Relativistic Momentum Calculator
At velocities approaching the speed of light, the classical formula p = mv must be replaced by the relativistic momentum p = gamma * m * v, where gamma = 1 / sqrt(1 - v^2/c^2) is the Lorentz factor. As v increases toward c, gamma grows without bound, so the momentum (and the energy required to accelerate the object further) increases without limit. This is why no massive object can ever reach the speed of light. In particle accelerators like CERN's LHC, protons are accelerated to 99.9999991% of c, where gamma exceeds 7,000 and their relativistic momentum is thousands of times their classical momentum. Enter the rest mass in kilograms and the velocity as a fraction of c.
Relativistic momentum formula
gamma = 1 / sqrt(1 - beta^2)
p = gamma * m * v = gamma * m * beta * c
Where beta = v/c, m is the rest mass, c = 2.99792458 x 10^8 m/s. Classical momentum is p_c = mv = m * beta * c. The ratio p/p_c = gamma, which shows how much the relativistic momentum exceeds the classical value.
Comparison: relativistic vs classical momentum
- At beta = 0.1: gamma = 1.005, relativistic momentum is 0.5% above classical.
- At beta = 0.5: gamma = 1.155, relativistic momentum is 15.5% above classical.
- At beta = 0.9: gamma = 2.294, relativistic momentum is 129% above classical.
- At beta = 0.99: gamma = 7.089, relativistic momentum is 609% above classical.
Relativistic momentum: frequently asked questions
Why is classical momentum insufficient at high speeds?
Classical momentum p = mv assumes mass is constant and velocities can add simply. Near the speed of light, this breaks down: adding velocity to an already fast-moving object requires more and more force. Relativistic momentum p = gamma m v grows without bound as v approaches c, correctly describing why objects cannot reach c.
What is the Lorentz factor gamma?
Gamma = 1 / sqrt(1 - v^2/c^2). At low speeds it is approximately 1 and classical mechanics is recovered. At 90% of c, gamma = 2.294; at 99% of c, gamma = 7.089. As v approaches c, gamma diverges, making the momentum and energy required to accelerate an object grow without limit.
How does relativistic momentum relate to force?
Newton's second law generalizes to F = dp/dt where p is the relativistic momentum. Because p = gamma m v, a force applied to an already fast-moving object increases both v and gamma, so the increase in velocity per unit force decreases as v approaches c.
Is relativistic momentum conserved?
Yes. Conservation of relativistic momentum holds in all inertial frames in special relativity. This is a fundamental symmetry (Noether's theorem applied to spatial translation invariance). It is verified in every collision experiment at high-energy physics facilities.
How does relativistic momentum differ from photon momentum?
For a massive particle, p = gamma m v. For a photon (m = 0), this formula gives 0/0 at v = c, but photon momentum is defined by p = h/lambda = E/c. These are consistent: the energy-momentum relation E^2 = (pc)^2 + (mc^2)^2 gives p = E/c for m = 0.
Official sources
- OpenStax University Physics Vol. 3: Relativistic Momentum.
- NIST: Speed of Light in Vacuum (CODATA 2018).
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.