Relativistic Kinetic Energy Calculator Formula
Understand the math behind the relativistic kinetic energy calculator. Each variable explained with a worked example.
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
| Variable | Description | Default |
|---|---|---|
rest_mass | Rest Mass(kg) | 1 |
velocity | Velocity(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.
- 01gamma = 2.294
- 02KE = (2.294 - 1) * 1 * (2.998e8)^2
- 03KE = 1.294 * 8.988e16
- 04KE = 1.163e17 J
- 05Classical: 0.5 * 1 * (2.698e8)^2 = 3.640e16 J
- 06Relativistic is 3.2x larger
Frequently Asked Questions
Why does classical mechanics fail at high speeds?
Classical mechanics assumes that velocities simply add and that kinetic energy grows as v^2. Relativity shows that the correct energy formula involves the Lorentz factor, which diverges as v approaches c.
At what speed does the relativistic correction matter?
At 10% of c, the relativistic KE is about 0.75% higher than classical. Above 30% of c, the difference exceeds 5%.
What happens to a particle accelerated to 0.99c?
gamma = 7.09, so the relativistic KE is 6.09 times the rest energy (m0 c^2). The classical formula would give only 1/7 of this.
Ready to run the numbers?
Open Relativistic Kinetic Energy Calculator