Last updated: 2026-05-18

Kinetic energy

Kinetic energy is the energy an object possesses because of its motion. It is defined as the work required to accelerate an object of mass m from rest to its current speed v, and for speeds well below the speed of light it equals ½mv². The accompanying quantity, momentum, is the product of mass and velocity, p = mv.

Given a mass and a velocity, the calculator returns both:

• Kinetic energy (KE) — the energy of motion, in joules, kilojoules, or kilocalories.
• Momentum (p) — the product of mass and velocity, in kg·m/s.

Both increase with mass and with speed, but speed enters the kinetic-energy formula squared, so it has a much larger effect than mass.

How the formula arises

The v² term follows from Newton's second law. The work done to accelerate an object from rest to speed v is the integral of force over the distance travelled, and carrying out that integral yields exactly ½mv². Because the speed appears squared, equal increments of speed add unequal amounts of energy: the energy gained going from 50 to 100 km/h is far larger than the energy gained going from 0 to 50 km/h.

A direct consequence is that doubling the speed multiplies the energy by four (2² = 4), and tripling it multiplies the energy by nine (3² = 9). For a 1 500 kg car:

This non-linear relationship is the reason crash severity and braking distance rise steeply with speed.

The formulas

Here m is mass in kilograms and v is speed in metres per second. One joule equals one kg·m²/s².

Worked examples

Baseball pitch

A baseball has a mass of 0.145 kg. A 40 m/s fastball carries:

KE = 0.5 × 0.145 × 40² = 0.5 × 0.145 × 1 600 = 116 J

p = 0.145 × 40 = 5.8 kg·m/s

Roller coaster at the bottom of a drop

A 500 kg coaster car released from a 30 m height gains speed as it falls. By energy conservation it reaches about $v = \sqrt{2 \times 9.81 \times 30} \approx 24.3$ m/s at the bottom:

KE = 0.5 × 500 × 24.3² ≈ 147 600 J ≈ 148 kJ

This equals the gravitational potential energy mgh = 500 × 9.81 × 30 = 147 150 J, the small difference being rounding.

Car at highway speed

A 1 500 kg car at 100 km/h (27.78 m/s):

KE = 0.5 × 1 500 × 27.78² ≈ 579 000 J ≈ 579 kJ

That energy is dissipated by the brakes — converted to heat in the pads and rotors — each time the car stops from highway speed.

Momentum compared with kinetic energy

Momentum and kinetic energy both describe a moving object, but they are different physical quantities:

In a perfectly elastic collision (two billiard balls), both momentum and kinetic energy are conserved. In an inelastic collision (two cars crumpling into each other), only momentum is conserved — the kinetic energy converts to heat, sound, and deformation.

The two quantities are linked by KE = p² / (2m).

The relativistic limit

The formula KE = ½mv² is the classical (Newtonian) approximation, valid when the object's speed is much less than the speed of light (c ≈ 3 × 10⁸ m/s). For everyday objects — cars, balls, aircraft — the error is negligible.

At speeds above roughly 10% of c (about 30 000 km/s), the full relativistic formula applies:

$KE = (\gamma - 1)mc^2$, where $\gamma = 1 / \sqrt{1 - v^2/c^2}$

Particle physics uses this form routinely; for engineering and sports applications the classical formula is accurate to many decimal places.