Kinetic and potential energy, work and power calculator

Kinetic energy formula: KE = ½mv²

Kinetic energy is half the mass times the speed squared: KE = ½mv², in joules when mass is in kilograms and speed in meters per second. A 2 kg ball moving at 3 m/s carries ½ × 2 × 3² = 9 J. This is the energy of the whole object moving, valid at speeds far below the speed of light.

Because speed is squared, doubling it quadruples the energy. OpenStax's kinetic energy lesson works through more examples.

A 1,000 kg car at 20 m/s (72 km/h) carries 200,000 J; at 40 m/s it carries 800,000 J. Convert km/h to m/s by dividing by 3.6 before squaring.

Speed or mass from kinetic energy

Rearranged, v = √(2KE/m) and m = 2KE/v². A 70 kg runner carrying 3,500 J of kinetic energy is moving at √(2 × 3,500 ÷ 70) = √100 = 10 m/s.

An object at rest has zero kinetic energy whatever its mass, so mass cannot be found from KE when the speed is zero.

Gravitational potential energy: PE = mgh

Lifting a mass m through a height h stores PE = mgh. With standard gravity, g = 9.80665 m/s², raising 50 kg by 10 m stores 50 × 9.80665 × 10 = 4,903.325 J, about 4.9 kJ.

Height is measured from a reference level you choose, such as the floor. Only differences in potential energy matter, so use the same reference for every position in a problem. OpenStax's potential energy lesson explains why.

Spring energy: E = ½kx²

A spring with stiffness k, stretched or compressed by x, stores E = ½kx². A 200 N/m spring compressed 10 cm (0.1 m) stores ½ × 200 × 0.1² = 1 J; compressed 20 cm, it stores 4 J.

The formula assumes Hooke's law, F = kx, which holds only while the spring stays in its elastic range and returns to its original length.

Work done by a force: W = Fd cos θ

A constant force F that moves an object a distance d does work W = Fd cos θ, where θ is the angle between the force and the direction of motion. For 100 N over 5 m, the work is 500 J at 0°, 250 J at 60°, zero at 90° and −500 J at 180°.

Negative work means the force opposes the motion, as friction and brakes do. The work-energy theorem says the net work on an object equals its change in kinetic energy.

Stopping a 1,000 kg car from 20 m/s therefore takes −200,000 J of net work. If the brakes supply a constant 8,000 N, the car stops in 200,000 ÷ 8,000 = 25 m.

Power formula: P = W/t

Power is the rate of doing work: P = W/t, in watts, where 1 W is 1 J per second. Delivering 3,600 J in 60 s is 3,600 ÷ 60 = 60 W. Rearranged, W = Pt and t = W/P.

A 100 W device running for one hour transfers 100 × 3,600 = 360,000 J, which is 0.1 kWh. One kilowatt-hour is 3,600,000 J. OpenStax's power lesson separates this average power from the instantaneous rate.

Speed after a fall: v = √(2gh)

Without air resistance, the potential energy lost in a fall becomes kinetic energy: mgh = ½mv², so the mass cancels. From rest, v = √(2gh). A 20 m drop ends at √(2 × 9.80665 × 20) ≈ 19.81 m/s.

With a starting speed v₀, the speed at the bottom is √(v₀² + 2gh). Impact speed grows with the square root of height: from 80 m, four times higher, it is about 39.61 m/s, only twice as fast. Real falls are slower because of drag.

Common mistakes

• Leaving speed in km/h. ½mv² gives joules only with speed in m/s. Using 72 instead of 20 for a car at 72 km/h makes its energy 12.96 times too large.
• Mixing reference levels. Measure every height in a PE problem from the same zero.
• Taking θ as the slope of the ground. In W = Fd cos θ, θ is the angle between the force and the displacement.
• Confusing power with energy. Watts measure a rate; joules and kilowatt-hours measure an amount.
• Using ½mv² for everything that moves. A spinning wheel also stores rotational energy, ½Iω², and speeds close to light need relativistic formulas.

Input formats and units

• Pick the formula, then the quantity to solve for; the remaining quantities become inputs.
• Potential energy and the fall use standard gravity, 9.80665 m/s², unless you enter another g.
• Energy accepts J, kJ, Wh, kWh or kcal, power accepts W, kW or hp, and force accepts N, kN or lbf. Results are in SI units.
• The work angle runs from 0° to 180°, in degrees or radians.

For signed motion and time, use the kinematics calculator; for collisions, the momentum and collision calculator. Electrical power uses P = VI in the Ohm's law calculator, and engine power from torque and rpm comes from the horsepower calculator.