Gravitational Potential Energy Calculator

Presets:

Solve for

Mass (m)

Height (h)

Gravity (g)

Energy (Ep)

Large Body (M)

M = 5.972 × 10²⁴ kg

Small Mass (m) — kg

Separation (r)

What Is Gravitational Potential Energy?

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It represents the work done against gravity to bring an object to that position from a reference point. When the object is released, this stored energy converts to kinetic energy.

GPE is a form of potential energy — it is latent, waiting to be released. A boulder perched on a cliff has enormous GPE. A compressed spring has elastic PE. Both store energy by virtue of position or configuration.

Ep = mgh

Near-surface formula (constant g)

U = −GMm/r

General two-body formula

G = 6.674×10⁻¹¹

N·m²/kg² — gravitational constant

Near-Surface Formula: Ep = mgh

Close to Earth's surface (or any planet), the gravitational field strength g is approximately constant. The change in GPE is simply:

Ep = m × g × h

m — mass (kg)

The object's mass. Heavier objects store more GPE at the same height.

g — gravity (m/s²)

Gravitational acceleration. Earth: 9.81, Moon: 1.62, Mars: 3.71 m/s².

h — height (m)

Height above the reference level. The choice of reference is arbitrary — only changes in GPE matter.

The formula is rearrangeable: m = Ep/(g·h), h = Ep/(m·g), g = Ep/(m·h). You can also find impact velocity: v = √(2·g·h).

General Formula: U = −GMm/r

When the separation between two masses is large — or when precision matters — use Newton's law of gravitation to compute the potential energy of the two-body system:

U = −G · M · m / r

Symbol	Meaning	Value / Unit
G	Universal gravitational constant	6.6743 × 10⁻¹¹ N·m²/kg²
M	Mass of large body (planet/star)	kg
m	Mass of small body	kg
r	Center-to-center separation	m

The negative sign indicates that gravity is attractive. U increases (becomes less negative) as r increases — the system has more energy when the masses are far apart. U = 0 at r = ∞.

Conservation of Energy & GPE

In the absence of friction and air resistance, mechanical energy is conserved:

KE₁ + PE₁ = KE₂ + PE₂ → ½mv₁² + mgh₁ = ½mv₂² + mgh₂

When an object falls from height h₁ to h₂ = 0, all GPE converts to kinetic energy. The final speed is:

v = √(2 · g · h)

This means a 70 kg person falling 10 m hits the ground at about 14 m/s (50 km/h) — regardless of mass. GPE is also the basis for hydroelectric power stations, pendulums, roller coasters, and spring-powered mechanisms.

Escape Velocity

Escape velocity is the speed at which an object's kinetic energy exactly equals the magnitude of its GPE at a given distance from a massive body:

v_esc = √(2GM / r)

Body	Mass (kg)	Radius (km)	v_esc (km/s)
Earth	5.972 × 10²⁴	6,371	11.19
Moon	7.342 × 10²²	1,737	2.38
Mars	6.417 × 10²³	3,390	5.03
Jupiter	1.898 × 10²⁷	69,911	59.5
Sun	1.989 × 10³⁰	695,700	617.5

Worked Examples

Example 1 — Person climbing stairs (70 kg, 10 m, Earth)

Given: m = 70 kg, h = 10 m, g = 9.81 m/s²

Formula: Ep = m × g × h

Step 1: Ep = 70 × 9.81 × 10

Answer: Ep = 6,867 J ≈ 6.87 kJ

Example 2 — Apple falling from 1.5 m (0.1 kg)

Given: m = 0.1 kg, h = 1.5 m, g = 9.81 m/s²

Formula: Ep = m × g × h

Step 1: Ep = 0.1 × 9.81 × 1.5

Answer: Ep = 1.47 J | Impact v = √(2×9.81×1.5) ≈ 5.42 m/s

Example 3 — Satellite at LEO (1000 kg, r = 6,771 km from Earth center)

Given: G = 6.6743×10⁻¹¹, M = 5.972×10²⁴ kg, m = 1000 kg, r = 6.771×10⁶ m

Formula: U = −G·M·m/r

Step 1: GMm = 6.6743×10⁻¹¹ × 5.972×10²⁴ × 1000 = 3.986×10¹⁷

Step 2: U = −3.986×10¹⁷ / 6.771×10⁶

Answer: U ≈ −5.887×10¹⁰ J ≈ −58.87 GJ

Example 4 — Height to reach if launched at 100 m/s

Given: v = 100 m/s, g = 9.81 m/s²

Formula: h = v² / (2g)

Step 1: h = 10,000 / (2 × 9.81) = 10,000 / 19.62

Answer: h ≈ 509.9 m

Planet & Object Reference Table

Body	g (m/s²)	Mass (kg)	Radius (km)
Earth	9.81	5.972 × 10²⁴	6,371
Moon	1.62	7.342 × 10²²	1,737
Mars	3.71	6.417 × 10²³	3,390
Jupiter	24.79	1.898 × 10²⁷	69,911
Sun	274	1.989 × 10³⁰	695,700
Venus	8.87	4.867 × 10²⁴	6,052

Frequently Asked Questions

Gravitational potential energy (GPE) is the energy stored in an object due to its position in a gravitational field. Near Earth's surface it is Ep = mgh, where m is mass, g is gravitational acceleration (9.81 m/s²), and h is height above the reference point. It represents the work done against gravity to lift the object to that height.

In the two-body formula U = −GMm/r, the potential energy is negative because the reference point (U = 0) is set at infinity. As two masses come closer together (r decreases), they release energy, so the system's potential energy decreases below zero. A negative value means the system is bound — you must add energy to separate the masses.

Use Ep = mgh for objects near Earth's surface where height changes are small compared to Earth's radius (below ~100 km). The formula treats g as constant. Use U = −GMm/r for large separations, orbital mechanics, space travel, or when dealing with two bodies of comparable mass, because it accounts for how gravity weakens with distance.

Escape velocity is the minimum speed an object needs to escape a planet's gravity without further propulsion. It equals v_esc = √(2GM/r), where G is the gravitational constant, M is the planet's mass, and r is the distance from the planet's center. For Earth at its surface, escape velocity ≈ 11.19 km/s.

No. Gravitational potential energy is a conservative quantity — it depends only on the object's position (height or distance from the center of mass), not on the path used to reach that position. This means the work done by gravity around any closed path is zero.

By the conservation of mechanical energy: Ep + KE = constant. When an object falls, GPE decreases and kinetic energy increases by the same amount. At impact, if we ignore air resistance, all GPE has become KE. The impact velocity is v = √(2gh) for the near-surface case.

GPE is measured in joules (J) in SI units. One joule equals 1 kg·m²/s² = 1 N·m. For everyday objects, kilojoules (kJ) are common. Other units include calories (1 cal = 4.184 J), kilocalories (kcal), and foot-pounds (ft·lb = 1.356 J).

Using the general formula U = −GMm/r, as r → ∞, U → 0. The gravitational potential energy of any object at infinite separation from all other masses is zero. This is why bound systems (planets, moons, satellites) always have negative total potential energy — they must absorb energy to reach U = 0 at infinity.