Particle Velocity Calculator

The Particle Velocity Calculator determines the velocity of a nuclear or subatomic particle from its kinetic energy and mass, providing both classical and relativistic results for comparison. This is the inverse problem to energy calculation and is frequently needed in nuclear physics for computing time-of-flight, reaction kinematics, magnetic rigidity, and detector response.

In the classical approximation, velocity is \(v = \sqrt{2 KE / m}\). This can yield values exceeding the speed of light for high-energy particles, which is physically impossible—a clear signal that relativistic mechanics is required.

The relativistic velocity uses the Lorentz factor \(\gamma = 1 + KE/(mc^2)\) and then \(\beta = \sqrt{1 - 1/\gamma^2}\), giving \(v = \beta c\). This always yields \(v < c\) regardless of how large the kinetic energy becomes. The approach uses energy units throughout: the rest mass energy \(mc^2 = m_{amu} \times 931.494\) MeV.

Comparing classical and relativistic results reveals when relativity matters. For a proton with 10 MeV kinetic energy (about 1% of its rest mass energy), the classical velocity is 4.38×10⁷ m/s (β = 0.146) while the relativistic velocity is 4.35×10⁷ m/s—only 0.5% different. But for a 938 MeV proton (KE = rest mass energy), classical gives \(v = c\) while relativistic gives \(v = 0.866c\)—a 15% error.

This calculator is particularly useful for time-of-flight spectroscopy (measuring particle velocity to determine mass), designing beam transport systems in accelerators, computing Cherenkov radiation thresholds, and analyzing nuclear reaction products. The Lorentz factor and beta parameters are essential inputs for relativistic kinematics calculations.

Input the kinetic energy in MeV and the particle mass in amu for instant results. Common particle masses: proton (1.00728 amu), neutron (1.00866 amu), alpha (4.00260 amu), electron (0.000549 amu).