Kinetic energy is ½ × mass × speed², and gravitational potential energy is mass × gravity × height. Choose which of the six quantities to find, give the others in any units, and the answer comes in the unit you choose: 10 kg at 5 m/s has 125 J.
The formula
Kinetic energy
KE = ½ × m × v²Potential energy
PE = m × g × hSolving for the others
m = 2 × KE ÷ v² · v = √(2 × KE ÷ m) · m = PE ÷ (g × h) · h = PE ÷ (m × g)Everything is converted to joules, kilograms, metres per second and metres, solved there, and converted back. The calorie is the thermochemical one (4.184 J) and the foot-pound is 1.355818 J, as the NIST Guide to the SI lists them. Standard gravity is 9.80665 m/s².
Worked example
For potential energy, 10 kg lifted 10 m on Earth has 10 × 9.80665 × 10 = 980.665 J, and the height 500 J lifts a 2 kg mass is 500 ÷ (2 × 9.80665) = 25.49 m.
When to use it, and the mistakes to avoid
Use it for physics problems, to compare the energy of moving things, to see how far a given energy lifts a mass, or to turn joules into watt-hours or food Calories.
- Forgetting to square the speed. The speed is squared, not doubled.
- Mixing units. Pounds with metres per second, or grams with joules, gives nonsense; pick the unit for each quantity here and let the calculator convert.
- Confusing mass and weight. Mass is in kilograms; weight is a force.
- Using Earth's gravity elsewhere. On the Moon or another planet, change g.
- Treating potential energy as absolute. It depends on the level you measure the height from.
FAQ
What is the formula for kinetic energy?
KE = ½ × m × v², half the mass times the speed squared. A 10 kg object at 5 m/s has ½ × 10 × 5² = 125 joules. The speed is squared, so doubling the speed gives four times the energy, which is why a crash at 100 km/h is far worse than one at 50.
What is the formula for potential energy?
For an object near the Earth, PE = m × g × h: mass times gravity times height. 10 kg lifted 10 m, with g = 9.80665 m/s², has 10 × 9.80665 × 10 = 980.665 joules. It is measured from the level you call zero height, so an object below that level has negative potential energy.
What units can I use?
Energy in joules, kilojoules, megajoules, watt-hours, kilowatt-hours, calories, kilocalories (food Calories) and foot-pounds; mass in kilograms, grams and pounds; speed in m/s, km/h, mph and ft/s; height in metres, centimetres, kilometres, feet and inches. You can mix them: a mass in pounds and a speed in mph, with the energy in kilojoules.
Does gravity change the result?
For potential energy, yes: it is directly proportional to g. The same 10 kg lifted 10 m has 980.665 J on Earth (g = 9.80665 m/s²) but about 162 J on the Moon (g ≈ 1.62 m/s²). Kinetic energy does not depend on gravity at all, only on mass and speed.
Why can the calculator not find the mass in some cases?
Because mass must be above zero, and some answers would not be. With no speed, any mass has no kinetic energy, so there is nothing to solve for; the same holds at a height of zero. And if the energy and the height have opposite signs, the mass would have to be negative. Change the values so the answer is a real mass.