Calculated Results
Results appear here after submission, directly below the header and above the form.
This calculator estimates relativistic momentum, total energy, kinetic energy, rest energy, gamma, and invariant energy consistency for particles moving close to light speed.
Example Data Table
| Particle | Rest Mass | Velocity | Gamma | Total Energy | Momentum |
|---|---|---|---|---|---|
| Electron | 9.109×10⁻³¹ kg | 0.80c | 1.6667 | 1.365×10⁻¹³ J | 3.642×10⁻²² kg·m/s |
| Proton | 1.673×10⁻²⁷ kg | 0.95c | 3.2026 | 4.818×10⁻¹⁰ J | 1.526×10⁻¹⁸ kg·m/s |
| Alpha particle | 6.645×10⁻²⁷ kg | 0.60c | 1.2500 | 7.465×10⁻¹⁰ J | 1.494×10⁻¹⁸ kg·m/s |
Formula Used
Lorentz factor: γ = 1 / √(1 − v²/c²)
Relativistic momentum: p = γm₀v
Total energy: E = γm₀c²
Rest energy: E₀ = m₀c²
Kinetic energy: K = (γ − 1)m₀c²
Invariant relation: E² = (pc)² + (m₀c²)²
Here, m₀ is rest mass, v is particle velocity, c is light speed, and γ increases sharply as velocity approaches c. The invariant relation is especially helpful for checking numerical consistency across energy and momentum outputs.
How to Use This Calculator
- Enter the particle rest mass and choose its input unit.
- Select whether velocity is entered as β or meters per second.
- Choose display units for energy, momentum, and mass outputs.
- Set significant figures for cleaner scientific notation.
- Click the calculate button to show results above the form.
- Use the CSV button for spreadsheet export.
- Use the PDF button to print or save a neat report.
Why This Calculator Helps
Classical momentum and energy formulas stop being accurate at high speed. This page adds the Lorentz factor automatically, supports common mass-energy units, handles multiple particles, and gives a quick invariant check for more confident physics work.
Frequently Asked Questions
1. What is relativistic momentum?
It is momentum corrected by the Lorentz factor. At high speed, momentum becomes p = γm₀v instead of the classical m₀v expression.
2. Why must velocity stay below light speed?
The Lorentz factor becomes undefined at or above light speed for massive particles. Required energy also grows without bound as v approaches c.
3. What does gamma represent?
Gamma measures how strongly relativity changes momentum, energy, and time-related effects. It equals 1 at rest and rises rapidly near light speed.
4. What is the difference between total and kinetic energy?
Total energy includes rest energy and motion energy. Kinetic energy is only the extra amount above rest energy.
5. Can I use MeV/c² for particle mass?
Yes. That unit is common in particle physics. The calculator converts it internally to kilograms before computing momentum and energy values.
6. Why include the invariant energy relation?
It helps verify that the reported energy and momentum are consistent. Small differences usually come from rounding rather than physics errors.
7. Is this useful for classroom and lab work?
Yes. It supports fast comparisons, unit changes, export options, and clear formulas, making it handy for assignments, demonstrations, and quick checks.