Energy of motion, solved instantly — compute kinetic energy, mass, or velocity from KE = ½mv². Free with steps shown.
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Anything that moves carries kinetic energy — the energy of motion. A sprinting athlete, a speeding car, a thrown baseball: all store energy in their movement, and that energy can do work when the motion stops (which is exactly why crashes are destructive). The formula KE = ½mv² quantifies it: kinetic energy equals half the mass times the velocity squared.
Our free Kinetic Energy Calculator solves the kinetic energy equation for any variable. Enter mass and velocity to find the energy in joules, or rearrange to find the mass behind a known energy, or the velocity needed to reach a target energy. Every result shows the substitution steps.
The squared velocity is the formula’s most dramatic feature: doubling your speed quadruples the energy. That single fact explains why highway speed limits matter so much, why a fastball hurts more than a changeup, and why spacecraft need staggering amounts of fuel — energy costs grow with the square of how fast you want to go.
Use SI units throughout: kilograms, meters per second, joules. Convert km/h to m/s by dividing by 3.6 before entering velocity.
Kinetic energy is the energy an object possesses because it is moving. It is a scalar quantity (no direction, just magnitude), measured in joules (J). When a moving object is stopped — by brakes, a wall, or a catcher’s mitt — its kinetic energy is converted into other forms: heat, sound, deformation, or work done on something else.
The formula KE = ½mv² comes from the work-energy theorem: the work done accelerating a mass from rest to velocity v equals the force times distance, which integrates to exactly ½mv². The ½ is not a fudge factor — it falls out of the calculus of constant acceleration, and it is confirmed by every experiment from air tracks to particle colliders.
Two features deserve emphasis. First, kinetic energy depends on the reference frame: a passenger sipping coffee on a train has nearly zero KE relative to the train but enormous KE relative to the ground. Energy is always ‘relative to’ something. Second, at everyday speeds the formula is essentially exact; only near the speed of light does Einstein’s relativistic correction matter, and our calculator (like all classical mechanics) assumes non-relativistic speeds.
Kinetic energy is also the reason regenerative braking works: electric cars convert the KE of their motion back into battery charge instead of wasting it as brake heat. The same ½mv² that makes crashes dangerous makes EVs efficient — energy is energy, and accounting for it precisely is what engineering is.
KE = ½ × m × v² • m = 2KE ÷ v² • v = √(2KE ÷ m)
KE = kinetic energy in joules (J) • m = mass in kilograms (kg) • v = velocity in meters per second (m/s). 1 joule = 1 kg·m²/s².
The rearranged forms follow from basic algebra. Solving for v requires a square root, and the calculator takes the positive root (speed is the magnitude of velocity).
Quick check: a 1,000 kg car at 20 m/s. KE = 0.5 × 1000 × 400 = 200,000 J (200 kJ). At 40 m/s — twice the speed — KE = 0.5 × 1000 × 1600 = 800,000 J. Double speed, quadruple energy.
A 1,200 kg car travels at 15 m/s (54 km/h). Find its kinetic energy.
Step 1: KE = ½ × m × v² = 0.5 × 1200 × 15².
Step 2: 15² = 225, so KE = 0.5 × 1200 × 225 = 135,000 J = 135 kJ.
That is roughly the energy of a 135 kW engine running for one second — and all of it must be absorbed by brakes or barriers in a stop.
Same car at 30 m/s (108 km/h).
Step 1: KE = 0.5 × 1200 × 30² = 0.5 × 1200 × 900.
Step 2: KE = 540,000 J = 540 kJ — exactly four times the 135 kJ at half the speed.
This is the physics behind speed enforcement: crash severity scales with v², not v.
A 0.145 kg baseball carries 150 J of kinetic energy. How fast was it thrown?
Step 1: v = √(2 × KE ÷ m) = √(2 × 150 ÷ 0.145).
Step 2: v = √(2068.97) ≈ 45.5 m/s ≈ 164 km/h — a genuine major-league fastball.
Transport engineers use KE = ½mv² to set speed limits and design barriers. Because energy grows with the square of speed, raising a limit from 100 to 120 km/h increases crash energy by 44 percent. Guardrails, crash cushions, and crumple zones are all dimensioned against the kinetic energies of vehicles at design speeds — get the v² scaling wrong and the barrier fails.
Coaches and equipment designers compute ball energies to balance performance and safety: a 150 J baseball (Example 3) versus a 0.43 kg soccer ball at 30 m/s carrying about 194 J. Helmet and padding standards are written around these energy figures, which is why youth leagues use lighter, slower balls — less mass and less velocity, dramatically less energy.
Kinetic energy KE = ½mv² is the scalar energy of motion in joules (1 J = 1 kg·m²/s²). Doubling velocity quadruples energy. Rearranged: m = 2KE/v², v = √(2KE/m). KE is frame-dependent and non-relativistic here; it converts to heat, sound, deformation, or work when motion stops. Distinct from momentum (mv), which is a conserved vector.
Kinetic energy is the energy an object has because it is moving, calculated as KE = ½mv². It is measured in joules and represents the capacity of the moving object to do work — for example, the damage in a crash or the distance a ball travels.
KE = ½ × m × v², where m is mass in kilograms and v is velocity in meters per second. The result is in joules. The velocity is squared, so doubling speed quadruples the energy.
Joules (J) in SI units, where 1 J = 1 kg·m²/s². Larger energies use kilojoules (kJ) or megajoules (MJ); our calculator also shows kWh for intuition.
It comes from the work-energy theorem: accelerating a mass requires force over distance, and the integration yields the ½mv² form. Physically, it means energy costs grow steeply with speed — the defining fact of crash safety.
Kinetic energy (½mv²) is a scalar measuring energy of motion; momentum (mv) is a vector measuring ‘quantity of motion’ and is conserved in collisions. KE is generally not conserved — some converts to heat and deformation.
No. Mass is positive and v² is always non-negative, so KE ≥ 0. An object at rest has zero kinetic energy. (Potential energy can be negative depending on the reference point, but kinetic cannot.)
Rearrange: v = √(2 × KE ÷ m). For example, 150 J in a 0.145 kg baseball gives v = √(2×150÷0.145) ≈ 45.5 m/s. Our calculator does this automatically in ‘solve for velocity’ mode.
No — KE is a scalar. A car moving north at 20 m/s has the same kinetic energy as one moving south at 20 m/s. Direction matters for momentum and velocity, not for energy.
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Last updated: October 11, 2026