Potential energy:
stored energy from height in a gravitational field

The formula and what each term means

Carry a 5 kg bag up onto a 2 m shelf and you do work against gravity. That work does not vanish; it sits in the bag as potential energy, ready to turn into motion if it falls. The amount is 5 × 9.81 × 2 = 98.1 joules.

It is a useful bookkeeping device: the energy depends only on where the object is now relative to a reference level, not on the path it took to get there. Carry it up a ladder or hoist it straight up, and the stored energy is the same.

Mass is in kilograms, not grams, and height in metres. The product kg × m/s² × m comes out in joules, since 1 J = 1 N·m. Some courses use g ≈ 10 m/s² for quick sums; stating which value you used keeps answers consistent.

Worked example: a water tank

A pump lifts 1,000 kg of water, one cubic metre, into a tank 12 m above the ground.

Run it backwards for a quick estimate of speed if the water fell freely: ½ × m × v² = m × g × h gives v = √(2 × 9.81 × 12) ≈ 15.3 m/s, ignoring air resistance.

Choosing the zero level

Height has to be measured from somewhere. The floor, a table top or sea level are all valid, and the choice changes the number of joules but not the physics. What matters physically is the difference in potential energy between two positions.

A negative value simply means the object is below the chosen zero. When the book falls from 1.0 m to the floor it loses 19.6 J whichever level you chose, because the difference is what counts.

Other worlds, and where mgh stops working

Only g changes from place to place. The same 5 kg mass lifted 2 m stores about 16.2 J on the Moon (g = 1.62 m/s²) and 37.1 J on Mars (g = 3.71 m/s²), against 98.1 J on Earth.

• The formula assumes g is constant. Over tens of kilometres, such as satellite altitudes, g falls noticeably and you must use PE = −G × M × m ÷ r.
• It describes gravitational energy only. A stretched spring stores ½ × k × x² and a charge in an electric field stores a different kind of potential energy.
• Potential energy belongs to the system of object and Earth together, though it is routinely attributed to the object.

Work done, power and the energy bill

Raising a mass at a steady speed requires a force equal to its weight, and the work done is that force times the distance, which is why the work equals m × g × h. Power is the rate: the same lift done in half the time needs twice the power.

A person climbing a 3 m flight of stairs, with a body mass of 70 kg, gains 70 × 9.81 × 3 = 2,060 J. If the climb takes 10 s, the average power going into the climb is 206 W. Muscles are only about 20 to 25% efficient, so the body burns around four to five times that energy as food energy, most of it released as heat.

A lift or hoist is more efficient, but the same figure sets the minimum: a motor cannot deliver less than m × g × h of useful lifting work.

Pumped storage and dams

Large-scale energy storage uses exactly this relation. Water pumped uphill when electricity is cheap stores energy and flows back through turbines when demand rises. A reservoir holding 1 million m³ of water, which is 10⁹ kg, raised 100 m stores 10⁹ × 9.81 × 100 = 9.81 × 10¹¹ J, about 272,500 kWh or 272 MWh.

• The figure is the upper limit; turbine, generator and pipe losses reduce what comes back, with round-trip efficiency commonly in the range of 70% to 80%.
• Doubling the height doubles the energy, and doubling the volume does the same, so both head and volume matter.
• The same arithmetic underpins hydroelectric dams, where the head of water above the turbines sets the available energy per cubic metre.

A tonne of water dropping 100 m releases 0.27 kWh, which is why dams need enormous volumes to make a meaningful quantity of energy.

Springs and other stored energy

Gravitational energy is only one kind of stored energy. A compressed or stretched spring stores elastic energy ½ × k × x², where k is the stiffness in N/m and x the displacement from rest. A spring with k = 200 N/m compressed 0.1 m holds ½ × 200 × 0.01 = 1.0 J. If that spring fired a 0.05 kg ball upward, all of it going to height, the ball would rise 1.0 ÷ (0.05 × 9.81) = 2.04 m, ignoring losses.

Chemical energy in fuel and batteries and electrical energy in a charged capacitor are also stored energy, and each has its own formula. What they share with gravitational energy is the principle: energy stored in a configuration can be released and converted, and the total is accounted for when you add every form.

From potential to kinetic and back

A pendulum, a roller coaster and a bouncing ball all swap energy between potential and kinetic forms. With friction ignored, the total stays constant: whatever PE is lost on the way down appears as kinetic energy ½ × m × v². Real systems lose some to heat and sound, which is why a bouncing ball never quite regains its starting height.