Gravitational Potential Energy Calculator

Q: Why is ΔU negative sometimes?

ΔU can be negative if Δh is negative (moving down), or because gravitational potential in space is negative by definition (U = −GMm/r) when zero is set at infinity.

Calculate potential energy near Earth (U = mgh), change in potential energy (ΔU = mgΔh), or use general gravity (U = −GMm/r) for orbits & space problems — with unit conversions, solve-for mode, quick picks, step-by-step, and a mini visual.

Background

Gravitational potential energy depends on mass, gravity, and height (near Earth), or on the distance r from a massive body (general gravity). In many homework problems, you only need ΔU — and the sign tells you whether energy increased or decreased.

Enter values

Calculator mode

Near Earth — U = mgh Change in PE — ΔU = mgΔh General gravity — U = −GMm/r (advanced) General gravity change — ΔU = GMm(1/r₁ − 1/r₂) (advanced)

Tip: Most class problems use ΔU = mgΔh. Use general gravity for satellites & space.

Solve for

Choose what you want to compute. Leave other fields blank only if they’re not required.

Gravity (g)

If you choose Custom, enter g below (supports m/s² or ft/s²).

Mass (m)

Height (h)

Near-Earth convention: choose U = 0 at your reference level, then compute U = mgh.

Height input

Use Δh (height change directly) Use h₁ and h₂ (calculator computes Δh = h₂ − h₁)

Change in height (Δh)

Positive Δh means moving up (energy increases). Negative Δh means moving down.

Energy (U or ΔU)

Only required when you choose a solve-for option like “solve for mass” or “solve for height”.

Options

Show step-by-step

Show mini visual

Display rounding

Rounding affects display only.

Quick picks

Chips prefill common scenarios and calculate immediately.

Result

No results yet. Enter values and click Calculate.

How to use this calculator

Pick a mode: Near Earth, ΔPE, General gravity, or General gravity Δ.
Choose Solve for (energy, mass, height/Δh, gravity, or orbital variables).
Use gravity presets for quick Earth/Moon/Mars/Jupiter values (or custom).
In Change in PE mode, you can input Δh directly or input h₁ & h₂ and the calculator will compute Δh.
Click Calculate for results, plus optional steps and visual.

How this calculator works

Near Earth: U = mgh
Change in PE: ΔU = mgΔh
General gravity: U = −GMm/r
General gravity change: ΔU = GMm(1/r₁ − 1/r₂)

Formula & Equation Used

Near Earth: U = mgh

Change in PE: ΔU = mgΔh

General gravity: U = −GMm/r

General gravity change: ΔU = GMm(1/r₁ − 1/r₂)

Example Problem & Step-by-Step Solution

Example 1 — Change in potential energy

A 2.0 kg book is lifted 1.5 m on Earth. Find ΔU.

Use ΔU = mgΔh.
m = 2.0 kg, g = 9.80665 m/s², Δh = 1.5 m.
ΔU = 2.0 × 9.80665 × 1.5 ≈ 29.4 J.

Example 2 — Near Earth potential energy

A 5.0 kg backpack sits on a shelf 2.0 m above your chosen reference level. Find U on Earth.

Use U = mgh.
m = 5.0 kg, g = 9.80665 m/s², h = 2.0 m.
U = 5.0 × 9.80665 × 2.0 ≈ 98.1 J.

Example 3 — General gravity (advanced)

A 1000 kg satellite is at distance r = 6.371×10⁶ m from Earth’s center. Use M = 5.972×10²⁴ kg. Find U.

Use U = −GMm/r.
Plug in: G = 6.67430×10⁻¹¹, M, m, r.
The result is negative because the reference is U = 0 at infinity.

Frequently Asked Questions

Q: Why is ΔU negative sometimes?

Because Δh can be negative (moving down), or because gravitational potential in space is negative by definition (U = −GMm/r) when zero is set at infinity.

Q: Should I use U = mgh or U = −GMm/r?

Use mgh near Earth for small height changes. Use −GMm/r when the distance from the planet’s center matters (satellites, orbits).