/m is stretched by 8 cm. How much elastic potential energy is stored?
4. A 0.2 kg ball is dropped from 3.2 m. What is its speed just before it hits the ground, ignoring air resistance?
5. A roller coaster car has 80,000 J of potential energy at the top of a hill and 60,000 J of kinetic energy at the bottom. Where did the rest go?
6. Describe lifting a box onto a shelf in energy-store language.

Download the formula sheet and the practice set (PDF)

Where to get help

A 1-on-1 online physics class with a tutor who teaches KS3 and GCSE physics, the Australian Years 8 to 11, NGSS middle and high-school science and CBSE Class 9 to 11 every week: energy taught with the units written on every line and the stores language matched to the mark scheme. The physics page lists the courses; the free 30-minute first class is a real lesson on this week's topic. Related pages: net force and Newton's second law, specific heat and Q = mcΔT and the five equations of motion.

Questions parents ask

See how we teach this, 1-on-1 online →