Relativistic Momentum Calculator

The relativistic momentum of a particle with rest mass $$m_0$$ moving at velocity v is:

$$p = \gamma m_0 v = \frac{m_0 v}{\sqrt{1 - v^2/c^2}}$$

This definition ensures that conservation of momentum holds in all inertial reference frames, which the classical expression $$p = m_0 v$$ fails to do at high speeds. Key properties:

• Reduces to classical at low speeds: When $$v \ll c$$, $$\gamma \approx 1$$ and $$p \approx m_0 v$$.
• Diverges as v → c: Since $$\gamma \to \infty$$, the momentum becomes infinite — consistent with the impossibility of accelerating massive objects to c.
• Conservation: In relativistic collisions, $$\sum \gamma_i m_{0i} \vec{v}_i$$ is conserved, not $$\sum m_{0i} \vec{v}_i$$.
• Energy-momentum relation: $$E^2 = (pc)^2 + (m_0 c^2)^2$$. For massless particles (photons), $$E = pc$$.

The momentum connects to force through the relativistic form of Newton's second law: $$\vec{F} = \frac{d\vec{p}}{dt} = \frac{d}{dt}(\gamma m_0 \vec{v})$$. This is more complex than $$F = ma$$ because γ also depends on v and changes with time when force is applied.

In particle physics, momentum is commonly expressed in units of eV/c (electron volts divided by the speed of light). This comes from the energy-momentum relation and the convention of setting natural units. A 1 GeV/c proton has momentum of about $$5.34 \times 10^{-19}$$ kg·m/s.

For ultra-relativistic particles ($$v \approx c$$, $$\gamma \gg 1$$), the momentum is approximately $$p \approx E/c$$, and the distinction between rest energy and kinetic energy becomes negligible compared to the total energy.

The relativistic/classical ratio equals the Lorentz factor γ, showing how much classical physics underestimates the momentum. A ratio of 2 means the true momentum is double the Newtonian prediction. In particle detector design, using classical momentum would give completely wrong predictions for particle trajectories in magnetic fields. The eV/c output is standard for comparing with particle physics data and accelerator specifications.