You're Lifting a Box onto a Shelf and Wonder How Much Energy You're Storing
When you lift an object against gravity, you store energy in it. The higher you lift it, the more energy it has. If you drop it, that stored energy converts to kinetic energy and heat on impact. This stored energy is called gravitational potential energy. It depends on three things: the object's mass, the height above a reference point, and the gravitational acceleration. The formula PE = mgh is simple but powerful-it explains why dropping a heavy object from a high shelf is so destructive.
What This Calculator Does
This calculator computes gravitational potential energy from mass, height, and gravitational acceleration (which varies by location—9.81 m/s² on Earth, 1.62 m/s² on the Moon, 3.71 m/s² on Mars). You can also find the missing variable: what height is needed for a certain energy, or what mass has a given PE at a specific height. It shows impact speed if the object falls, and compares energy to familiar amounts like food calories.
How to Use This Calculator
Mass (m): Enter the object's mass in kilograms (kg) or pounds (lb).
Height (h): Enter the height above the reference level (usually the ground) in meters (m), feet (ft), or other units. The reference height is where PE = 0.
Gravitational Acceleration (g): Select the location: Earth (9.81 m/s²), Moon (1.62 m/s²), Mars (3.71 m/s²), or enter a custom value. This is the local gravitational field strength.
Potential Energy (PE): Enter the potential energy in joules (J), kilocalories (kcal), or watt-hours (Wh).
Enter any three values, and the calculator solves for the fourth. It also shows the impact speed if the object falls freely from that height, and energy equivalents.
The Formula Behind the Math
Gravitational potential energy is:
PE = m × g × h
Where:
Rearranging to solve for each variable:
h = PE / (m × g) (height for a given PE)
m = PE / (g × h) (mass needed for a given PE at a height)
g = PE / (m × h) (gravitational field strength)
When an object falls freely from height h, all potential energy converts to kinetic energy (ignoring air resistance):
PE = KE = ½mv²
Solving for impact velocity:
v = √(2 × g × h)
This is the speed at which an object hits the ground after falling from height h.
Worked Example:
A 2 kg book sits on a shelf 1.5 meters above the ground. What is its potential energy?
Now suppose the book falls. What is its speed at impact?
The book hits the ground at about 5.4 m/s (about 19 km/h or 12 mph). This energy is absorbed as deformation and sound.
Our calculator does all of this instantly, but now you understand exactly what it's computing.
Water and Hydroelectric Power
Hydroelectric dams store water at height. The potential energy depends on the water's mass and the height (head) of the dam. As water falls through turbines, PE converts to kinetic energy, which drives the turbine and generates electricity. A dam storing 1 million cubic meters of water (1 × 10⁹ kg) at a height of 100 meters stores about 1 × 10¹⁴ joules of potential energy-equivalent to roughly 25 megatons of TNT. This is why dams are such efficient power sources.
Roller Coasters and Amusement Park Design
A roller coaster climbs to a high point (high PE) then drops (converting PE to KE). The highest point on the track determines the maximum speed at the lowest point (assuming no friction). Designers calculate PE at each point to ensure safety and maximize thrills. Loop heights, drop angles, and speeds all depend on potential energy conversions.
Tidal and Gravitational Potential Energy
The Moon orbits Earth, creating tides. The gravitational potential energy between Earth and Moon affects their orbital mechanics. Larger bodies at greater distances have different PE. This concept extends to astrophysics: the binding energy of galaxies and the energy released in supernovae depend on gravitational PE.
Energy Storage and Pumped Hydro
Pumped hydro storage is a battery technology: during low-demand hours, water is pumped uphill (using excess electricity). During high-demand hours, water flows downhill through turbines, generating electricity. The stored energy is PE. A system pumping water 500 meters high stores enormous energy cheaply and can dispatch it on demand.
Tips and Things to Watch Out For
PE depends on the reference height. If you set the ground as height 0, then objects on the ground have PE = 0. If you set a higher reference (like a table), objects on the ground have negative PE. This doesn't matter for physics calculations-what matters is the change in PE (ΔPE) between two heights, which is independent of reference.
PE is always relative to a reference point. You can't say an object "has PE = 500 J" without specifying the reference. It's more precise to say "PE = 500 J relative to ground level."
Don't confuse PE and KE. As an object falls, PE decreases and KE increases. Total mechanical energy (PE + KE) stays constant (ignoring air resistance). At maximum height, all energy is PE. At impact, all energy is KE (and heat/sound).
Gravitational field strength varies with location and altitude. On Earth's surface, g ≈ 9.81 m/s². At high altitude, g is slightly smaller. On the Moon, g ≈ 1.62 m/s². Always use the correct value for your location.
The formula PE = mgh is only valid for constant g. Near Earth's surface (up to a few kilometers), g is nearly constant. For great heights (hundreds of km into space), use the gravitational potential energy formula from general relativity: PE = −G × m₁ × m₂ / r.
Frequently Asked Questions
What's the relationship between potential energy and kinetic energy?
They're complementary forms of mechanical energy. As an object falls, PE converts to KE. Total mechanical energy = PE + KE = constant (with no air resistance). At the start, all energy is PE. At impact, all energy is KE.
Why is the impact speed v = √(2gh) independent of mass?
The formula comes from PE = KE: mgh = ½mv². Canceling m gives v = √(2gh). Interestingly, the impact speed doesn't depend on mass-only on height. A feather and a bowling ball dropped from the same height hit the ground at the same speed (in a vacuum). Air resistance affects them differently, but pure gravity treats them identically.
How much energy does a meteorite release?
Depends on mass and impact speed. A 1000 kg meteorite at 20 km/s (typical impact velocity) has KE = ½ × 1000 × (20,000)² = 2 × 10¹¹ joules. This is equivalent to about 50 megatons of TNT or roughly 2.5 times the Hiroshima bomb. Large meteorites are catastrophically dangerous.
Can potential energy be negative?
Yes, if you set the reference above the object. If an object is below the reference point, h is negative, so PE is negative. This is just a convention; what matters is the change in PE.
How does potential energy relate to work?
Work done against gravity equals the change in PE: W = ΔPE = mgh (if height changes by h). Lifting a 2 kg object 3 meters high requires work of 2 × 9.81 × 3 ≈ 59 joules. This work is stored as PE.
What's the escape velocity?
Escape velocity is the speed needed to leave a planet's gravitational field entirely. It's derived from setting kinetic energy equal to the negative of gravitational PE: ½mv² = G × m × M / R. For Earth, escape velocity is about 11.2 km/s. It depends on the planet's mass and radius, not on the object's mass.
Related Calculators
Use our Kinetic Energy Calculator to explore conversions between KE and PE. The Acceleration Calculator helps understand how gravity accelerates falling objects. The Gravity Calculator provides gravitational parameters for different celestial bodies. For more physics concepts, explore our Pressure and Density Calculators.