About the Potential Energy Calculator
Gravitational potential energy is the energy an object has because of its height. Lift a book onto a shelf and you store energy in it; knock it off and that energy turns into motion as it falls. The same idea powers hydroelectric dams, grandfather clocks with hanging weights and roller coasters, and it is the energy you spend every time you climb a flight of stairs.
This potential energy calculator uses PE = m × g × h to find the energy stored by lifting a mass to a height, or works backwards to the mass or height for a given energy. It uses Earth's standard gravity by default, and can also calculate for the Moon, Mars, Venus or Jupiter. It gives the energy in several units and the speed the object would reach if it fell the same height.
How to Use the Potential Energy Calculator
Choose Find energy, Find mass or Find height.
Enter the known values and choose their units.
Choose the gravity: Earth, the Moon, Mars, Venus or Jupiter.
The Formulas
PE = m × g × h
m = PE ÷ (g × h) h = PE ÷ (m × g)
g on Earth = 9.80665 m/s² (standard gravity)
falling speed from rest, no air resistance: v = √(2 g h)Step-by-Step Examples
Lifting 10 kg by 5 metres.
PE = 10 × 9.80665 × 5 = 490.33 J
if dropped: v = √(2 × 9.80665 × 5) = 9.903 m/sClimbing a flight of stairs. A 70 kg person climbing 3 metres:
PE = 70 × 9.80665 × 3 = 2,059.4 JThe same lift on the Moon. With g = 1.62 m/s², lifting 10 kg by 5 m stores only 10 × 1.62 × 5 = 81 J.
Height for 1,000 J. Raising 10 kg to store 1,000 J needs h = 1,000 ÷ (10 × 9.80665) = 10.197 m.
Pumped storage. Raising 1,000 kg of water by 100 m stores 980,665 J, about 0.2724 kWh.
Potential Energy and Falling
When an object falls freely, its potential energy turns into kinetic energy. Setting m g h equal to ½ m v² shows that the landing speed, v = √(2 g h), does not depend on the mass: without air resistance, a heavy and a light object dropped from the same height land at the same speed, as Galileo argued. Air resistance changes this in real life, slowing light, spread-out objects much more than dense, compact ones.
Only the Change in Height Matters
Potential energy is always measured from a chosen reference level, such as the floor, the ground or sea level, and only differences in height matter. Lifting a box one metre stores the same energy whether you start on the ground floor or the tenth. The path does not matter either: carrying a load up a long gentle ramp stores exactly the same energy as lifting it straight up to the same height, although the ramp lets you use a smaller force over a longer distance.
Potential Energy in the Real World
Hydroelectric power turns the potential energy of water held behind a dam into electricity as it falls through turbines. Pumped storage plants do the reverse when electricity is plentiful, pumping water uphill into a reservoir, then releasing it when demand peaks; the largest can store energy measured in gigawatt-hours. Roller coasters use a motor to haul the cars to the top of the first hill, then let potential energy drive the rest of the ride. Even a pendulum clock is powered by slowly falling weights or a wound spring.
Other Kinds of Potential Energy
Gravity is not the only source of potential energy. A stretched spring or a drawn bow stores elastic potential energy, ½ k x², released when it snaps back. Chemical bonds store chemical potential energy, released when fuel burns or food is digested, and charged particles store electrical potential energy. This calculator deals with gravitational potential energy near a planet's surface, where gravity is roughly constant; for heights comparable to a planet's radius, such as satellite orbits, a different formula is needed because gravity weakens with distance.
Energy Changes on a Roller Coaster
A roller coaster is a clear example of energy changing form. A chain lift does work to raise the train to the top of the first hill, giving it its greatest potential energy. As the train plunges down, potential energy turns into kinetic energy and the train speeds up; as it climbs the next hill, kinetic energy turns back into potential energy and it slows. Because some energy is always lost to friction and air resistance, each later hill must be lower than the first, and the ride ends with brakes turning the remaining kinetic energy into heat. Engineers use m g h and ½ m v² to work out the speed at every point on the track, and to check that the train will clear each hill and loop safely with a comfortable margin.
Understanding Your Result
The headline is the energy, mass or height you asked for.
The energy in other units line gives the energy in J, kJ, kWh and kcal.
The gravity line shows the gravity used, and the Earth equivalent for other bodies.
The if it fell line gives the landing speed after falling the same height, ignoring air resistance.
When Should You Use This Calculator?
Use it for physics homework on energy.
Use it to estimate the energy stored in raised water or weights.
Use it to compare gravity on different planets.
Common Mistakes
Using weight in newtons as the mass. Enter the mass in kilograms.
Measuring height from the wrong level. Use the change in height.
Forgetting air resistance. Real falling speeds are lower for light objects.
Using PE = mgh for great heights. Gravity weakens far from the surface.