This potential energy calculator covers both stored-energy classics: gravitational potential energy (PE = mgh) for lifted masses, and elastic potential energy (PE = ½kx²) for stretched or compressed springs.

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Energy in the Bank

Potential energy is energy by position, spendable on demand. Gravitational PE = mgh prices height: linear in mass and height, with h measured from whatever reference you declare (only PE changes ever matter — the zero is bookkeeping). Elastic PE = ½kx² prices springs quadratically: double the stretch, quadruple the storage — the same square law that powers the Hooke’s Law Calculator‘s force side. When stored energy lets go it usually becomes the Kinetic Energy on the sibling page; conservation (PE lost = KE gained) solves half of mechanics with one equation. Part of our Physics Calculators collection.

Worked Examples (Verified)

  • A 70 kg climber 10 m up: PE = 70 × 9.81 × 10 = 6,867 J relative to the ground — the energy the rope system must absorb in a fall (plus the fall’s dynamics, which is why climbing gear is stretchy)
  • A spring, k = 200 N/m, stretched 0.15 m: PE = ½ × 200 × 0.0225 = 2.25 J, with 30 N of peak restoring force
  • Scale check: a roller coaster car (500 kg) cresting a 30 m hill banks 147,150 J ≈ 0.041 kWh — thrill rides run on pocket change worth of electricity; the art is in the trading between PE and KE, not the total

Reference Levels and Honest Caveats

  • PE can be negative (below your chosen zero) — that’s fine; physics reads the differences.
  • mgh assumes constant g: flawless near Earth’s surface, wrong for orbit (the exact form uses −GMm/r — gravitational PE across space is its own calculator’s business).
  • Springs store ½kx² only while Hookean: stretch past the elastic limit and the energy bookkeeping includes permanent deformation — and a much worse day.

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Frequently Asked Questions

How do you calculate potential energy?

Gravitational: PE = mgh (mass × 9.81 × height). Elastic: PE = ½kx² (half the spring constant times stretch squared). The 70 kg climber at 10 m stores 6,867 J; the 200 N/m spring at 0.15 m stores 2.25 J.

Why is there a ½ in the spring formula?

Because the spring’s force grows from 0 to kx as you stretch it — the average force over the stretch is kx/2, and work = average force × distance = ½kx·x. The triangle under the force-distance graph, exactly.

What are the units of potential energy?

Joules (kg·m²/s²) — the same as every energy form, which is what makes energy conservation arithmetic work across gravity, springs, motion and heat.

Can potential energy be negative?

Yes, relative to any chosen zero below the object. PE values are like altitudes relative to sea level: the zero is conventional, differences are physical. (In the exact gravitational formula, PE also approaches the famous −GMm/r form for spaceflight.)

How does PE turn into KE?

Release the system and gravity (or the spring) does work: PE falls, KE rises, and in the no-friction ideal they trade exactly — a falling mass from height h reaches √(2gh) regardless of mass, as the widget’s note shows. Friction and drag skim their cut on the way.

Is this potential energy calculator free?

Yes — free, instant, no sign-up.