Relativistic Kinetic Energy Calculator (K = (γ − 1)mc² vs ½mv²)

Understanding the relativistic kinetic energy calculator — concept and principle

This calculator finds the kinetic energy of a moving body using special relativity and compares it with the classical ½mv². It also gives the Lorentz factor, total energy, rest energy, momentum and the energy in MeV. It is used in physics courses and for particle beams, cosmic rays and accelerators.

Near light speed γ climbs steeply, so the true energy is over 12 times the classical estimate.

How it is calculated

γ = 1 ÷ √(1 − v²/c²)

K = (γ − 1) m c²

Total energy E = γ m c²; rest energy E₀ = m c²; momentum p = γ m v

Classical estimate ½ m v²

Mass in kg, electron or proton masses; speed in m/s, km/s or % of c. 1 MeV = 1.602176634 × 10⁻¹³ J.

The principle

In relativity the energy of a body is γmc², and at rest it is mc². Kinetic energy is the difference. At low speeds γ − 1 is almost exactly v²/2c², so K reduces to ½mv². As v approaches c, γ grows without limit, so the kinetic energy needed to go faster grows without limit too. That is why no massive object can reach the speed of light, and why classical formulas badly underestimate energies near c.

Worked example

Default inputs: 1 electron mass, 99% of c.

γ = 1 ÷ √(1 − 0.99²) = 7.08881.
Rest energy = mₑc² = 8.187 × 10⁻¹⁴ J.
K = (7.08881 − 1) × 8.187 × 10⁻¹⁴ = 4.985 × 10⁻¹³ J = 3.111 MeV.
Classical ½mv² = 4.012 × 10⁻¹⁴ J, so relativistic ÷ classical = 12.42.
Total energy = 5.804 × 10⁻¹³ J; momentum = 1.917 × 10⁻²¹ kg·m/s.

How to read the result

The ratio to the classical value shows how important relativity is: about 1 at everyday speeds, already noticeable above a few percent of c, and enormous near c. Particle physicists usually quote energies in MeV or GeV and momenta in MeV/c.

Common mistakes

Using ½mv² at high speeds.
Forgetting that total energy includes rest energy.
Entering a speed equal to or above c.

Keep in mind: The formulas apply to particles moving freely at constant speed. They do not include radiation losses or forces.

Frequently asked questions

Why can't an electron reach the speed of light?

Its kinetic energy would become infinite.

When does relativity matter?

Above roughly 10% of c the classical formula starts to be noticeably off.

What is the Lorentz factor?

γ = 1/√(1 − v²/c²), the factor by which energy, time dilation and length contraction scale.

Is momentum also different?

Yes; p = γmv, larger than the classical mv.

Sources: OpenStax, University Physics Volume 3 (2016), section 5.9 "Relativistic Energy"

Checked 9 Nov 2026. OpenStax, University Physics Volume 3 (2016), §5.9 “Relativistic Energy” (K_rel = (γ − 1)mc², E = γmc², E₀ = mc², low-speed limit ½mu²; Example 5.12: electron at 0.990c, 3.12 MeV vs 0.251 MeV classical). Constants: c = 299,792,458 m/s, e = 1.602176634 × 10⁻¹⁹ C, k_B = 1.380649 × 10⁻²³ J/K, h = 6.62607015 × 10⁻³⁴ J·s and N_A = 6.02214076 × 10²³ mol⁻¹ are exact in the 2019 SI.

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