About gravitational potential energy
Gravitational potential energy (GPE) is the energy an object stores purely because of its height above a chosen reference level. Lift a book onto a shelf and you do work against gravity; that work does not vanish, it is banked in the book as potential energy and is released the instant the book falls. The calculator above turns the three quantities that set this energy, mass, gravity, and height, into a single number in joules, and also shows the speed the object would reach if it fell, because potential energy and the kinetic energy of a fall are two views of the same thing.
The concept matters far beyond the physics classroom. It is the reason a hydroelectric dam can power a city: water held behind the dam carries GPE that turbines convert to electricity as it drops. It governs the energy a roller coaster trades between its highest hill and its fastest dip, the impact a falling tool delivers on a construction site, and the work a pump must do to raise water to a tank. Any time something is lifted, held high, or allowed to fall, gravitational potential energy is the bookkeeping that tells you how much energy is in play.
Because GPE depends on a reference height, the value is always relative. The energy of a book "five metres up" only means five metres above whatever floor, table, or sea level you decide to measure from. In practice you pick the lowest point the object can reach, usually the ground or the bottom of a fall, and measure height from there. The calculator follows that convention, treating the height you enter as the drop available to convert into motion.
How it works
Gravitational potential energy is the product of three quantities. The work needed to raise a mass through a height equals force times distance, and near a planet's surface the gravitational force on a mass is simply its weight, mass times gravitational acceleration. Multiply by the height lifted and you have the stored energy:
The second formula comes from conservation of energy. As the object falls, every joule of potential energy converts into kinetic energy, 1/2 m v squared. Setting the two equal and solving for speed gives v = sqrt(2 g h), and notice the mass cancels out: a heavy and a light object dropped from the same height hit the ground at the same speed in a vacuum, exactly as Galileo argued. Air resistance is what breaks that equality in the real world.
Worked example
Take a 10 kg mass held 5 metres above the ground on Earth, where g = 9.81 m/s squared.
- Multiply mass by gravity: 10 x 9.81 = 98.1 newtons (this is the object's weight).
- Multiply by height: 98.1 x 5 = 490.5 joules of stored potential energy.
- Find the impact speed: v = sqrt(2 x 9.81 x 5) = sqrt(98.1) = 9.9 m/s, about 22 mph.
- Check the energy balance: kinetic energy at impact = 1/2 x 10 x 9.9 squared = 490.5 J, matching the potential energy exactly.
Reference: energy and impact speed by height
For a 10 kg mass on Earth (g = 9.81 m/s squared), the stored energy and the speed it would reach in a vacuum at the end of the fall.
| Height | Potential energy (10 kg) | Impact speed |
|---|---|---|
| 1 m | 98 J | 4.4 m/s (10 mph) |
| 5 m | 491 J | 9.9 m/s (22 mph) |
| 10 m | 981 J | 14.0 m/s (31 mph) |
| 20 m | 1,962 J | 19.8 m/s (44 mph) |
| 50 m | 4,905 J | 31.3 m/s (70 mph) |
| 100 m | 9,810 J | 44.3 m/s (99 mph) |
A falling human reaches a terminal velocity of roughly 53 m/s (about 120 mph) in air, so beyond a few hundred metres real-world fall speeds stop tracking the vacuum formula as drag balances gravity.
Common pitfalls
- Using the wrong gravity. The 9.81 m/s squared default is Earth at sea level. The Moon is 1.62, Mars is 3.72, and Jupiter is 24.8. Change g for any other body, or the energy will be off by the same ratio.
- Mixing units. The formula needs kilograms and metres for an answer in joules. Pounds, grams, or feet will give a number that is not in joules; convert first.
- Forgetting the reference level. Height must be measured from the point the object falls to, not from sea level or some arbitrary zero. The "height" is the drop available, not the altitude.
- Expecting heavier objects to fall faster. Impact speed from the formula does not depend on mass at all. A bowling ball and a marble reach the same speed in a vacuum; only air resistance separates them.
- Assuming the vacuum speed in air. v = sqrt(2gh) ignores drag. For light objects or long falls, real impact speeds are lower because air resistance caps them at terminal velocity.
Frequently asked questions
What is the formula for gravitational potential energy?
Gravitational potential energy is PE = m x g x h, where m is mass in kilograms, g is gravitational acceleration (9.81 m/s squared on Earth), and h is height in metres above the reference level. The result is in joules. A 10 kg mass at 5 m holds 10 x 9.81 x 5 = 490.5 J.
Does a heavier object hit the ground faster?
Not in a vacuum. The impact speed from a fall is v = sqrt(2 g h), and mass cancels out entirely, so a heavy and a light object dropped from the same height reach the same speed. In air, drag slows lighter objects more, which is the only reason a feather falls slower than a stone on Earth.
What value of gravity should I use for the Moon or Mars?
Replace the gravity field with the local value: the Moon is 1.62 m/s squared, Mars is 3.72, and Jupiter is 24.8, against Earth's 9.81. The same mass at the same height stores proportionally less energy on the Moon, about one sixth of its Earth value, because g is roughly six times smaller.
Why does doubling the height not double the impact speed?
Potential energy grows in direct proportion to height, so doubling the height doubles the energy. Speed, however, follows v = sqrt(2 g h), so it grows with the square root of height. Doubling the height multiplies the impact speed by sqrt(2), about 1.41 times, not 2.
Why is potential energy always relative to a reference height?
Only differences in height carry physical meaning, so you must choose a zero level to measure from. The same object has different GPE values depending on whether you measure from the floor, a table, or sea level. In practice the reference is the lowest point the object can reach, which makes the GPE equal to the energy released in the fall.