Kinetic Energy Calculator

Kinetic energy is KE = ½ m v²: half the mass multiplied by the square of the velocity. Choose which quantity to solve for above and the calculator rearranges the equation and converts between joules, kilojoules, calories and foot-pounds.

The square is the important part

Mass appears once, velocity appears twice. That asymmetry is the single most consequential fact in road safety, and it is worth seeing in numbers.

A 1500 kg car at 20 m/s, about 72 km/h, carries ½ × 1500 × 400 = 300,000 J, or 300 kJ. The same car at 40 m/s carries ½ × 1500 × 1600 = 1,200,000 J. Twice the speed, four times the energy.

Every joule has to be removed by the brakes before the car stops, so stopping distance grows with the square of speed too. Doubling your speed does not double the distance needed to stop; it quadruples it, before reaction time is even counted.

A worked example

Take a mass of 1500 kg travelling at 20 m/s.

• Square the velocity: 20² = 400
• Multiply by the mass: 1500 × 400 = 600,000
• Halve it: 300,000 J, which is 300 kJ

For comparison, 300 kJ is roughly the energy in 17 grams of sugar, or what a 100 W bulb uses in 50 minutes. Motion stores a surprising amount of energy.

Where the half comes from

Kinetic energy is the work done to accelerate an object from rest, and work is force multiplied by distance. Using F = ma and the distance covered under constant acceleration, the algebra produces the factor of a half rather than it being chosen.

The same factor appears in every energy expression of this shape, including the energy stored in a spring, ½kx², and in a capacitor, ½CV². It comes from the average of a quantity rising linearly from zero, which is half its final value.

Comparing things that move

Object	Mass	Speed	Kinetic energy
Rifle bullet	8 g	900 m/s	3240 J
Cyclist	90 kg	8 m/s	2880 J
Car in town	1500 kg	13.9 m/s	145 kJ
Car on motorway	1500 kg	31 m/s	721 kJ

A rifle bullet and a cyclist carry comparable energy, which is unintuitive until you remember the squaring. The bullet does far more damage because that energy is delivered into a few square millimetres rather than spread over a whole body.

Energy and momentum answer different questions

Momentum tells you how hard something is to stop. Kinetic energy tells you how much damage it can do. They are not interchangeable, and picking the wrong one is a common error.

Doubling speed doubles momentum and quadruples energy. That is why a modest increase in impact speed produces a disproportionate increase in injury, and why speed limits are set against energy rather than momentum.

Braking and heat

Brakes work by converting kinetic energy into heat through friction. Removing 300 kJ from a car raises the temperature of 8 kg of steel discs by roughly 45 degrees, assuming a specific heat capacity near 460 J per kg per degree.

Repeat that on a long descent and the discs cannot shed heat fast enough, which is brake fade. It is also why heavy vehicles use engine braking on hills: the energy has to go somewhere, and the friction brakes alone cannot absorb it indefinitely.

Regenerative braking

An electric vehicle recovers part of this energy instead of discarding it as heat, by running the motor as a generator. A car shedding 300 kJ might return 200 kJ to the battery at typical efficiencies.

The benefit is largest in stop-start driving, where that energy would otherwise be lost repeatedly, which is why electric and hybrid vehicles often show better consumption in town than on a motorway. That reverses the usual pattern for a combustion engine.

Where it gets used

Vehicle safety. Crash tests are specified at set speeds because energy, not force, determines the severity.

Ballistics. Muzzle energy is the standard measure of a cartridge, quoted in joules or foot-pounds.

Wind power. The energy in moving air scales with the cube of wind speed, since the mass arriving per second also rises with speed, which is why turbine siting matters so much.

Sport. The energy a ball carries determines both how far it travels and what it does on impact.

Putting the numbers in context

Joules are hard to picture, so it helps to compare motion against energy people already have a feel for.

Quantity	Energy	Equivalent motion
1 gram of sugar	17 kJ	a 1500 kg car at 4.8 m/s
An AA battery	about 10 kJ	a 1500 kg car at 3.7 m/s
1 litre of petrol	about 34 MJ	113 times the car at 20 m/s

The last row is the one worth sitting with. A single litre of fuel holds enough energy to accelerate a family car to 72 km/h more than a hundred times over. Almost all of it is lost as heat instead, which is why engine efficiency figures are so low and why regenerative braking is worth the complication.

Rotational kinetic energy

Spinning objects store energy too, given by ½Iω², where I is the moment of inertia and ω the angular velocity. The form is identical to the linear version with mass and speed replaced by their rotational counterparts.

This is how a flywheel works. Energy goes in as the wheel is spun up and comes back out as it slows, which smooths the delivery of a piston engine and, in some vehicles, recovers braking energy without a battery. A wheel on a moving car carries both kinds at once: linear energy from moving along the road, and rotational energy from spinning about its axle.

Units of energy

The joule is the SI unit and equals one newton acting through one metre. Everything else is a convenience.

A calorie is 4.184 J, and the food calorie printed on packaging is a kilocalorie, a thousand of those. A kilowatt hour is 3.6 million joules, which is why it suits electricity billing where a joule would be uselessly small. The foot-pound survives in ballistics, where a joule is 0.7376 of one.

Mixing them is a common source of error, particularly the food calorie, which is a thousand times larger than the scientific one despite sharing the name.

Work is how energy gets in and out

Kinetic energy changes when work is done, and work is force multiplied by the distance it acts through. This is the work-energy theorem, and it is often the quickest route to an answer.

A 1500 kg car at 20 m/s holds 300 kJ. Braking to rest means removing all of it through the brakes. Over a 40 m stopping distance the average force is 300,000 / 40 = 7500 N. Halve the distance and the force doubles, which the tyres may not be able to supply.

The same relation run in reverse gives the distance needed to reach a speed under a known force, which is how acceleration figures and runway lengths are worked out.

Common mistakes

Forgetting to square the velocity. The most frequent error, and it understates the answer badly at speed.

Squaring the whole expression. Only v is squared, not ½m.

Using km/h directly. Convert to metres per second first, or the answer is out by a factor of about 13.

Confusing energy with momentum. They scale differently with speed and answer different questions.