## Friday, 16 July 2021

### Kinetic Energy ≠ ½mv²

While studying $E=mc^2$ I keep noticing that kinetic energy is not $\frac{1}{2}mv^2$ as we were always taught in school. The relativistic formula is, in various forms, $$K_{rel}=\frac{c^3}{\sqrt{c^2-v^2}}-c^2=c^2\left(\frac{1}{\sqrt{1-\beta^2}}-1\right)=c^2\left(\frac{1}{2}\beta^2+\frac{3}{8}\beta^4+\ldots\right)$$where $\beta=v/c$.

On May 2nd 2021 the Parker Solar Probe achieved a velocity of 532,000 km/h which is about $0.0005c$. That gives
\begin{align}\frac{K_{rel}}{c^2}=1.25000023\times{10}^{-7}\ ,\ \frac{K_{cl}}{c^2}=1.25\times{10}^{-7}&\phantom {10000}(8)\nonumber\end{align}Not much difference. The mass of the probe is about 600kg so the difference is $2.3\times{10}^3\ Joules$ which might keep a phone going for an hour.

The difference is more noticeable at much higher velocities. At $0.1c$ the difference is about 1%. It goes wild as you approach the speed of light.
A bit more here: 2.2 Kinetic energy.pdf (1 page)