How this calculator works
An electron starting from rest gains kinetic energy equal to the magnitude of its charge times the accelerating potential. Classical mechanics uses mv²/2; relativity uses (gamma-1)mc². The relativistic result stays below c for physical finite voltages, while the classical comparison can exceed c and then loses physical validity. Only potential magnitude is used: the electrode polarity must accelerate the negative electron along the intended direction. Energy losses, initial motion and field emission are omitted. The electron mass is CODATA 2018.
Formula
eV = K; v_classical = sqrt(2eV/me); gamma = 1+eV/(me c²); v_rel = c sqrt(1-gamma^-2).
Worked example
At zero accelerating voltage, both models give zero speed. At about 510998.950 V, the gained kinetic energy equals the electron rest energy, gamma is 2 and speed is about 0.8660254c.
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Frequently asked questions
Why do the models disagree?
The classical kinetic-energy formula is only the low-speed limit of the relativistic formula.
Can the classical result exceed light speed?
The formula can produce that number, which shows that the approximation is no longer physically valid.
Does this include the electron initial energy?
No. The electron is assumed to start from rest and gain energy without losses.