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Formula Used
For a particle with rest mass m moving at speed v, the Lorentz factor is:
γ = 1 / √(1 − v²/c²)
Total relativistic energy, rest energy, and kinetic energy:
E = γmc², E₀ = mc², K = (γ − 1)mc²
If momentum p is known instead, energy is computed using:
E = √((pc)² + (mc²)²)
How to Use This Calculator
- Select Use velocity or Use momentum.
- Enter the rest mass and choose its unit.
- Provide the matching input (velocity or momentum) with units.
- Press Calculate to view results above the form.
- Use Download CSV or Download PDF for records.
Example Data Table
| Rest mass | Input | γ | Total energy (J) | Kinetic energy (J) |
|---|---|---|---|---|
| 9.109×10−31 kg (electron) | v = 0.80c | 1.667 | 1.365×10−13 | 5.117×10−14 |
| 1.673×10−27 kg (proton) | v = 0.50c | 1.155 | 1.735×10−10 | 2.335×10−11 |
| 1 u | p = 500 MeV/c | ≈ 1.133 | ≈ 1.693×10−10 | ≈ 1.989×10−11 |
Examples are illustrative and rounded for readability.
This overview explains how total energy behaves at relativistic speeds and how to interpret the outputs.
1) Understanding relativistic total energy
Relativistic total energy is the full energy budget of a moving particle, including its rest energy and motion energy. At everyday speeds, classical formulas work well, but at high fractions of light speed the Lorentz factor boosts energy strongly. This calculator reports total energy, rest energy, and kinetic energy together for clear comparisons.
2) Choosing inputs: mass with velocity or momentum
You can compute energy from rest mass and velocity, or from rest mass and momentum. The velocity method is intuitive for spacecraft and lab experiments, while the momentum method is common in particle beams. Both routes produce the same total energy when units are consistent and the inputs describe the same physical state.
3) Lorentz factor and the speed limit
The Lorentz factor γ = 1/√(1−v²/c²) quantifies time dilation and energy amplification. As v approaches c, the denominator shrinks and γ grows rapidly. That is why the calculator blocks speeds equal to or exceeding c. Even 0.99c corresponds to γ≈7.09, already multiplying energies by more than seven.
4) Rest energy and mass–energy equivalence
Rest energy E0 = mc² is present even when v = 0. For an electron with m≈9.109×10−31 kg, E0 is about 8.19×10−14 J (≈0.511 MeV). For a proton with m≈1.673×10−27 kg, E0 is about 1.50×10−10 J (≈938 MeV).
5) Kinetic energy growth near light speed
Relativistic kinetic energy is K = (γ−1)mc². Unlike classical ½mv², it does not scale simply with v² at high speed. For a proton at 0.50c, γ≈1.155 and K is roughly 2.34×10−11 J. At 0.80c, γ≈1.667 and K becomes about 1.00×10−10 J.
6) Momentum–energy relation for beams
When momentum is known, the relation E = √((pc)² + (mc²)²) is preferred. Beamline diagnostics often report p in MeV/c or GeV/c, so this calculator converts those values internally. The same equation also lets you infer an effective velocity and γ for reporting and sanity checks.
7) Units, conversions, and reporting
To reduce mistakes, select mass units such as kilograms, grams, atomic mass units, or MeV/c². Velocity can be entered in m/s, km/s, or as a fraction of c. Results are shown in joules and electronvolts for convenience, and exports are available for lab notes and documentation.
8) Common use cases and quick validation
Use this tool for accelerator energies, cosmic‑ray estimates, relativistic flight studies, and teaching demonstrations. A fast validation is to set v = 0 and confirm total energy equals rest energy, with K = 0. Another check is to compare momentum outputs between modes for the same scenario.
1) Why must velocity be less than c?
Special relativity requires massive particles to remain below light speed. As v approaches c, γ grows without bound, so required energy becomes enormous. The calculator prevents unphysical inputs by rejecting v ≥ c.
2) What happens when v equals zero?
When v = 0, γ = 1. Total energy equals rest energy mc², kinetic energy becomes zero, and momentum is zero. This is a quick way to validate units and confirm your mass entry.
3) Which mode should I use for particle physics data?
If your source reports beam momentum in MeV/c or GeV/c, use the momentum mode. If you have a measured speed or a fraction of c, use the velocity mode. Both produce the same total energy for consistent inputs.
4) Can I enter mass in MeV/c²?
Yes. MeV/c² is a common rest‑mass unit in high‑energy physics. The calculator converts MeV/c² to kilograms internally before computing mc² and total energy, then reports energies in joules and eV.
5) Why do results appear in both joules and eV?
Joules are standard SI units, while electronvolts are convenient for atomic and particle scales. Showing both makes it easier to compare with lab specifications, beam energies, and textbook values.
6) How is momentum reported when I use velocity mode?
The calculator computes p = γmv using your mass and velocity. This helps compare against spectrometer readings or to switch into the momentum‑energy relation without re‑entering data.
7) How accurate are the example rows?
The example table is rounded for readability and uses typical constants. Your computed outputs may differ slightly due to rounding and chosen units. For precise work, rely on the calculator’s numeric results and exports.