Potential Energy Calculator
Calculate gravitational potential energy from mass, gravity, and height, or solve for height from energy. Useful for lift and energy balance problems.
Potential Energy Calculator
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What the potential energy calculator does
Lift something up and you store energy in it. Let it go and that energy comes back as motion. This stored energy of height is gravitational potential energy, and this calculator works it out from the mass, the strength of gravity, and the height. It can also solve for any one of those from the others.
Because gravity is a value you can set, it works for any planet, not just Earth. Below is what this energy is, the equation behind it, why the height needs a reference point, and a worked example.
How to use it
- Choose what to find: potential energy, mass, gravitational acceleration, or height.
- Enter the others. Gravity starts at Earth's value and can be set in m/s² or in g; height and mass each have their own units.
- Press Calculate for the answer across the energy units, or Reset to clear it.
What gravitational potential energy is
Potential energy is stored energy, energy an object has by virtue of where it is rather than how it moves. Raising a mass against gravity takes work, and that work does not vanish, it is banked in the object's height. The higher you lift it, and the heavier it is, the more is stored, ready to be released the moment it is allowed to fall.
When it does fall, that stored energy turns into kinetic energy, the energy of motion. This trade is one of the cleanest examples of energy conservation: the potential energy lost in falling equals the kinetic energy gained, which is why a heavier object or a greater drop hits harder. It is also the energy a hydroelectric dam taps, letting raised water fall to spin turbines.
The equation it solves
Gravitational potential energy, often written PE or U, depends on the mass m, the acceleration of gravity g, and the height h:
PE = m g h
Rearranged, the same relationship gives the other quantities the calculator can find: the height for a given energy, h = PE ÷ (m g), along with the mass, m = PE ÷ (g h), and the gravity, g = PE ÷ (m h).
Height from where? The reference level
Potential energy always depends on a choice: height measured from what? The energy is not an absolute number pinned to an object, it is the energy of being at one height rather than another. So before the formula means anything, you pick a level to call zero, the floor, the ground, the bench top, and measure height from there.
This sounds like a complication but it is a freedom. Only changes in potential energy have physical effect, the energy gained or released as something moves up or down, and those changes come out the same whatever zero you chose. Pick the level that makes your problem simplest, usually the lowest point the object will reach, and the height you enter is the rise above it.
Units, gravity, and precision
The calculator works in SI, converting mass to kilograms and height to metres, and reports the energy in joules and a range of other energy units. Gravity defaults to Earth's value of about 9.81 m/s², which you can enter directly or as a multiple of g, and change for the Moon, Mars, or anywhere else, since the same mass at the same height stores less energy where gravity is weaker. Results are rounded for display while the calculation runs at full precision.
A worked example
How much energy is stored by lifting a 5 kg object onto a shelf 2 metres high, on Earth?
From the formula, PE = 5 × 9.81 × 2 = 98.1 J. That is the work it took to lift it, now banked as potential energy. Let the object fall back from the shelf and it arrives at the floor with very nearly that same 98.1 J, now as kinetic energy, having traded its height for speed.
Questions people ask
What is the formula for gravitational potential energy?
It is mass times gravity times height, PE = mgh. Knowing any three of energy, mass, gravity, and height fixes the fourth.
Height measured from where?
From whatever level you choose to call zero. Only changes in potential energy matter physically, and those come out the same regardless of the zero you pick, so choose the level that makes the problem simplest.
What are the units of potential energy?
The SI unit is the joule (J), the same as for all energy, since potential and kinetic energy are interchangeable forms of the same quantity.
Does it change on the Moon or other planets?
Yes. Potential energy is proportional to gravity, so the same mass at the same height stores less energy where gravity is weaker. Set the gravity value to match the world you are on.
References
A quick note on where the physics comes from. Gravitational potential energy as mgh, the role of the reference level, and its conversion to kinetic energy are standard mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The joule and the value of standard gravity follow the US National Institute of Standards and Technology.
- OpenStax, University Physics Volume 1, Section 8.1, Potential Energy of a System. https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system
- HyperPhysics, Georgia State University, Gravitational Potential Energy. http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html
- National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811
Bibek Lal Karna is a PhD student and graduate teaching assistant at the University of Mississippi, with deep interests in theoretical and gravitational physics. He is also the founder of NRCC and is strongly engaged in scientific teaching and communication. At Eon Tools, he reviews physics tools.