Last updated: 2026-05-18

Definition

Gravitational potential energy is the energy an object possesses because of its position above a reference level. Raising a book to a higher shelf does work against gravity, and that work is stored as potential energy in the book-Earth system, ready to be released when the book falls.

Potential energy is energy of position, not energy of motion. A boulder at the edge of a cliff has large potential energy even when it is completely still. The moment it falls, that stored energy converts into kinetic energy.

The formula: PE = mgh

Gravitational potential energy near Earth's surface is calculated with a single equation:

PE = m × g × h

Symbol	Quantity	SI unit
PE	Gravitational potential energy	Joule (J)
m	Mass of the object	Kilogram (kg)
g	Standard gravitational acceleration	9.80665 m/s²
h	Height above the reference plane	Metre (m)

The formula follows directly from the definition of work: raising a mass m by height h against gravity requires a force mg over distance h, so work done = mgh. That work is stored as potential energy.

The standard value of g

The constant 9.80665 m/s² is the standard acceleration of gravity adopted by the 3rd General Conference on Weights and Measures (CGPM) in 1901. It represents the mean gravitational acceleration at sea level at a geodetic latitude of approximately 45°.

In practice, surface gravity varies slightly:

• Equator: ~9.764 m/s² (weaker — farther from Earth's centre, centrifugal effect)
• Poles: ~9.834 m/s² (stronger — closer to Earth's centre)

For everyday calculations, laboratories, and engineering work, 9.80665 m/s² is the universally accepted standard, and it is what this calculator uses.

Contrast with kinetic energy

Kinetic energy is the energy of motion: KE = ½mv², where v is speed. Potential and kinetic energy are two sides of the same coin in classical mechanics:

Property	Potential energy	Kinetic energy
Source	Position	Motion
Formula	PE = mgh	KE = ½mv²
Zero when	h = 0 (reference level)	v = 0 (at rest)
Unit	Joule (J)	Joule (J)

The total mechanical energy of a system is E = PE + KE. When no non-conservative forces (friction, air resistance) act, E is constant — a principle called conservation of mechanical energy.

Energy conservation: PE converts to KE

When an object falls freely from rest at height h, its potential energy converts entirely into kinetic energy. Setting PE = KE:

mgh = ½mv²

The mass cancels, giving the impact speed:

$v = \sqrt{2gh}$

For h = 30 m: $v = \sqrt{2 \times 9.80665 \times 30} \approx 24.3$ m/s (about 87 km/h).

Roller coaster example

A roller coaster car at the top of a 40 m hill (relative to the lowest point of the track) has:

PE = m × 9.80665 × 40 = 392.3 m joules

where m is the mass of the car and passengers. At the bottom of the hill, ignoring friction, all of that has become kinetic energy. The car reaches $v = \sqrt{2 \times 9.80665 \times 40} \approx 28$ m/s (about 100 km/h).

In reality, friction and air resistance dissipate some energy as heat, so the actual speed is slightly lower. Engineers account for this with efficiency factors, but PE = mgh gives the theoretical maximum.

Pendulum example

A pendulum swings between two extreme positions where it is momentarily at rest (KE = 0, all energy is PE) and a lowest point where PE = 0 (taking the lowest point as the reference) and all energy is kinetic. At every intermediate angle, PE + KE equals the total energy at the release point. This is why a pendulum swings to the same height on both sides (if undamped) — energy conservation demands it.

The reference plane

PE = mgh only gives a meaningful absolute value once the reference plane (h = 0) is fixed. Physics only ever requires differences in PE, so the choice of reference cancels out.

Example: A 2 kg book sits on a 1 m high table. The table stands on a floor that is 5 m above sea level.

• Relative to the table: PE = 0 (h = 0 by definition)
• Relative to the floor: PE = 2 × 9.80665 × 1 = 19.6 J
• Relative to sea level: PE = 2 × 9.80665 × 6 = 117.7 J

Each value is correct for its chosen reference. When the book falls off the table onto the floor, it loses exactly 19.6 J of PE regardless of which reference was used, and that energy appears as kinetic energy (and then sound and heat on impact).

A common convention is to set h = 0 at the lowest point in the problem, so all PE values are positive and easy to track.

Worked example

A 5 kg object is lifted from the floor to a shelf 2.4 m above the floor. How much gravitational PE has it gained?

PE = m × g × h = 5 × 9.80665 × 2.4 = 117.68 J

That is also the minimum work done against gravity to lift it. If the object were released and fell freely back to the floor, it would arrive with a kinetic energy of 117.68 J and a speed of:

$v = \sqrt{2 \times 9.80665 \times 2.4} = \sqrt{47.07} \approx$ 6.86 m/s

Limitations of PE = mgh

The formula assumes g is constant. This holds accurately when the height change is small compared to Earth's radius (~6371 km):

• Heights up to a few kilometres: error < 0.1 % — use PE = mgh freely.
• Satellite orbits or spacecraft trajectories: g decreases significantly with altitude; use the full gravitational potential energy formula PE = −GMm/r, where G is the gravitational constant, M is Earth's mass, and r is the distance from Earth's centre.

For everything from buildings and bridges to sports biomechanics and roller coaster design, PE = mgh is the right tool.

Frequently Asked Questions (FAQ)

What is the formula for gravitational potential energy?

Gravitational potential energy is PE = mgh, where m is the mass in kilograms, g is the standard gravitational acceleration (9.80665 m/s²), and h is the height above the reference point in meters. The result is in joules (J). For example, a 5 kg object raised 2 m has PE = 5 × 9.80665 × 2 ≈ 98.07 J.

Why is g = 9.80665 m/s²?

The value 9.80665 m/s² is the standard acceleration of gravity defined by the 3rd General Conference on Weights and Measures (CGPM) in 1901. It represents the mean gravitational acceleration at sea level at a latitude of approximately 45°. Actual surface gravity varies from about 9.764 m/s² at the equator to 9.834 m/s² at the poles due to Earth's rotation and flattening, but 9.80665 m/s² is the universally accepted standard for calculations.

How does potential energy convert to kinetic energy?

By conservation of energy, gravitational PE converts entirely into kinetic energy (KE = ½mv²) when an object falls freely in the absence of friction and air resistance. A roller coaster car at the top of a 30 m hill has PE = mgh = m × 9.80665 × 30 ≈ 294 m joules. At the bottom, that same energy appears as kinetic energy, giving a speed of v = √(2gh) = √(2 × 9.80665 × 30) ≈ 24.3 m/s (about 87 km/h).

Does potential energy depend on the reference point?

Yes. Gravitational PE is always measured relative to a chosen reference plane (where h = 0). Only differences in PE matter physically — the absolute value depends on where the zero is placed. For a ball on a table, h = 0 can be set at the floor (PE relative to floor) or at the table surface (PE = 0 on the table, negative below it). Physics problems always use the same reference for initial and final states, so the choice cancels out.