Relativistic Mechanics Calculator
Calculate time dilation, length contraction, relativistic mass, kinetic energy, and momentum from velocity as a fraction of the speed of light.
About this calculator
This calculator applies the core equations of Einstein's special relativity to a single input velocity, expressed as a fraction of light speed (beta), which it also caps just under 1 to keep the math from producing infinities. Everything flows from the Lorentz factor, gamma = 1/√(1 − beta²): time intervals in a moving frame dilate by that factor (a clock on a fast-moving object ticks slower as seen from a stationary frame), lengths along the direction of motion contract by the inverse of that factor, and both relativistic mass and momentum scale up with gamma as an object's effective inertia increases near light speed. Kinetic energy is computed the fully relativistic way — (gamma − 1) times rest mass times c² — rather than the classical one-half m v² formula, which only approximates the true value at low speeds and diverges badly as velocity approaches c; total energy adds the rest energy (m c²) on top.
One simplification worth knowing: the calculator uses c = 3×10⁸ m/s rather than the exact 299,792,458 m/s, which is accurate to about three significant figures and fine for conceptual work but will introduce small errors in precision applications. The accompanying chart traces the Lorentz factor's characteristic near-vertical climb as beta approaches 1, illustrating why reaching light speed itself would require infinite energy for anything with mass.
Inputs
Results
Lorentz Factor (γ)
2.29
Dilated Time
2.29 s
Contracted Length
0.44 m
≈ 5 credit cards
How to Use This Calculator
- Enter Velocity (fraction of c), Proper Time, and Proper Length.
- Set Rest Mass.
- Review Lorentz Factor (γ), Dilated Time (s), and Contracted Length (m).
- Use Relativistic Mass (kg) and Kinetic Energy (J) to inform your decision.
- Use the chart to visualize the results and explore different scenarios by adjusting inputs.
How the result changes with Velocity (fraction of c)
| Velocity (fraction of c) | Lorentz Factor (γ) | Dilated Time | Contracted Length |
|---|---|---|---|
| 0.45 | 1.12 | 1.12 s | 0.89 m |
| 0.68 | 1.36 | 1.36 s | 0.74 m |
| 1 | 2,236.07 | 2,236.07 s | 0 m |
What each input means
- Velocity (fraction of c)
- Velocity as a fraction of the speed of light (0 to 1). For example, 0.9 = 90% the speed of light.
- Proper Time
- Time interval measured in the moving object's rest frame.
- Proper Length
- Length of the object measured in its rest frame.
- Rest Mass
- Mass of the object at rest. Electron ≈ 9.109×10⁻³¹ kg, proton ≈ 1.673×10⁻²⁷ kg.
How this is calculated
Worked example, using the default values
- Identify Input Parameters4 parametersVelocity (fraction of c) = 0.9, Proper Time = 1, Proper Length = 1, Rest Mass = 1 = 4 input(s) provided
- Calculate Lorentz FactorLorentz Factor = g2.29416 = 2.29416
- Calculate Dilated TimeDilated Time2.29416 = 2.29416
- Calculate Contracted LengthContracted Length0.43589 = 0.43589
- Calculate Relativistic MassRelativistic Mass2.294157338705618 = 2.294157338705618
- Calculate Kinetic EnergyKinetic Energy116474160483505620 = 116474160483505620
Engine last updated . Checked against 1 independently-derived test — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why is the input velocity capped at 0.9999999 times the speed of light?
The Lorentz factor is 1/√(1 − beta²), so as beta approaches exactly 1 the term under the square root approaches zero and gamma approaches infinity. The calculator caps beta just below 1 (and the input's max is set to 0.9999999) so the math always produces a large but finite number rather than dividing by zero or returning Infinity.
Why is relativistic kinetic energy so much larger than the classical ½mv² formula at high speed?
Classical kinetic energy (½mv²) is only the low-speed approximation of the true relativistic expression, (gamma − 1)mc². At everyday speeds gamma is barely above 1 and the two formulas nearly agree, but as velocity climbs toward c, gamma grows without bound while the classical formula keeps growing only with the square of velocity — so the relativistic value diverges sharply upward and correctly reflects that reaching c would take infinite energy.
Why does the calculator use c = 3×10⁸ m/s instead of the exact value?
3×10⁸ m/s is a rounded approximation of the true speed of light, 299,792,458 m/s, accurate to about three significant figures. That's precise enough for conceptual results like the Lorentz factor and general energy/momentum trends, but it will introduce small (roughly 0.07%) errors in any output — like rest energy or velocity in m/s — that depends directly on the value of c.
How are relativistic mass and momentum related to the Lorentz factor?
Both scale directly with gamma: relativistic mass is gamma times rest mass, and relativistic momentum is gamma times rest mass times velocity. Because gamma itself grows sharply as beta approaches 1, both an object's effective mass and its momentum increase far faster than velocity alone would suggest as it nears the speed of light.
Related Calculators
The questions that sit next to this one — chosen by subject, including calculators filed under a different category.
Electromagnetic Wave Calculator
Calculate the wavelength and photon energy of an electromagnetic wave from its frequency using c = fλ and E = hf.
PhysicsProjectile Motion Calculator
Calculate the range, maximum height, and flight time of a projectile given its initial velocity, launch angle, and initial height.
Science & EngineeringMomentum Calculator
Calculate momentum from mass and velocity. Use the formula p = mv.
Science & EngineeringPower Calculator
Calculate power from work and time, or force and velocity. Use formulas P = W/t or P = Fv.
Math & StatisticsScientific Calculator
Scientific calculator with multiple modes and expression input
More in Science & Physics.