Why ½mv² Is Only an Approximation

Einstein's kinetic energy is mc²(1/√(1 − v²/c²) − 1). Expand the square root with the binomial series and the first term is the familiar ½mv²; the next terms are tiny at everyday speeds and huge near light speed.

How it works

The binomial series

In one sentence

For any real $k$ and $|u| < 1$ , $(1 + u)^k = \sum_{n=0}^\infty\binom{k}{n}u^n$ , where $\binom{k}{n} = \frac{k(k-1)\cdots(k-n+1)}{n!}$ .

General formula

{\htmlData{sym=0}{K}} = {\htmlData{sym=1}{m}}{\htmlData{sym=2}{c}}^2\left[\left(1 - \frac{{\htmlData{sym=3}{v}}^2}{{\htmlData{sym=2}{c}}^2}\right)^{-1/2} - 1\right] = \frac12{\htmlData{sym=1}{m}}{\htmlData{sym=3}{v}}^2 + \frac38\frac{{\htmlData{sym=1}{m}}{\htmlData{sym=3}{v}}^4}{{\htmlData{sym=2}{c}}^2} + \frac{5}{16}\frac{{\htmlData{sym=1}{m}}{\htmlData{sym=3}{v}}^6}{{\htmlData{sym=2}{c}}^4} + \cdots

The binomial theorem works for any power, not just whole numbers, as long as you accept an infinite series that converges when $|u| < 1$ .

What the symbols mean

$K$
Kinetic energy
Energy of motion.
$m$
Mass
The object's rest mass.
$c$
Speed of light
About 300,000 km/s.
$v$
Speed
How fast the object moves.

The binomial series: $(1 + u)^k = 1 + ku + \frac{k(k-1)}{2!}u^2 + \cdots$ for $|u| < 1$ , for any power $k$ .
Use $k = -\frac12$ and $u = -\frac{v^2}{c^2}$ : $(1 - v^2/c^2)^{-1/2} = 1 + \frac12\frac{v^2}{c^2} + \frac38\frac{v^4}{c^4} + \cdots$ .
Subtract 1 and multiply by $mc^2$ : the first term is $\frac12mv^2$ .
At a car's speed the next term is a trillion times smaller; near light speed the series converges slowly and you need the exact formula.

Learn the math

The Binomial SeriesCalc II · Lesson 41, Power Series and Taylor Series