Relativistic Kinetic Energy Calculator

Frequently Asked Questions

What is relativistic kinetic energy?

KE = (γ-1)mc², where γ is the Lorentz factor. It diverges as v approaches c.

When should I use this instead of ½mv²?

By about 0.1c the difference is close to 1%, and it grows rapidly from there; by 0.5c it is enormous.

What is the Lorentz factor?

γ = 1 / √(1 - v²/c²). At rest γ = 1; near c, γ blows up.

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Speed Comparison: β, γ, and the Energy Ratio

This table shows how the Lorentz factor γ and the ratio of relativistic to classical kinetic energy (KE_rel ÷ KE_classical) grow with speed. Even at 0.1c the classical formula is already off by nearly 1%, and by 0.99c it underestimates the true energy more than twelvefold.

Speed (β = v/c)

γ = 1/√(1−β²)

KE_rel ÷ KE_classical

0.1c

1.005

1.008

0.5c

1.155

1.238

0.9c

2.294

3.195

0.99c

7.089

12.42

The ratio equals 2(γ − 1) ÷ β², which tends to 1 at low speed (where the two formulas agree) and diverges as β approaches 1.

Worked Example

An electron (m = 9.11 × 10⁻³¹ kg) at v = 2 × 10⁸ m/s (β ≈ 0.667). γ ≈ 1.342. KE = (1.342 − 1) × 9.11e−31 × (3e8)² ≈ 2.8 × 10⁻¹⁴ J ≈ 0.18 MeV. The classical formula gives ≈ 0.114 MeV - the classical value is about 35% too low (the relativistic result is ~54% higher). This example rounds c to 3 × 10⁸ m/s for readability; the calculator uses the exact c = 299,792,458 m/s, so its digits will differ slightly from the round numbers above.

Common Mistakes

Using ½mv² at relativistic speeds - it underestimates badly above ~0.1c.

Forgetting v must be strictly less than c (3 × 10⁸ m/s).

Confusing rest mass and relativistic mass; this calc uses rest mass (the modern convention).

Reporting total energy E = γmc² instead of kinetic energy - this calculator subtracts the rest energy mc².

Tips & FAQ

When does relativity matter? By ≈ 0.1c the classical formula is off by nearly 1% (about 0.76% at exactly 0.1c), and the error climbs rapidly at higher speeds. Particle accelerators, cosmic rays, and electrons in CRTs need relativistic treatment.

What is γ? The Lorentz factor. At low speed γ ≈ 1; at v = c, γ → ∞.

Why can't v reach c? Because energy diverges - reaching c would require infinite energy.

Relativistic Kinetic Energy Calculator

Frequently Asked Questions

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