Link energy and momentum using special relativity accurately. Select units and solve for any variable. Get rest energy, speed, and kinetic energy instantly here.
The relativistic energy–momentum relation connects total energy E, momentum p, and rest mass m:
E² = (pc)² + (mc²)²
| Case | Momentum p (MeV/c) | Rest Mass m (MeV/c²) | Computed Total Energy E (MeV) | Computed Kinetic Energy K (MeV) |
|---|---|---|---|---|
| Electron-like | 1.000 | 0.511 | 1.123 | 0.612 |
| Moderate momentum | 3.000 | 0.938 | 3.144 | 2.206 |
| High momentum | 10.000 | 0.938 | 10.044 | 9.106 |
These examples assume units commonly used in particle physics. Results are rounded for display.
Einstein’s energy–momentum relation links total energy E, momentum p, and rest mass m through E² = (pc)² + (mc²)². It unifies classical motion and relativity, ensuring energy stays consistent at high speeds. The calculator lets you solve any one variable when the other two are known, using clear unit conversions. It is handy for laboratory checks too.
For everyday speeds, E ≈ mc² + p²/(2m) works well, but it breaks down as velocity approaches the speed of light. Relativistic effects become noticeable when kinetic energy is a meaningful fraction of rest energy. In particle beams, nuclear decays, and cosmic rays, the exact relation prevents large errors.
This tool uses c = 299,792,458 m/s. Because c is large, relativistic momentum p = γmv grows rapidly, and small mass particles can carry huge energies. Keeping c explicit also helps unit checking: pc has energy units, matching E and mc² in every system.
In particle physics, energies are often expressed in eV, keV, MeV, or GeV. Momentum may be written as MeV/c or GeV/c, and mass as MeV/c² or GeV/c². The calculator supports these formats and converts internally, so you can mix inputs safely and still obtain consistent outputs.
If you solve for energy, the result is E = √((pc)² + (mc²)²). Solving for momentum uses p = √(E² − (mc²)²)/c, valid only when E ≥ mc². Solving for mass uses m = √(E² − (pc)²)/c², requiring E ≥ pc; otherwise the inputs are nonphysical.
Total energy includes rest energy and kinetic energy: E = mc² + K. The calculator reports rest energy mc² and kinetic energy K = E − mc² for intuition. For a proton, mc² ≈ 938.272 MeV; for an electron, mc² ≈ 0.511 MeV, showing why electrons become relativistic easily at modest energies.
For massless particles, m = 0 and the relation reduces to E = pc (photons). In the ultra‑relativistic limit where pc ≫ mc², total energy is approximately pc and kinetic energy dominates. In the non‑relativistic limit, K is small and classical approximations return smoothly, which is useful for cross‑checking.
A quick check is E should never be below mc² for massive particles, and E should be at least pc for any particle. If you enter values that violate E² ≥ (mc²)² or E² ≥ (pc)², the calculator flags the constraint. When measurements have uncertainty, vary inputs slightly to see sensitivity, record ranges, and keep significant figures consistent.
It solves the relativistic energy–momentum equation for E, p, or m, and also reports rest energy and kinetic energy when possible.
You can enter SI-style values (J, kg, N·s) or particle-physics units (eV, MeV, GeV with /c and /c²). The tool converts them consistently before calculating.
The square‑root terms require E² ≥ (mc²)² and E² ≥ (pc)². If your inputs violate these inequalities, no real solution exists, so the tool shows a validation warning.
Set m = 0 and solve for energy or momentum. The relation simplifies to E = pc, which is appropriate for photons and other effectively massless particles.
Momentum p is not energy by itself. Multiplying by c gives pc, which has energy units and appears directly in the relation alongside E and mc².
Not directly. You can estimate velocity using p = γmv if you also compute γ from E = γmc², then v = c√(1 − 1/γ²). Use consistent units.
Yes. After you calculate, the export buttons capture the displayed inputs and outputs, producing a simple CSV row list and a printable PDF layout.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.