Relativistic Kinetic Energy Calculator Formula

Understand the math behind the relativistic kinetic energy calculator. Each variable explained with a worked example.

Use the Relativistic Kinetic Energy Calculator

Formulas Used

Relativistic KE

ke_rel = (1 / sqrt(1 - pow(velocity, 2) / pow(2.998e8, 2)) - 1) * rest_mass * pow(2.998e8, 2)

Classical KE (comparison)

ke_classical = 0.5 * rest_mass * pow(velocity, 2)

Relativistic / Classical

ratio = ((1 / sqrt(1 - pow(velocity, 2) / pow(2.998e8, 2)) - 1) * pow(2.998e8, 2)) / (0.5 * pow(velocity, 2))

Variables

VariableDescriptionDefault
rest_massRest Mass(kg)1
velocityVelocity(m/s)269800000

How It Works

Relativistic Kinetic Energy

At high speeds, the classical KE formula (0.5mv^2) underestimates the true kinetic energy.

Formula

KE = (gamma - 1) * m0 * c^2

This approaches 0.5 m0 v^2 for v much less than c, and grows without bound as v approaches c.

Worked Example

1 kg at 0.9c.

rest_mass = 1velocity = 269800000
  1. 01gamma = 2.294
  2. 02KE = (2.294 - 1) * 1 * (2.998e8)^2
  3. 03KE = 1.294 * 8.988e16
  4. 04KE = 1.163e17 J
  5. 05Classical: 0.5 * 1 * (2.698e8)^2 = 3.640e16 J
  6. 06Relativistic is 3.2x larger

Ready to run the numbers?

Open Relativistic Kinetic Energy Calculator