Relativistic Momentum Calculator
γmv, and the ratio to classical is exactly γ.
Work out Relativistic Momentum. γmv, and the ratio to classical is exactly γ. Names the misconception directly.
A proton is 1.673×10⁻²⁷ kg; an electron is 9.109×10⁻³¹.
Relativistic momentum (kg·m/s)
3.5191 × 10^-18
7.0888× the classical value · γ = 7.08881
Relativistic momentum is γmv rather than mv, and the ratio between the two is exactly γ. Below about 10% of light speed the difference is under half a percent and Newton is fine; at 99% the momentum is seven times what classical mechanics predicts. This is not a correction factor bolted on. Momentum conservation only holds in every reference frame if momentum is defined this way, which is how the form is derived rather than fitted. Particle accelerators are where it becomes an engineering constraint. At the LHC protons reach about 0.999999991c, where γ exceeds 7,000 — so almost all the energy poured in goes into momentum rather than speed, and the last fraction of a percent of c costs more than everything before it.
How the Relativistic Momentum Calculator works
Relativistic momentum and kinetic energy for a particle at any sub-light speed, against the classical values. The ratio between them is exactly the Lorentz factor, which makes the departure from Newton immediate.
Also known as: momentum near the speed of light · relativistic kinetic energy · gamma factor for a proton · when does newton stop working
Frequently asked questions
What is relativistic momentum?
γmv rather than mv. The ratio to the classical value is exactly γ, so at 99% of light speed the momentum is about seven times what Newton predicts.
Why isn't momentum just mv?
Because momentum conservation only holds in every reference frame if momentum is defined with the γ. The form is derived from that requirement rather than fitted to data.
What is relativistic kinetic energy?
(γ−1)mc², which reduces to ½mv² at low speed. At γ = 2 the kinetic energy equals the entire rest energy of the particle.
How fast do particles go in an accelerator?
At the LHC, protons reach about 0.999999991c, where γ exceeds 7,000. Almost all the energy poured in goes into momentum rather than speed — they are already at 99.9999% of c long before full energy.
When can I ignore relativity?
Below about 10% of light speed, where the correction is under half a percent. Above 50% the classical formulas are badly wrong, and above 90% they are not even the right order of magnitude.
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