# Where does the equation $p=\frac{1}{c}\sqrt{T^2 +2mTc^2}$ come from?

Where does the relativistic formula

$$p~=~\frac{1}{c}\sqrt{T^2 +2mTc^2}$$

come from? What is the derivation from Einstein's formula? $T$ is the kinetic energy $m$ is the mass $p$ is the momentum.

• Can you add a bit more context? what is $T$ here? – By Symmetry Jul 26 '15 at 12:40
• Some more context on where you saw this equation would be better as well. – Kyle Kanos Jul 26 '15 at 14:45
• In /gamma decay – physnolimits Jul 26 '15 at 14:46

We can write total energy $E$ two ways: $$E^2=p^2c^2+m^2c^4 \\ E=T+mc^2,$$ where $T$ is kinetic energy. Eliminating $E$ from those two equations will give you the desired result.
Hint: $T = E - E_0 = m\gamma c^2 - mc^2 = mc^2(\gamma -1)$ and $p = |\vec p| = m\gamma |\vec v| = m\gamma v$