The Potential Energy Formula, Explained with Examples

What is potential energy?

Potential energy is stored energy: the energy an object has because of its position or its shape, not because it is moving. It is measured in joules (J), and it represents work that has already been done on the object and can be paid back later. A book on a high shelf, water held behind a dam, and a stretched rubber band all store potential energy.

Lift a heavy box onto a table and you give it gravitational potential energy. Let go and that stored energy turns into motion as the box falls. Energy is never created or destroyed, only transferred from one form to another, which is why potential energy, kinetic energy and work all share the same unit.

Two kinds show up in almost every physics course. Gravitational potential energy comes from an object's height in a gravitational field. Elastic potential energy is stored when you stretch or compress something springy, like a spring, a bow, or a trampoline. Each has its own formula, and the rest of this guide covers both.

The gravitational potential energy formula

Each symbol has a specific meaning and a specific unit, and getting the units right is half the battle. The table below lays them out.

On Earth, g is roughly 9.8 m/s squared, though many textbooks round it to 9.81 for extra precision. The internationally agreed standard value is 9.80665 m/s squared, and the real local value varies slightly with latitude and altitude, which is why 9.8 is good enough for schoolwork. Plug kilograms, 9.8 and metres into the formula and the answer lands in joules with no extra conversion, because one joule is exactly one kilogram times metres squared per second squared.

Notice what is not in the formula: speed, direction, shape, and the path taken. Gravitational potential energy depends only on mass and height. Carry a suitcase up a spiral staircase or a straight ladder to the same floor and you have stored exactly the same energy in it.

Why is the potential energy formula m x g x h?

The formula comes straight from the definition of work. Work equals force times distance, the force needed to lift an object steadily is its weight (m x g), and the distance is the height h, so the work done is m x g x h. That work does not vanish, it is stored in the object as potential energy, which is why PE = m x g x h.

This is worth remembering because it explains the units for free. A newton times a metre is a joule, and weight in newtons times height in metres is exactly what m x g x h calculates. If you want the fuller treatment of force times distance, the work formula guide covers it in detail.

The reference point, and why height is a choice

Height h is always measured from a reference point that you pick, usually the floor or the ground. There is no absolute zero of gravitational potential energy, so only the difference between two heights is physically meaningful. A coin on a tenth-floor windowsill has far more potential energy relative to the street than the same coin on a ground-floor table, even though both are sitting perfectly still.

This also means potential energy can legitimately be negative. Choose the tabletop as your zero and a coin on the floor below it has negative potential energy. Nothing is broken: the negative sign just says the coin sits below your chosen zero. Physicists exploit this deliberately in the general form of the formula described further down.

Gravity is different on other worlds

Because g appears directly in the formula, the same object at the same height stores different amounts of energy on different planets. The values below come from NASA's planetary fact sheets, with the last column showing the potential energy of a 2 kg book raised 1.5 m.

The pattern is easy to read: lifting the book on the Moon stores about a sixth of the energy it would on Earth, so it also hits the ground far more gently. Nothing about the formula changes, only the number you put in for g.

How to calculate potential energy step by step

To calculate gravitational potential energy, convert the mass to kilograms and the height to metres, use g = 9.8 m/s squared, multiply mass by gravity, then multiply that result by the height. The answer is in joules.

1. Make sure the mass is in kilograms and the height is in metres. Convert first if needed.
2. Choose a value for gravity: 9.8 m/s squared on Earth (or 9.81 for more precision).
3. Multiply the mass by gravity. This is the object's weight in newtons.
4. Multiply that result by the height. The answer is in joules.

Mixing units is the number one source of wrong answers. If a mass is given in grams, divide by 1000 to get kilograms. If a height is given in centimetres, divide by 100 to get metres. A scientific calculator keeps the arithmetic tidy, and an energy converter turns the joules into calories, kilowatt hours or foot-pounds if your assignment asks for a different unit.

Worked example: a book on a shelf

A 2 kg book sits on a shelf 1.5 m above the floor. Find its gravitational potential energy, using g = 9.8 m/s squared.

1. Check the units: mass is 2 kg and height is 1.5 m, both in SI units already.
2. Multiply mass by gravity: 2 x 9.8 = 19.6 newtons.
3. Multiply by the height: 19.6 x 1.5 = 29.4.
4. State the answer: the potential energy is 29.4 joules.

Now raise the same book to a shelf 3 m high. The calculation becomes 2 x 9.8 x 3 = 58.8 joules. The height doubled and so did the energy, exactly as expected from a formula with no squared terms.

Worked example: converting grams and centimetres

A 250 g mug is lifted onto a counter 90 cm high. Neither figure is in SI units, so convert first: 250 g is 0.25 kg, and 90 cm is 0.9 m. Then PE = 0.25 x 9.8 x 0.9 = 2.205 joules, or about 2.2 J. Had you left the numbers as 250 and 90, the answer would have been out by a factor of 100,000.

The elastic potential energy formula

Elastic potential energy is the energy stored in a stretched or compressed spring, and its formula is PE = 1/2 x k x x squared, where k is the spring constant in newtons per metre and x is the displacement in metres. Because x is squared, stretching a spring twice as far stores four times the energy.

The spring constant k measures stiffness: a high k means a stiff spring that resists stretching. The formula assumes the spring obeys Hooke's law, meaning the restoring force grows in proportion to the stretch. Push a real spring far enough and it deforms permanently, at which point neither Hooke's law nor this formula applies.

Worked example: a stretched spring

A spring with a spring constant of 200 N/m is stretched 0.1 m from its rest position. Find its elastic potential energy.

1. List the values: k = 200 N/m, x = 0.1 m.
2. Square the displacement: 0.1 x 0.1 = 0.01.
3. Multiply by the spring constant: 200 x 0.01 = 2.
4. Multiply by one half: 0.5 x 2 = 1.
5. State the answer: the elastic potential energy is 1 joule.

Stretch that same spring to 0.2 m and the energy becomes 0.5 x 200 x 0.04 = 4 joules. Twice the stretch, four times the energy. This is the squared term at work, and it is why a bow drawn all the way back is dramatically more dangerous than one drawn halfway.

What are the types of potential energy?

Gravitational and elastic are the two you calculate by hand, but potential energy is a family. Any time a force can push something back toward a lower-energy arrangement, there is potential energy stored in that arrangement.

• Gravitational. Stored by height in a gravitational field. Formula: PE = m x g x h. Example: water behind a hydroelectric dam.
• Elastic. Stored by deforming something springy. Formula: PE = 1/2 x k x x squared. Example: a drawn bow or a compressed car suspension.
• Electric. Stored by a charge's position in an electric field. Example: the separated charge in a charged capacitor.
• Chemical. Stored in the arrangement of atoms within molecular bonds. Example: petrol, a battery, or the food you ate this morning.
• Nuclear. Stored in the forces binding an atomic nucleus together. Example: the energy released by fission in a reactor.

The last three are not calculated with m x g x h, but they follow the same logic: energy parked in a configuration, waiting to be released. Wikipedia's potential energy article is a solid reference if you want the formal treatment of each type.

When m x g x h stops working

PE = m x g x h assumes g is constant, which is only true near a planet's surface. Climb far enough and gravity weakens noticeably, so satellite and orbital problems use the general form instead: PE = -G x M x m / r, where G is the gravitational constant (about 6.674 times 10 to the power minus 11), M is the planet's mass, m is the object's mass, and r is the distance between the centres.

The minus sign appears because this version sets zero potential energy at infinite distance rather than at ground level, so anything gravitationally bound sits below zero. For homework about books, shelves and roller coasters, m x g x h remains exactly right and far easier to use.

How does potential energy become kinetic energy?

When a raised object is released, gravity pulls it down and its gravitational potential energy converts into kinetic energy, the energy of motion. Ignoring air resistance, the potential energy lost equals the kinetic energy gained, so at the bottom of the fall almost all the stored energy has become movement.

That neat trade is a genuinely useful shortcut. Setting m x g x h equal to 1/2 x m x v squared lets the masses cancel, which is the reason a feather and a hammer hit the ground at the same speed in a vacuum regardless of weight. It is also the principle behind roller coasters, waterfalls and hydroelectric dams: raise something, then cash the height in for speed.

You can predict how fast a falling object will be moving with the kinetic energy formula, and study how forces build that speed up with the acceleration formula guide. Springs do the same thing in reverse: release a compressed spring and its elastic potential energy launches whatever it pushes.

To see the conversion in action, try the calculator below. Enter a mass and a speed and it returns the kinetic energy an object would carry, which is exactly what a falling object gains as it trades away its potential energy.

Common mistakes to avoid

• Using the wrong units. Mass must be in kilograms and height in metres for the answer to come out in joules. Convert grams or centimetres first.
• Forgetting the reference point. Height h is measured from a chosen zero level, usually the ground. Always state where you are measuring from.
• Mixing up the two formulas. Use m x g x h for gravity and 1/2 x k x x squared for springs. They are not interchangeable.
• Leaving off the one half in the spring formula. The 1/2 factor is a fixed part of the elastic equation, not optional.
• Forgetting to square the displacement. In the spring formula, x is always squared. Stretching twice as far stores four times the energy.
• Using the total height instead of the height gained. If an object drops from 5 m to 2 m, the energy released comes from the 3 m difference, not from 5 m.
• Measuring h along a slope. Height means vertical height. For an object on a ramp, use the vertical rise, not the length of the ramp.

Once you are comfortable with both potential energy formulas, you can analyse almost any everyday energy situation, from a dropped phone to a drawn bow. Identify whether the energy is gravitational or elastic, pick the matching formula, keep your units consistent, and the joules will follow. For the wider picture of motion, the momentum formula guide is a natural next read.