Input parameters
Calculated results
Gamma (γ): -
Relativistic momentum p: - kg·m/s
Relativistic momentum p: - MeV/c
Classical momentum m v: - kg·m/s
Relative difference: -
Velocity v: - m/s
β = v / c: -
Total energy E: - J, - MeV
Kinetic energy K: - J, - MeV
| # | Rest mass (kg) | β (v/c) | Velocity (m/s) | Gamma (γ) | Relativistic p (kg·m/s) | Classical p (kg·m/s) | Difference (%) | Total energy (J) | Kinetic energy (J) |
|---|
Example data table
The table below shows relativistic momentum for an electron at several speeds. Values are approximate and given in scientific notation.
| Particle | Rest mass (kg) | β (v/c) | Velocity (m/s) | Gamma (γ) | Relativistic p (kg·m/s) | Classical p (kg·m/s) |
|---|---|---|---|---|---|---|
| Electron | 9.1094×10⁻³¹ | 0.1 | 2.9979×10⁷ | 1.005 | 2.74×10⁻²³ | 2.73×10⁻²³ |
| Electron | 9.1094×10⁻³¹ | 0.5 | 1.4989×10⁸ | 1.155 | 1.58×10⁻²² | 1.37×10⁻²² |
| Electron | 9.1094×10⁻³¹ | 0.9 | 2.6981×10⁸ | 2.29 | 5.64×10⁻²² | 2.46×10⁻²² |
Formula used
In special relativity, the relativistic momentum of a particle is
defined as
p = γ m v, where m is the rest mass,
v is the velocity, and γ (gamma) is the
Lorentz factor.
The Lorentz factor is given by
γ = 1 / √(1 − β²), with
β = v / c and c the speed of light.
As velocity approaches light speed, γ grows rapidly, making
relativistic momentum much larger than its classical counterpart.
For comparison, classical momentum is simply
pclassical = m v. At low velocities
(small β), the relativistic and classical expressions agree
closely, but they diverge significantly at high velocities.
Relativistic total energy is E = γ m c², while
kinetic energy is K = (γ − 1) m c². This calculator
reports both in joules and converts them to mega-electronvolts
using 1 MeV = 1.602176634×10⁻¹³ J for convenience.
The relativistic momentum can also be expressed in units of MeV/c,
using the relation p (MeV/c) = p (kg·m/s) × c / (1.602176634×10⁻¹³).
This is useful for high energy physics, where natural units dominate.
When solving for velocity from a known relativistic momentum, the
calculator uses the relation
p = γ m v together with
γ = 1 / √(1 − β²) to obtain a closed-form expression
for β, then recovers the velocity as v = β c.
How to use this calculator
- Choose the rest mass of your particle and select the appropriate mass unit (kilograms, grams, or MeV/c² for high energy physics applications).
- Optionally pick a preset particle such as an electron, proton, neutron, or alpha particle to auto-fill a realistic rest mass value in kilograms.
- Select whether you want to compute relativistic momentum from mass and velocity, or find the velocity from a known relativistic momentum.
- For the first mode, choose a velocity input format: either β = v/c or direct velocity in meters per second. Enter the corresponding value within the allowed range.
- For the second mode, provide the relativistic momentum in kg·m/s. The calculator will infer β and the associated velocity, ensuring it remains below light speed.
- Optionally adjust the speed of light constant if you are using a different convention, then click the calculate button to generate the results.
- Review the outputs, including γ, relativistic momentum, classical momentum, percentage difference, velocity, β, total energy, and kinetic energy. Export results with the CSV and PDF buttons, or reset to begin again.
Relativistic momentum in context
Understanding relativistic motion
At everyday speeds, objects obey Newtonian mechanics and momentum is simply mass times velocity. As a particle accelerates toward light speed, relativistic effects grow and the classical picture breaks down. This calculator highlights exactly where relativistic corrections matter most.
Classical versus relativistic momentum
Classical momentum increases linearly with velocity. Relativistic momentum, however, is scaled by the Lorentz factor γ. When β is small, both formulas nearly coincide. Near light speed, γ increases rapidly and relativistic momentum becomes significantly larger than the Newtonian prediction.
The role of the gamma factor
The Lorentz factor measures how strongly time, length, and momentum differ from classical expectations. By reporting γ for each calculation, the calculator allows you to see how small velocity changes at high β greatly increase momentum and energy, emphasizing relativistic behavior clearly.
Working with β and velocity
Many high energy physics problems are expressed in terms of β rather than direct velocity. This tool accepts either representation. Converting from β to v occurs internally using the chosen value for c, keeping units consistent while offering flexible input modes.
Preset particle options
Built-in presets for electrons, protons, neutrons, and alpha particles help you start quickly. Selecting a preset automatically fills a realistic rest mass. You can then vary β to explore how momentum and energy scale across different particle species.
Energy outputs for deeper insight
Beyond momentum, total relativistic energy and kinetic energy are displayed in joules and MeV. These quantities connect momentum calculations with energy budgets in accelerators, detectors, and astrophysical environments, making the tool useful for numerical experiments and coursework.
Applications in study and teaching
Instructors can use the calculator to generate classroom examples, while students can test homework scenarios or research problems. Exported CSV and PDF files make it simple to document results, compare cases, and embed numbers inside lab reports or digital lecture notes.
Frequently asked questions
When do I really need relativistic momentum?
You need relativistic momentum whenever a particle’s speed is a noticeable fraction of light speed. As a quick rule, if β exceeds about 0.1, classical momentum begins to deviate measurably from relativistic predictions.
Why does classical momentum fail at high speeds?
Classical momentum assumes time and space are absolute. At high speeds, time dilation and length contraction change how motion behaves. The Lorentz factor accounts for these effects, making relativistic momentum consistent with experimental observations and special relativity.
Can I use this tool for massless particles?
No, this calculator assumes a nonzero rest mass. Massless particles, such as photons, obey different relationships where energy and momentum are linked directly by E = pc, and rest mass is identically zero.
Which units should I choose for rest mass?
Use kilograms or grams for general physics problems and MeV per c squared for high energy physics. The calculator internally converts all supported units into kilograms before evaluating momentum and energy expressions.
How accurate are the numerical results?
The calculator uses double precision arithmetic available in modern browsers. For typical educational and laboratory problems, this precision is more than sufficient, although extreme parameter values may accumulate rounding errors.
Can I export data for plotting elsewhere?
Yes. Use the CSV export button to download a comma separated table of all calculated cases. You can import this file into spreadsheets, plotting tools, or numerical software to build detailed graphs and comparisons.