Relativistic Speed Calculator

Most relativity problems hand you $v$ and ask for $\gamma$. The harder and more common direction in practice is the reverse — you measure a dilation, an energy or a decay length, and you want the speed. Invert the Lorentz factor:

$\gamma = \frac{1}{\sqrt{1-\beta^2}} \quad\Longrightarrow\quad \beta = \frac{v}{c} = \sqrt{1 - \frac{1}{\gamma^2}}$

with $c = 2.998 \times 10^8$ m/s. The three routes into $\gamma$:

• From a time dilation or lifetime: $\gamma = \Delta t / \Delta t_0$
• From a total energy: $\gamma = E/(mc^2)$
• From a kinetic energy or an accelerating voltage: $\gamma = 1 + K/(mc^2)$, with $K = qV$ in eV for a charge accelerated through $V$ volts

The assumption people forget: $\beta$ approaches 1 but never reaches it. Doubling the accelerator voltage roughly doubles $\gamma$ but barely moves $v$ — from $0.979c$ to $0.995c$ — which is why beam energy, not beam speed, is the number accelerator physicists quote.

Velocities do not simply add. If $S'$ moves at $v$ relative to $S$, and an object moves at $u'$ in $S'$, then in $S$:

$u = \frac{u' + v}{1 + u'v/c^2}$

all speeds in m/s along the same axis. The denominator is the whole story: it is close to 1 at everyday speeds, so the classical sum survives, and it grows just fast enough to keep $u < c$ whenever $u'$ and $v$ are.

Check the limits. Put $u' = c$ and the expression returns exactly $c$ for any $v$ — the second postulate falls out of the algebra. Put $u' = v = 0.6c$ and you get $0.882c$, not $1.2c$.

A practical consequence is the decay length of an unstable particle. Its lifetime in the lab is $\gamma\tau_0$, so it travels

$d = \beta c \gamma \tau_0$

metres before decaying — the reason atmospheric muons reach the ground at all.

When it applies: inertial frames and collinear motion. Perpendicular components need the full Lorentz transformation.

• Solving $\gamma$ for $\beta$ without squaring — it is $\beta = \sqrt{1 - 1/\gamma^2}$, not $\sqrt{1 - 1/\gamma}$.
• Using $K = \tfrac12 mv^2$ to get a speed — at MeV energies this returns values above $c$, which is the giveaway that the classical formula has broken down.
• Forgetting that $\gamma = 1 + K/mc^2$ needs the rest energy in the same units — $0.511$ MeV for an electron, $938.3$ MeV for a proton.
• Applying time dilation with the wrong clock — $\Delta t_0$ is the proper time, measured where the two events happen at the same place.
• Adding velocities classically and getting a result above $c$.
• Quoting a speed to four figures near $c$ — $0.9999c$ and $0.99999c$ differ enormously in energy but almost not at all in speed.