How to Calculate Potential Energy: Physics Guide
Master gravitational potential energy PE = mgh, elastic energy in springs, and mechanical energy conservation. Worked examples cover heights, roller coasters, and springs, with common mistakes and links to the Potential Energy Calculator.
How to Calculate Potential Energy
Potential energy is energy stored because of position or configuration. A book on a shelf, water behind a dam, and a compressed spring all hold energy that can later become motion, heat, or work.
Two forms dominate introductory physics: gravitational potential energy (height in a gravity field) and elastic potential energy (stretch or compression of a spring). Both feed into the same conservation story: when non-conservative work is negligible, mechanical energy is conserved as kinetic and potential forms trade off.
Gravitational potential energy
Near Earth's surface, where g is nearly constant:
PE_g = mgh
Where m is mass in kilograms, g is gravitational acceleration (about 9.81\,\mathrmm/s^2), h is height in metres above a chosen reference, and PE_g is in joules (J).
Height is relative. Only differences Δ PE_g = mgΔ h have physical meaning for work. You may set h = 0 at the floor, the ground, or the bottom of a track — pick whatever makes the algebra simplest, then stay consistent.
Example — book on a shelf. A 2 kg book sits 1.5 m above the floor:
PE_g = 2 × 9.81 × 1.5 = 29.43 \text J
Lift it another 0.5 m and the increase is:
Δ PE_g = 2 × 9.81 × 0.5 = 9.81 \text J
That increase equals the work you do against gravity (ignoring air resistance).
Choosing g and units
Convert mass to kg and height to metres before multiplying. A 150 lb person is about 68 kg; 10 feet is about 3.05 m.
Elastic potential energy
For an ideal spring obeying Hooke's law F = -kx:
PE_s = (1)/(2)kx^2
Where k is the spring constant in N/m and x is the displacement from the unstretched length in metres. Energy depends on x^2, so stretching 4 cm stores four times the energy of stretching 2 cm (same spring).
Example — garage-door spring. k = 200\,\mathrmN/m, stretched by 0.30\,\mathrmm:
PE_s = (1)/(2)(200)(0.30)^2 = 100 × 0.09 = 9 \text J
Compression uses the same formula: x is the magnitude of the displacement from equilibrium.
Conservation of mechanical energy
When only conservative forces do work (gravity, ideal springs), total mechanical energy is constant:
KE + PE = \textconstant
Or between two points:
(1)/(2)mv_1^2 + mgh_1 + (1)/(2)kx_1^2 = (1)/(2)mv_2^2 + mgh_2 + (1)/(2)kx_2^2
Omit spring terms when there is no spring. If friction or air drag does work W_\mathrmnc, then:
KE_1 + PE_1 + W_\mathrmnc = KE_2 + PE_2
Friction typically makes W_\mathrmnc negative (energy leaves the mechanical account as heat).
Worked example: falling object
Problem. A 0.50 kg apple falls from a 3.0 m branch. Find its speed just before it hits the ground. Ignore air resistance; take g = 9.81\,\mathrmm/s^2 and h = 0 at the ground.
At the branch: v_1 = 0, h_1 = 3.0\,\mathrmm.
PE_1 = mgh_1 = 0.50 × 9.81 × 3.0 = 14.715 \text J, \quad KE_1 = 0
At the ground: h_2 = 0, so PE_2 = 0.
KE_2 = PE_1 = 14.715 \text J = (1)/(2)mv_2^2
v_2 = √(\frac2 × 14.715)0.50 = √(58.86) = 7.67 \text m/s
Same result from v = √(2gh) for free fall from rest — conservation of energy and kinematics agree when gravity alone acts.
Worked example: roller-coaster crest to valley
Problem. A 450 kg coaster car rolls (no friction) from a 25 m crest, starting at rest, into a valley at h = 5\,\mathrmm. Find the speed in the valley.
mgh_1 = (1)/(2)mv_2^2 + mgh_2
Mass cancels:
gh_1 = (1)/(2)v_2^2 + gh_2
v_2^2 = 2g(h_1 - h_2) = 2(9.81)(20) = 392.4
v_2 = 19.8 \text m/s
About 71 km/h — and independent of mass. That is why empty and full cars reach similar speeds on a frictionless track; real tracks lose some energy to rolling resistance and air drag.
Worked example: spring and mass
Problem. A 0.20 kg block compresses a spring (k = 400\,\mathrmN/m) by 5.0 cm on a frictionless horizontal surface, then is released from rest. Find the speed when the spring returns to its natural length.
At maximum compression: KE = 0, PE_s = (1)/(2)k x^2.
PE_s = (1)/(2)(400)(0.05)^2 = 0.50 \text J
At the natural length: PE_s = 0, so all of that energy is kinetic:
(1)/(2)mv^2 = 0.50 \implies v = √(\frac1.0)0.20 = √(5) = 2.24 \text m/s
Gravity far from Earth
When height is not "small" compared with Earth's radius R_E, use the Newtonian form:
PE = -(GMm)/(r)
Where G = 6.674× 10^-11\,\mathrmN· m^2/kg^2, M is Earth's mass, and r is distance from Earth's centre. Escape speed from the surface follows from setting total energy to zero:
v_\mathrmesc = √(\frac2GM)R_E ≈ 11.2 \text km/s
For buildings, sports, and most engineering heights, mgh remains the right tool.
Applications
Hydropower. Water mass m dropping through head h can deliver up to mgh of energy; turbines convert a fraction to electricity. Doubling head roughly doubles energy per kilogram.
Roller coasters and ski jumps. Designers set crest heights so that mgΔ h supplies the kinetic energy needed for loops and landings, then add margin for friction.
Pumped storage. Off-peak electricity lifts water; later release converts PE_g back to power — a grid-scale battery made of height.
Vehicle suspension and archery. Springs and limbs store (1)/(2)kx^2 (or analogous elastic energy) and return it as motion.
Safety. Fall energy scales with height. A fall from 2 m stores twice the PE of a fall from 1 m for the same mass — and that energy must be absorbed by the body or a harness system.
Common mistakes
Using the wrong zero of height. Mid-problem switches of reference level ruin Δ h. State h = 0 once.
Forgetting that PE_g can be negative if your zero is above the object. That is fine; differences still work.
Mixing cm and m in spring problems. x = 5\,\mathrmcm must enter as 0.05\,\mathrmm when k is in N/m.
Assuming mechanical energy is always conserved. Sliding friction, inelastic collisions, and drag remove mechanical energy.
Using weight in kg as mass. Kilograms are mass; weight is mg in newtons. PE = mgh needs mass in kg.
Thinking heavier objects fall faster in vacuum. Energy scales with m, but so does inertia — free-fall speed from a height does not depend on mass.
Frequently asked questions
Is potential energy "real" energy? Yes. It is a bookkeeping form for work that conservative forces can do. The joules in mgh are the same unit as kinetic energy and heat.
Why does PE = mgh use height, not path length? Gravity is a conservative force: work between two heights depends only on the vertical change, not on the path. Climbing a ramp of length L at angle θ still changes PE by mg(L\sinθ).
Can potential energy be zero? Only relative to your reference. Absolute gravitational potential in Newtonian gravity is often taken as zero at infinity, which makes PE negative for bound orbits — a convention, not a claim that energy is "missing."
How is PE related to work? The work done by gravity when something falls a distance Δ h downward is +mgΔ h; the change in gravitational PE is -mgΔ h. Work by the conservative force equals minus the change in PE.
What if there is both a spring and gravity? Add both: PE_\mathrmtotal = mgh + (1)/(2)kx^2. Use one consistent coordinate system for h and x.
Does temperature affect potential energy? Gravitational mgh does not depend on temperature. Real springs can change k slightly with temperature, and materials can store chemical or thermal energy separately from mechanical PE.
Summary
Gravitational potential energy near Earth is:
PE_g = mgh
Elastic energy in an ideal spring is:
PE_s = (1)/(2)kx^2
Choose a clear height reference, keep units SI, and use conservation of mechanical energy when non-conservative work is absent or accounted for. Height differences and spring displacements — not path lengths — set how much energy is stored.
Compute gravitational potential energy with our Potential Energy Calculator.