Kinetic Energy Calculator

Solve for kinetic energy, mass, or velocity — Newtonian or relativistic mechanics, 10+ unit options.

Physics model

Newtonian (classical) Relativistic (special relativity)

Solve for

Kinetic energy Mass Velocity

Mass

Velocity

Kinetic Energy

E_k

About kinetic energy

What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion. Any object with mass that is moving has kinetic energy. It is a scalar quantity — it has magnitude but no direction — and it is always non-negative.

Newtonian formula (classical mechanics)

In classical mechanics, valid when the object's speed is much less than the speed of light (v ≪ c), kinetic energy is:

E_k = ½mv²

where m is mass in kilograms and v is speed in metres per second. The result is in joules (J). This formula is accurate to within 1% for speeds below about 4 000 km/s (1.3% of c).

Relativistic formula (special relativity)

When an object's speed approaches the speed of light, the Newtonian formula breaks down. Einstein's special relativity gives the correct expression:

γ = 1 / √(1 − v²/c²) (Lorentz factor)
E_k = (γ − 1)mc²

Here c = 299 792 458 m/s is the speed of light (exact). At low speeds, Taylor-expanding γ recovers the classical result: (γ−1)mc² ≈ ½mv². As v→c, the Lorentz factor γ→∞ and kinetic energy→∞, which explains why no massive object can reach or exceed c.

When to use each model

Use the Newtonian model for everyday speeds (cars, planes, even spacecraft like Voyager travel at roughly 17 km/s — 0.006% of c). Use the relativistic model for particle physics (electrons in an accelerator can reach 99.9999999% c), cosmic rays, and any scenario where v > ~0.1c (30 000 km/s).

	Newtonian (classical)	Relativistic (special relativity)
Formula	`½mv²`	`(γ−1)mc²`
Valid when	v ≪ c	always valid
Error at v = 0.1c	0.25%	0%
Error at v = 0.5c	6.7%	0%
When v→c	underestimates	E_k→∞

Frequently asked questions

Can kinetic energy be negative?

No. Kinetic energy is always ≥ 0. It equals zero only when an object is at rest (v = 0). Since mass is always positive and v² is always non-negative, ½mv² ≥ 0 and (γ−1)mc² ≥ 0 always.

What is the Lorentz factor γ?

The Lorentz factor γ = 1/√(1−v²/c²) quantifies how much special relativity departs from classical mechanics. At rest, γ = 1. At 90% c, γ ≈ 2.29. At 99% c, γ ≈ 7.09. At 99.9% c, γ ≈ 22.4. Kinetic energy grows without bound as v→c.

How do I convert kinetic energy to temperature?

For a monatomic ideal gas, the average kinetic energy per molecule is (3/2)k_B T, where k_B = 1.380649×10⁻²³ J/K is the Boltzmann constant. So T = 2E_k / (3k_B).

What is the difference between kinetic energy and momentum?

Momentum p = mv is a vector quantity; kinetic energy E_k = p²/(2m) = ½mv² is a scalar. Two objects can have the same kinetic energy but opposite momenta (e.g. in a collision). Momentum is conserved in all collisions; kinetic energy is conserved only in elastic collisions.

Why does doubling velocity quadruple kinetic energy?

In the Newtonian formula E_k = ½mv², kinetic energy is proportional to v². So if you double v, E_k increases by 2² = 4×. This is why highway speeds are so much more dangerous than city speeds: a car at 120 km/h has four times the kinetic energy of the same car at 60 km/h.

What are the SI units of kinetic energy?

The SI unit is the joule (J = kg·m²/s²). Other common units: 1 kJ = 1 000 J; 1 cal = 4.184 J; 1 kWh = 3 600 000 J; 1 eV = 1.602×10⁻¹⁹ J (used in atomic and particle physics).