Potential Energy Calculator finds gravitational PE from mass, height and gravity or from weight, solves for mass, height or gravity, and gives the drop speed and mechanical energy.
Enter the mass and its height above the level you treat as zero.
Speed and Units
Formula Triangle
Cover the part you want. PE on top is the three bottom parts multiplied; a bottom part is PE divided by the other two. The shaded part is the one solved here.
Step-by-Step Working
- Formula: PE = m × g × h
- Weight: m × g = 10 kg × 9.80665 m/s² = 98.0665 N
- Multiply by height: PE = 98.0665 N × 50 m = 4,903.325 J
Energy by Height
| Height (m) | Energy (J) | Drop speed (m/s) |
|---|---|---|
| 12.5 | 1,226 | 15.66 |
| 25 | 2,452 | 22.14 |
| 37.5 | 3,677 | 27.12 |
| 50 | 4,903 | 31.32 |
| 75 | 7,355 | 38.35 |
| 100 | 9,807 | 44.29 |
| 150 | 14,710 | 54.24 |
Potential Energy Calculator for Height and Gravity
The Potential Energy Calculator finds the gravitational potential energy stored in a raised object from its mass, its height and the strength of gravity. It also solves the formula for any one of those three when you know the energy, and shows how the stored energy turns into speed when the object falls.
It works as a gravitational potential energy calculator for any planet or height you enter. The Calculate menu has five modes. Potential Energy (PE) is the usual one. PE from Weight and Height takes a force in newtons or pounds-force instead of a mass. The last three modes solve for mass, height or gravity from a known energy.
Mass can be typed in kilograms, grams, metric tons, pounds or ounces. Height can be in meters, centimeters, kilometers, feet, inches or miles. Gravity starts at 9.80665 m/s² and can be changed to any value in m/s², ft/s², standard gravities or cm/s², so the same tool works for the Moon, Mars or a lab measurement.
The default is a 10 kg mass held 50 m up. The Potential Energy Calculator shows 4,903 J stored, a weight of 98.07 N and a speed of 31.32 m/s if the mass is dropped from that height.
Potential Energy Formula
Near the ground, gravitational potential energy is the weight of an object times its height above a chosen zero level. NCERT’s Class 11 physics text writes it as V(h) = mgh in its chapter on work, energy and power, and most textbooks use PE or U for the same quantity.
$$PE = m g h$$
With m in kilograms, g in m/s² and h in meters, PE comes out in joules. To calculate potential energy by hand, multiply the mass by gravity to get the weight, then multiply by the height. For the default, 10 × 9.80665 = 98.07 N, and 98.07 × 50 = 4,903 J.
The step list in the Potential Energy Calculator prints those two lines with your own numbers. The tool uses standard gravity, 9.80665 m/s², the exact value fixed in NIST’s conversion tables, rather than the rounded 9.81 or 9.8 many examples use. Class 11 problems often use 9.8 m/s², so type that into the gravity field when you want to match a textbook answer exactly.
How to Calculate Potential Energy From Force
If you know an object’s weight as a force rather than its mass, the formula is even shorter. Weight already includes gravity, so potential energy is simply the weight times the height.
$$PE = F h$$
That is the PE from Weight and Height mode. A 100 N load raised 50 m stores 5,000 J. The tool also works out the mass behind that weight, 10.2 kg on Earth, which changes if you change gravity, because the same weight means a different mass on another planet.
Potential Energy Triangle
The Formula Triangle block under the results draws the potential energy triangle, with PE on top and m, g and h along the bottom. Cover the quantity you want and the others show how to get it. The part being solved is shaded.
$$m = \frac{PE}{g h} \quad h = \frac{PE}{m g} \quad g = \frac{PE}{m h}$$
With 2,000 J stored at 50 m, the mass is 4.08 kg. With 2,000 J in a 10 kg mass, the height is 20.39 m. With 2,000 J in 10 kg at 50 m, gravity must be 4.0 m/s², well under Earth’s value. Those are the default figures in the three solve-for modes.
What Is Zero Potential Energy?
Potential energy has no fixed starting point. You choose the level that counts as zero height, and every height is measured from there. NCERT says this plainly: the zero of potential energy is arbitrary, but once set it must be used consistently.
Only changes in potential energy have physical meaning. A 10 kg mass lifted from a 2 m table to a 5 m shelf gains 10 × 9.80665 × 3 = 294 J, whether you call the floor zero or the table zero. To find the change between two heights, enter the difference between them as the height.
Negative heights work too. An object below your zero level has negative potential energy, and the first result card then shows the launch speed it would need to climb back up to zero. This is where the lowest potential energy sits in any problem, at the lowest point the object can reach.
Kinetic and Potential Energy
Kinetic energy is energy of motion, KE = ½mv², and potential energy is energy stored by position. The difference between KE and PE is that one depends on speed and the other on height, but both are measured in joules and they turn into each other as an object rises or falls.
$$m g h = \frac{1}{2} m v^2$$
Mass cancels from both sides, so the drop speed is v = √(2gh) for any mass. The first result card shows it. The default 50 m drop reaches 31.32 m/s, which is 112.7 km/h, after 3.19 s with no air resistance.
To calculate KE and PE together, add a current speed in the options panel, and the Potential Energy Calculator works as a mechanical energy calculator. The third result card then shows the mechanical energy, the sum of the two. At 3 m/s the 10 kg mass carries 45 J of kinetic energy on top of its 4,903 J of potential energy, for 4,948 J in all.
OpenStax University Physics explains that with only conservative forces like gravity acting, this total stays fixed. So the moving mass reaches the zero level at 31.46 m/s, a little faster than the dropped one. The Kinetic Energy Calculator handles the motion side on its own.
Potential Energy Unit
The SI unit of both kinetic and potential energy is the joule. One joule is one newton acting through one meter, which is why weight times height gives joules directly. The gravitational potential energy unit is the same joule.
The result unit menu of the Potential Energy Calculator also offers kilojoules, megajoules, foot-pounds, kilocalories, watt-hours and kilowatt-hours. The default 4,903 J reads as 3,617 ft·lbf, 1.172 kcal or 1.362 Wh in the fourth result card.
Potential Energy in kWh
The kilowatt-hour option shows how small everyday potential energy is next to electrical energy. One kWh is 3.6 million joules. Raising one metric ton of water by 100 m stores 980,665 J, only 0.272 kWh.
Turned around, storing a single kilowatt-hour this way means lifting about 3,671 kg of water through 100 m. That is why pumped-storage power plants need whole reservoirs and large height differences to hold useful amounts of energy.
When mgh Stops Working
The formula mgh assumes gravity is the same at every height. OpenStax notes this works well only if g does not change much between the two heights. Far from the ground, the general form takes over, with the zero set at infinite distance.
$$U = -\frac{G M m}{r}$$
OpenStax works the case of lifting a 9,000 kg Soyuz capsule to the 400 km height of the International Space Station. The general formula gives 3.32 × 10¹⁰ J. Plain mgh gives 3.53 × 10¹⁰ J, about 6 percent too high, because gravity at 400 km is weaker than at the surface.
The error grows with height. It is about 0.16 percent at 10 km, 1 percent at 64 km and 1.6 percent at 100 km. For aircraft, mountains and buildings, mgh is fine. The Potential Energy Calculator flags this for you. Once the height passes 63.71 km, 1 percent of Earth’s radius, an amber alert gives the size of the overstatement and the corrected energy for Earth’s gravity. For the Soyuz case it reports 6.28 percent and 3.32 × 10¹⁰ J, matching OpenStax.
The corrected figure uses mgh × R ÷ (R + h), with R = 6,371 km, which equals the exact change in U from the surface. For other gravity values, heights below zero and the solve-for modes, the alert only warns that the result is approximate. For rockets and satellites in general, the Gravitational Force Calculator works with distance from a planet’s center instead.
Types of Potential Energy
Potential energy is stored energy, energy held in a position or arrangement that can be released later. The Potential Energy Calculator deals with gravitational potential energy, the most common type, but there are others.
Elastic potential energy is stored in a stretched or squeezed spring, ½kx², which the Elastic Potential Energy Calculator covers. Electric potential energy is stored by charges held apart. Chemical and nuclear potential energy are stored in the bonds between atoms and inside atomic nuclei.
Lists of three, six or seven types usually split these further, adding magnetic potential energy or treating chemical energy in food and fuel separately. When a question asks for two types, it usually means gravitational and elastic, the two met first in mechanics. When it asks for three, electric is usually added.
Potential Energy Examples
What is a good example of potential energy?
Water held behind a dam is the classic example of gravitational potential energy. A book on a shelf, a car parked at the top of a hill, a raised pile-driver hammer and a roller coaster at the top of its first drop all store energy by height.
A 60 kg person climbing a 3 m flight of stairs stores 60 × 9.80665 × 3 = 1,765 J. That is only 0.42 kcal of food energy, which is why climbing stairs burns far more energy in the muscles than it stores in height.
How can we use potential energy?
Hydroelectric dams release stored water through turbines. Pumped-storage plants lift water uphill when power is cheap and let it fall back when it is needed. Grandfather clocks, pile drivers and gravity-fed water tanks all run on falling weight.
When the stored energy is released in a collision, the Impact Energy Calculator turns it into the force of the hit. The Free Fall Calculator follows the fall itself, height by height.
How do you increase or decrease potential energy?
Raise the object, add mass or move to stronger gravity, and its potential energy rises. Lower it, remove mass or move to weaker gravity, and it falls. Doubling any one of the three doubles the energy, which the Energy by Height table shows for heights from a quarter to three times your entry.
Gravity is the one people forget. The same 10 kg at 50 m stores far less on the Moon. The Weight on Other Planets Calculator gives surface gravity for other worlds, and that value can be typed straight into the gravity field.