Relativistic Energy Calculator
Calculate total relativistic energy using E = γmc² = √((pc)² + (mc²)²).
What Is Relativistic Energy Calculator?
The formula Calculate total relativistic energy using E = γmc² = √((pc)² + (mc²)²). is fundamental to this area of physics.
It follows from first principles and has wide application in science and engineering.
Pay careful attention to units — results are only valid when SI units are used consistently.
See the worked examples for typical values and real-world applications of this formula.
Formula Reference Table
| Solve For | Formula | Notes |
|---|---|---|
| Total energy | E = γmc² = mc²/√(1−v²/c²) | J or MeV |
| Relativistic KE | KE = (γ−1)mc² | J; reduces to ½mv² at low v |
| Rest energy | E₀ = mc² | J; always present |
| Energy-momentum | E² = (pc)² + (mc²)² | Useful when p is known |
| Lorentz factor | γ = 1/√(1−β²) | β = v/c |
| Threshold energy | E ≥ mc² | Minimum total energy |
3 Worked Examples
m_e = 9.109×10⁻³¹ kg, v = 0.9c.
- γ = 1/√(1−0.81) = 1/0.436 = 2.294
- E = γm_ec² = 2.294×9.109×10⁻³¹×(3×10⁸)²
- E = 1.881×10⁻¹³ J = 1.174 MeV
- Rest energy = 0.511 MeV; KE = 1.174−0.511 = 0.663 MeV
Proton at 99.9999991%c (γ ≈ 7,461).
- E = γm_pc² = 7461×938.3 MeV = 7,000,000 MeV = 7 TeV
- KE ≈ E (rest energy negligible at this γ)
Electron total energy = 2.0 MeV.
- γ = E/(m_ec²) = 2.0/0.511 = 3.914
- v = c√(1−1/γ²) = c√(1−0.0654) = 0.9664c
Real-World Applications
Common Mistakes to Avoid
SI units.
Correct variant.
Ideal conditions.
Check direction.
Verify.
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Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.