Kinetic and potential energy: the formulas
Updated on June 30, 2026 · by Rafael Rossi
Kinetic energy is the energy of motion; gravitational potential energy is the energy of position. Together they explain why a car at 60 mph is four times as dangerous as one at 30 — not twice.
The two formulas
Kinetic energy: KE = m·v² ÷ 2
Gravitational potential energy: PE = m·g·h
Where m is mass in kg, v is speed in m/s, h is height in metres and g ≈ 9.8 m/s². The result comes out in joules (J).
Note the asymmetry: in potential energy everything is linear — double the height, double the energy. In kinetic energy, speed is squared. It is the single most important difference in everyday physics.
Worked example
A 2,200 lb car (1,000 kg) at 30 mph:
- Convert to m/s: 30 mph = 13.41 m/s
- KE = 1,000 × 13.41² ÷ 2 = 89,900 J
Now the same car at 60 mph: v = 26.82 m/s, and KE = 359,600 J. Doubling the speed quadrupled the energy.
The translation that changes how risk feels
A figure in joules means nothing to most people. But kinetic energy converts into an equivalent fall height — the height from which the car would need to drop to hit the ground with that same energy:
equivalent height = v² ÷ (2g)
| Speed | Energy (2,200 lb) | Equivalent to falling from |
|---|---|---|
| 20 mph | 39,900 J | 13 ft — 1st floor |
| 30 mph | 89,900 J | 30 ft — 3rd floor |
| 40 mph | 159,800 J | 54 ft — 5th floor |
| 60 mph | 359,600 J | 120 ft — 12th floor |
| 70 mph | 489,500 J | 164 ft — 16th floor |
Hitting a fixed obstacle at 40 mph is the energetic equivalent of dropping the car off a fifth floor. At 70 mph, a sixteenth. Going from 60 to 70 mph — 17% more speed — adds 36% more energy, and that energy is what the vehicle structure and the occupants' bodies have to absorb.
The conversion between the two
Energy is not lost, it transforms. An object dropped from height h starts with PE = mgh and no KE; on reaching the ground it has KE = mv²/2 and no PE. Setting them equal:
m·g·h = m·v² ÷ 2 → v = √(2gh)
Notice what cancels out: the mass. That is why, ignoring air resistance, a feather and a hammer fall together — the hammer's greater potential energy is exactly offset by the greater inertia it has to overcome.
Where the calculation shows up outside an exam
- Braking distance. Brakes dissipate kinetic energy. Since it grows with the square of speed, braking distance does too: doubling your speed quadruples the space needed to stop — before adding reaction time, which grows linearly.
- Fall protection on site. Working-at-height standards are built on this relationship; the impact energy of a dropped tool grows with the height.
- Hydroelectric power. The whole operation converts PE into KE and then into electricity: available energy is proportional to flow rate and to the head of water.
Energy is not the same as momentum
There is a second quantity of motion that gets confused with energy: linear momentum, given by p = m·v. Here speed enters linearly, and mass counts just as much.
The difference has practical consequences. A 40,000 lb truck at 20 mph has twenty times the momentum of a 2,000 lb car at the same speed, and also twenty times the energy. But a car at 80 mph has only four times the momentum of one at 20 mph — and sixteen times the energy.
Momentum explains who pushes whom in a collision; energy explains how much the material deforms. That is why speed is the dominant factor in how severe a crash is, and mass is the dominant factor in who walks away from it.
Frequently asked questions
Why is speed squared but height is not? Because kinetic energy is the work done by a force through an acceleration, and the distance covered during that acceleration itself grows with time — the square appears in the integration. Intuition does not follow it, but the practical consequence is measurable in any crash test.
Where does the "÷ 2" come from? From the same place: during constant acceleration from rest, the average speed is half the final speed. The factor of 2 is that average.
Can potential energy be negative? It can, and that is fine: it is always measured relative to a chosen reference. Set zero at street level and an object in a basement has negative PE. Only differences in energy have physical meaning.
How much energy is a car at 60 mph, in familiar terms? The 359,600 J of a 2,200 lb car is about 86 kilocalories — the energy in half a granola bar. Which reveals something else uncomfortable: food is extraordinarily energy-dense, and the human body is far too fragile to absorb that much at once.