Potential Energy Calculator

Calculate gravitational potential energy, mass, or height using PE = mgh. Supports all common units with instant conversion.

🏔️ Gravity📐 PE = mgh⚡ Energy
Mass (m)
Unit
Height (h)
Unit
Gravitational acceleration g = 9.8 m/s² (Earth). For other bodies: Moon = 1.62, Mars = 3.72, Jupiter = 24.8 m/s².
⚠️ Please enter valid positive numbers for all required fields.

What Is Potential Energy?

Potential energy (PE) is stored energy that an object possesses due to its position or configuration relative to other objects. Unlike kinetic energy, which is energy of motion, potential energy represents the capacity to do work — it hasn't been converted to motion yet, but it will be when conditions allow.

The most common form in introductory physics is gravitational potential energy, given by PE = mgh, where m is mass (kg), g is gravitational acceleration (9.8 m/s² on Earth), and h is height above a reference level (m). The reference level is arbitrary — you choose it. Often the ground, the table surface, or the lowest point of the problem is set as h = 0.

A second important form is elastic potential energy (stored in springs and elastic materials): PE_elastic = ½kx², where k is the spring constant and x is the compression or extension from equilibrium. This formula mirrors the kinetic energy equation KE = ½mv².

The power of potential energy comes from the law of conservation of mechanical energy: in the absence of friction and other dissipative forces, the total mechanical energy (KE + PE) remains constant throughout motion. This is why a roller coaster car at the top of a hill (maximum PE, minimum KE) moves fastest at the bottom (minimum PE, maximum KE) — the energy simply changes form. Knowing PE at any height instantly gives you the velocity anywhere else on the track.

Formula Reference Table

FormFormulaVariablesSI Unit
Gravitational PEPE = m·g·hm = mass, g = 9.8 m/s², h = heightJ
Find Massm = PE / (g·h)PE = energy, h = heightkg
Find Heighth = PE / (m·g)PE = energy, m = massm
Elastic PE (spring)PE = ½·k·x²k = spring constant, x = displacementJ
Conservation of EnergyPE₁ + KE₁ = PE₂ + KE₂At any two points in the systemJ
Velocity from PE dropv = √(2gh)When KE₁ = 0 (starts at rest)m/s

3 Worked Examples

Example 1
Person on a Cliff — Find Gravitational PE

A 70 kg person stands at the edge of a 45-meter cliff. What is their gravitational PE relative to the base?

  • Given: m = 70 kg, g = 9.8 m/s², h = 45 m
  • Apply: PE = m × g × h = 70 × 9.8 × 45
  • Result: PE = 30,870 J = 30.87 kJ
✓ Gravitational PE = 30,870 J (30.87 kJ)
Example 2
Roller Coaster — Find Speed at Bottom Using Conservation

A coaster car (500 kg) starts from rest at 40 m height. Ignoring friction, what is its speed at the bottom?

  • At top: PE = mgh = 500 × 9.8 × 40 = 196,000 J; KE = 0
  • Conservation: Total energy = 196,000 J at all points
  • At bottom: h = 0, so PE = 0; all energy is KE = ½mv²
  • Solve for v: v = √(2gh) = √(2 × 9.8 × 40) = √784 = 28 m/s
✓ Speed at bottom = 28 m/s (100.8 km/h)
Example 3
Water Tower — Find Mass from PE

A water tank stores 5 × 10⁷ J of gravitational PE at a height of 25 m. What mass of water does it hold?

  • Rearrange: m = PE / (g × h)
  • Substitute: m = (5 × 10⁷) / (9.8 × 25)
  • Calculate: m = 5 × 10⁷ / 245 = 204,082 kg ≈ 204 tonnes
  • At 1 kg/L, that's ≈ 204,000 litres of water
✓ Mass of water ≈ 204,082 kg (204 tonnes)

Real-World Applications

💧
Hydroelectric Power
Dams convert water's gravitational PE into electricity. The stored PE = mgh with enormous m (billions of kg) and h (dam height). The Hoover Dam generates ~4 billion kWh/year from this principle.
🎢
Roller Coasters
Engineers calculate PE at the first hill to ensure all subsequent hills are shorter, guaranteeing enough speed throughout (PE converts to KE and back, minus friction losses).
🏗️
Construction & Cranes
PE = mgh tells engineers the minimum energy needed to lift materials. A 500 kg beam raised 20 m requires at least 98 kJ — used to size crane motors and cables.
🌍
Tidal & Pumped Storage
Pumped hydro is the world's largest form of energy storage: excess grid electricity pumps water uphill (storing PE) and releases it through turbines when demand spikes.
🪂
Skydiving & Free Fall
A 90 kg skydiver at 4,000 m has ~3.5 MJ of gravitational PE relative to the ground. This converts entirely to KE and heat (via air drag) by the time they land.

Common Mistakes to Avoid

⚠️
Using an inconsistent reference level

PE is always relative to a chosen zero height. If you set h = 0 at the ground for one object but at the table for another in the same problem, your energy comparisons will be wrong. Pick one reference and stick to it throughout.

⚠️
Forgetting to convert units

PE = mgh requires m in kg, g in m/s², and h in meters to get joules. Using grams for mass or feet for height without conversion produces wrong answers — often by factors of 1,000 or more.

⚠️
Confusing potential energy with force

Weight (W = mg) is force in newtons; potential energy (PE = mgh) is energy in joules. They share the factors m and g but are different quantities with different units and physical meanings.

⚠️
Applying conservation of energy when friction is present

PE + KE = constant only when there are no non-conservative forces (like friction or air resistance). In real problems, energy is lost to heat: PE₁ + KE₁ = PE₂ + KE₂ + E_loss.

⚠️
Using g = 10 instead of 9.8 m/s²

g = 10 is a useful approximation for mental math, but it introduces a 2% error. In exam problems and engineering calculations, use g = 9.8 m/s² (or 9.81 m/s² for higher precision).

⚠️
Negative height causing confusion

PE can be negative if h is measured below the reference level (e.g., a mine shaft below ground level). This is physically valid — it just means the object would need to gain PE to reach the reference point.

Frequently Asked Questions

What is the difference between potential energy and kinetic energy? ▾
Kinetic energy (KE = ½mv²) is the energy an object has because it is moving. Potential energy is energy stored due to position or configuration — gravitational PE from height, elastic PE from compression, chemical PE in bonds. In a conservative system, they convert into each other while their sum (total mechanical energy) remains constant.
Why does PE = mgh only work near Earth's surface? ▾
PE = mgh assumes g is constant, which is a good approximation within a few hundred km of Earth's surface (g varies less than 0.5% from sea level to 10 km altitude). For objects far from Earth (satellites, space probes), the correct formula is gravitational PE = −GMm/r, which accounts for g weakening with distance.
What is elastic potential energy and how is it different? ▾
Elastic PE (½kx²) is stored in deformed springs, rubber bands, and elastic materials. Unlike gravitational PE (depends on height), elastic PE depends on how much the object is stretched or compressed from its natural length (x) and the spring constant k (stiffness). Both types can convert to kinetic energy — a compressed spring launches a projectile just as a dropped weight accelerates toward the ground.
How is gravitational PE related to work? ▾
The change in gravitational PE equals the negative of work done by gravity: ΔPE = −W_gravity. When you lift a book, you do positive work on it (W = mgh); the book gains PE = mgh. When the book falls, gravity does positive work and PE decreases as KE increases. This is why PE and work are measured in the same unit (joules).
Does PE depend on the path taken to reach a height? ▾
No — gravitational PE is path-independent. Climbing straight up vs. taking a winding ramp to reach the same height gives the same PE = mgh. This is a hallmark of conservative forces. Friction, however, is non-conservative: the work done against friction depends on path length.
How does g differ on other planets? ▾
This calculator lets you enter a custom g value for exactly this. Surface g values: Moon = 1.62 m/s² (17% of Earth), Mars = 3.72 m/s², Venus = 8.87 m/s², Jupiter = 24.8 m/s², Saturn = 10.4 m/s², Sun = 274 m/s². A 70 kg person on the Moon has PE = 70 × 1.62 × h — much less PE per meter of height.
What is gravitational potential (V) vs. gravitational PE? ▾
Gravitational potential V = PE/m = gh (J/kg), the PE per unit mass at a given height. It's a field property — independent of the mass of the object placed there. Gravitational PE = m × V is the energy for a specific mass m. Potential (V) is used in field theory and when analyzing how different masses would behave in the same gravitational field.
Can potential energy be negative? ▾
Yes. If an object is below your reference height (h is negative), PE = mgh is negative. This is physically meaningful — it means the object is in a lower-energy state than the reference. In the rigorous gravitational formula PE = −GMm/r, PE is always negative and approaches zero only at infinite separation. Negative PE simply means the object is bound to the gravitational field.

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