Kinetic Energy Calculator: Solve for Energy, Mass, Speed

Kinetic Energy Calculator

Kinetic energy is the energy a body has because it is moving. This calculator works out that energy from a body's mass and its speed, and it also runs the other way: given an energy and one of the other two figures, it returns the third.

What the calculator does

The page has three fields: kinetic energy in joules, mass, and speed. Mass and speed each have a unit menu. Two fields are filled in, one is left blank, and the blank one is the answer. All three figures are then shown together, so the pair that was entered can be checked beside the result.

Mass can be entered in kilograms, grams, milligrams, metric tonnes, pounds, ounces, stone, US short tons or long tons. Speed can be entered in metres per second, kilometres per hour, miles per hour, feet per second or knots. Energy has no unit menu: it is read and shown in joules.

Before any arithmetic runs, the mass is converted to kilograms and the speed to metres per second by the site's shared unit registry. The answer is converted back into the unit chosen for that field. Results appear to eight significant figures. Only the display is rounded, and the arithmetic runs on the values as typed.

The kinetic energy formula

$E_k = \frac{1}{2} m v^2$

where $E_k$ is the kinetic energy in joules (J), $m$ is the mass in kilograms (kg) and $v$ is the speed in metres per second (m/s).

The same relation, rearranged, gives the other two answers:

$m = \frac{2 E_k}{v^2}$

$v = \sqrt{\frac{2 E_k}{m}}$

The speed is squared and the mass is not. Doubling the mass doubles the energy. Doubling the speed multiplies it by four, so a car at 60 km/h carries four times the kinetic energy it carries at 30 km/h.

Speed here is a size, not a direction. A negative entry is refused rather than squared into a positive one, and the square root above is the non-negative root only.

Which constants are exact

Every conversion on this page is a definition rather than a measurement, so no better experiment will ever revise one.

Constant	Value	Exact?	Source
The half in the formula	1/2	exact; it is a number, not a measurement	Halliday, Resnick and Walker, Fundamentals of Physics, work-energy theorem
Joule	1 kg⋅m²/s²	exact by definition of the SI derived unit	SI Brochure, 9th edition
Pound	0.45359237 kg	exact by definition	1959 International Yard and Pound Agreement; NIST SP 811, Appendix B
Ounce, stone, short ton, long ton	1/16 lb, 14 lb, 2,000 lb, 2,240 lb	exact; each follows from the pound	NIST SP 811, Appendix B
Metric tonne	1,000 kg	exact by definition	SI Brochure, 9th edition
Mile per hour	0.44704 m/s	exact; one mile is 1,609.344 m by definition	1959 Agreement; NIST SP 811, Appendix B
Foot per second	0.3048 m/s	exact by definition	1959 Agreement; NIST SP 811, Appendix B
Knot	1,852 m per hour	exact; the nautical mile is 1,852 m by definition	International Hydrographic Conference, Monaco, 1929
Kilometre per hour	1,000 m per hour	exact by definition	SI Brochure, 9th edition

Two of these have no exact decimal form once written per second, because 3,600 does not divide 1,000 or 1,852 evenly. The registry keeps each as that division instead of storing a truncated decimal.

Worked example

A car with a mass of 1,500 kg travelling at 72 km/h.

A speed of 72 km/h is 72 times 1,000 divided by 3,600, which is 20 m/s exactly. Squaring 20 gives 400. Multiplying by 1,500 gives 600,000. Half of that is 300,000, so the car carries 300,000 J, or 300 kilojoules.

The same figures run backwards. Leave the speed blank and enter 300,000 J with 1,500 kg: doubling the energy gives 600,000, dividing by the mass gives 400, and the square root of 400 is 20 m/s, shown as 72 km/h. Leave the mass blank instead: 300,000 divided by 20 twice is 750, and doubling that gives 1,500 kg.

A rounder-looking case that does not come out even is 1,500 kg at 100 km/h. That speed is 27.777… m/s, and the energy is 578,703.7 J to eight significant figures.

An imperial case, worked from the two defining constants above: 1 lb at 1 mph. Squaring 0.44704 gives 0.1998447616 exactly. Multiplying by 0.45359237 gives 0.090648059046228992, and half of that is 0.045324029523114496 J. The page shows 0.04532403 J.

How the arithmetic is arranged

Eight significant figures is fine enough to show rounding dust, so each form is written to cancel exactly at round values.

The energy is worked out as the mass times the speed twice, and that product is then halved. Halving is exact in binary arithmetic, so 2 kg at 3 m/s gives 9 J with nothing trailing.

The mass is worked out by dividing the energy by the speed twice and then doubling, rather than by dividing by the square of the speed. Doubling is exact, and two separate divisions carry speeds whose square would overflow or fall to zero on its own. Feeding 9 J at 3 m/s back in returns exactly 2 kg.

The speed is worked out by dividing the energy by the mass and doubling before the square root is taken. For 9 J and 2 kg that leaves exactly 9 under the root, and exactly 3 m/s comes out.

What the calculator refuses

An entry box will pass through any number typed or pasted into it, so the checks that matter are the ones in the calculation code.

Situation	Why there is no answer
Mass of zero or less	A body has mass above zero.
Negative speed	Speed here is a size, not a direction.
Negative energy	Mass is positive and a squared speed is positive, so half their product never falls below zero.
Working out a mass at a speed of zero	A body at rest has no kinetic energy whatever it weighs, so no mass can be recovered from one. Dividing by the zero would report infinity, which is a confident wrong answer.
A mass that works out to zero	Zero energy at a speed above zero would need a body with no mass at all. An energy tiny beside the speed also divides down to zero even though the true mass is above it. Neither is a mass this page can report.
An answer too large to hold	The figures are outside the range the page's numbers carry, so the result is not an energy.
A unit the page does not have	A shared or hand-edited link can name any unit, including one belonging to another quantity. The page says so rather than breaking.

Two blank fields, or none, are treated as guidance rather than as a fault. The page asks for exactly one blank field and works that one out.

Where this formula stops working

The formula here is the classical one, and it describes a body moving from place to place. It fits everyday speeds closely. It becomes wrong as a speed approaches the speed of light, where the relativistic expression is needed. It also leaves out the extra energy stored in a body that is spinning, which is counted separately as rotational kinetic energy.

Frequently asked questions

What is kinetic energy? It is the energy a body has because it is moving. A body at rest has none. The more mass it has and the faster it goes, the more it carries. The unit is the joule.

How is kinetic energy calculated? Multiply the mass by the speed, multiply by the speed again, then halve the result. For 1,500 kg at 20 m/s, 1,500 times 20 times 20 is 600,000, and half of that is 300,000 J.

What happens to kinetic energy when the speed doubles? It becomes four times as large, because the speed is squared. Doubling the mass instead only doubles the energy. This is why the speed of a vehicle matters more to its energy than its weight does.

Can kinetic energy be negative? No. Mass is above zero and a squared speed is never below zero, so half their product is never below zero either. A negative energy entry is refused.

How is speed found from a known kinetic energy? Double the energy, divide by the mass, then take the square root. For 300,000 J and 1,500 kg, doubling gives 600,000, dividing gives 400, and the square root of 400 is 20 m/s.

Why will the page not work out a mass when the speed is zero? A body at rest has zero kinetic energy whatever its mass, so the energy says nothing about the mass. There is nothing to recover, and the division needed would have zero on the bottom.

Kinetic Energy Calculator: Solve for Energy, Mass, Speed