The potential energy calculator above solves U = mgh in whichever direction you need. Give it mass, height and gravity for the energy; give it the energy and any two of the others to recover the third. Alongside the result it reports what happens if the object is simply let go — the speed it reaches at the bottom and how long the fall takes.
Arb Digital builds free tools that pick one job and finish it properly. This page handles gravitational potential energy near a planetary surface, where gravity can be treated as constant. The boundary against the site's other energy pages is clear: the kinetic energy calculator covers energy of motion, the work calculator covers energy transferred by a force through a distance, and an energy converter only rescales a joule figure into other units.
What This Potential Energy Calculator Does
Gravitational potential energy is the work that would have to be done against gravity to lift an object to its current height, and equivalently the energy that becomes available if it falls back. Near a planetary surface it is the product of three things: mass, the local gravitational acceleration and height above a chosen reference.
All four rearrangements are implemented. Solving for mass suits problems where an energy is measured and the object is unknown. Solving for height is the pumped-storage question — how high you must raise a given mass to bank a given amount of energy. Solving for gravity is the planetary comparison, and is the form used when the same experiment gives different answers on different worlds.
The supporting figures come from energy conservation. If the object falls freely from that height, all the potential energy becomes kinetic, so its speed on arrival is the square root of twice gravity times height — a result that does not involve the mass at all. The fall time follows from the same constant acceleration, and the Moon comparison shows how much of the answer was gravity rather than the object.
How to Use It
- Choose what you are solving for. Energy is the usual case; the other three are rearrangements that appear more often than people expect.
- Decide your zero level and stick to it. Height is always above something. The floor, the ground, sea level and the bottom of a shaft are all legitimate choices, but mixing them in one problem is not.
- Use mass in kilograms, not weight. A figure in newtons or in kilograms-force is a weight, and it already contains gravity, so entering it as mass counts gravity twice.
- Change gravity for other worlds. The two preset buttons set lunar and Martian values, and any other figure can be typed in directly.
- Treat the impact speed as a vacuum figure. It ignores air resistance entirely, which is a good approximation for a dense compact object over short drops and a poor one for anything light or over long ones.
The Formula: How Potential Energy Is Calculated
The relation is U = mgh. OpenStax University Physics Volume 1, section 8.1 on potential energy of a system, gives the gravitational potential energy function near Earth's surface as U(y) = mgy plus a constant, and works through an example computing the energy of a 75 kilogram hiker at different elevations relative to a chosen base.
That trailing constant is the important detail. It encodes the freedom to choose where zero sits, and it cancels out of every physical prediction because only differences in potential energy do anything. Rearranged, m = U/(gh), h = U/(mg) and g = U/(mh).
The default value for gravity is standard gravity, which is a defined rather than measured quantity. The NIST CODATA entry for the standard acceleration of gravity gives it as exactly 9.80665 metres per second squared with no uncertainty.
Work the defaults. A 70 kilogram person 10 metres up has a potential energy of 70 × 9.80665 × 10 = 6,865 joules, or 6.86 kilojoules. Released, they would arrive at the bottom at √(2 × 9.80665 × 10) = 14.0 metres per second, which is 50.4 kilometres per hour, after a fall lasting √(2 × 10 ÷ 9.80665) = 1.43 seconds. The same drop on the Moon would store only 70 × 1.625 × 10 = 1,138 joules.
Why Impact Speed Does Not Depend on Mass
The grid figure for impact speed contains no mass term, and that surprises people who have just watched the energy figure triple when the mass tripled. Both are correct, and reconciling them is one of the more useful exercises in elementary mechanics.
Setting potential energy equal to kinetic energy gives mgh = ½mv2. The mass appears on both sides and cancels, leaving v = √(2gh). A heavier object stores more energy at the same height, but it also needs more energy to reach any given speed, and those two effects cancel exactly.
This is the same result Galileo is credited with, and it holds precisely in a vacuum. In air it fails, not because the physics changes but because drag is an additional force that depends on shape and speed rather than on mass. A feather and a hammer fall at the same rate on the Moon and manifestly do not on Earth. For dense compact objects over modest drops, the vacuum answer is close enough to be useful; the free fall calculator covers the motion in more detail and the drag force calculator covers what air does to it.
The Zero Point Is Yours to Choose
Ask what the potential energy of a book on a table is and there is no answer until someone nominates a reference. Relative to the table, zero. Relative to the floor, a few joules. Relative to the bottom of the building, rather more. Relative to the centre of the Earth, a very large number indeed, and one this formula is not valid for anyway.
None of those is more correct than the others, because no experiment can measure a potential energy — only changes in it, which appear as work done or kinetic energy gained. Any consistent choice of zero produces identical predictions. The practical rule is to put zero wherever it makes the arithmetic simplest, which is usually the lowest point the object will reach.
Negative values are legitimate and often useful. A submarine 300 metres below sea level has a negative potential energy relative to the surface, which correctly says that raising it to the surface requires work. This calculator takes positive heights for clarity, but the physics is symmetric and a descent simply reverses the sign of every energy transfer.
Where mgh Stops Being Valid
The formula assumes gravity is constant over the height involved, which it is not. Gravitational attraction falls with the square of distance from the centre of the planet, so at altitude the true value is slightly lower than at the surface. At the height of a low orbit, roughly 400 kilometres, gravitational acceleration has already dropped by about eleven per cent.
For everyday heights the error is invisible. Over one kilometre of altitude the change is about three hundredths of a per cent, far smaller than the uncertainty in most inputs. The rule of thumb is that mgh is safe while the height is a small fraction of the planet's radius, which for Earth means anything up to a few tens of kilometres.
Beyond that, the correct expression involves the full inverse-square law and the potential energy approaches a finite limit rather than growing without bound. That limit is what makes escape velocity a finite number. The gravitational force calculator handles the inverse-square force and the escape velocity calculator handles the energy needed to leave entirely.
Storing Energy by Lifting Things
Pumped-storage hydroelectricity is gravitational potential energy used industrially, and running the numbers shows why it needs mountains and lakes rather than warehouses. One kilowatt hour is 3.6 million joules. Raising a mass through 100 metres stores about 981 joules per kilogram, so banking a single kilowatt hour needs roughly 3,670 kilograms of water lifted 100 metres — nearly four cubic metres for one unit of electricity.
That is a poor energy density by any chemical standard, and it is why gravity storage is only ever built at enormous scale with cheap working mass and free terrain. It is also why the technology persists despite that: water is free, the round-trip efficiency is high, the plant lasts for decades and nothing degrades with cycling.
The same arithmetic scales down usefully. A grandfather clock weight, a trebuchet counterweight and a raised loading platform are all the same calculation. Solving for height in the tool answers the design question directly: given a mass you have and an energy you need, how far up does it have to go? Use the energy converter to move between joules, kilowatt hours and calories, and the weight converter for mass units.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering a weight as a mass — a figure in newtons or kilograms-force already contains gravity, so using it here multiplies by gravity a second time.
- Changing the zero level mid-problem — the reference is free to choose but must stay fixed, or the energy difference you compute will be meaningless.
- Expecting the impact speed to depend on mass — it does not, because mass cancels when potential energy is equated with kinetic energy.
- Using mgh at orbital altitudes — gravity falls with the square of distance, and the constant-g assumption breaks down once the height is a noticeable fraction of the planet's radius.
- Treating the fall figures as real-world predictions — they assume a vacuum, and air resistance matters enormously for anything light, large or falling a long way.
Related Free Tools From Arb Digital
For the energy of motion use the kinetic energy calculator, and for energy transferred by a force the work calculator. Falling motion is covered by the free fall calculator and the drag force calculator, while the gravitational force calculator and the escape velocity calculator extend beyond constant gravity. The momentum calculator covers the other conserved quantity in a collision, the Hooke's law calculator covers energy stored in a spring instead of a height, and the energy converter and weight converter handle units. The full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
Potential energy equals mass multiplied by gravitational acceleration multiplied by height, written U = mgh. Mass is in kilograms, gravity in metres per second squared and height in metres, giving energy in joules.
From any level you choose, as long as you use the same one throughout. Only differences in potential energy have physical meaning, so the reference cancels out of every prediction. The lowest point the object reaches is usually the simplest choice.
Because setting mgh equal to half m v squared lets the mass cancel from both sides, leaving v as the square root of twice gravity times height. A heavier object stores more energy but also needs more energy to reach any given speed.
Standard gravity is defined as exactly 9.80665 metres per second squared and is the right default for Earth. The true local value varies by roughly half a per cent between the equator and the poles, and by altitude, which rarely matters at everyday precision.
Yes. Anything below the chosen zero level has negative potential energy, which correctly indicates that work must be done to bring it back up. The sign is a consequence of where the reference was placed, not a property of the object.
Only while gravity is effectively constant, which means heights that are a small fraction of the planet's radius. At 400 kilometres, roughly low orbit, gravitational acceleration has already fallen by about eleven per cent and the full inverse-square treatment is needed.
A kilowatt hour is 3.6 million joules, and raising one kilogram by 100 metres stores about 981 joules, so roughly 3,670 kilograms — nearly four cubic metres of water — lifted 100 metres. That poor energy density is why gravity storage is only built at very large scale.
This tool is provided for educational and estimating use. The fall figures assume a vacuum and constant gravity, and nothing on this page is engineering, safety or working-at-height guidance.
