Question

At what speed does the classical momentum, p=mv, give an error, when compared with the relativistic momentum, of 1.35 % ?

Answer 1

The classical momentum is given by:

`p_c=mv`
where m is the particle mass and v its velocity, while the relativistic momentum is given by:

`p_r=\gamma mv`
where

`\gamma = \frac{1}{ \sqrt{1- (v^2)/(c^2) } }` (1)
is the relativistic factor, with c being the speed of light.

The condition given by the problem (error of 1.35%) can be rewritten as

`(p_r - p_c)/(p_r) = 0.0135`
which means

`p_r = (p_c)/(0.9865)`
and since
`p_r = \gamma p_c`, this also means that

`\gamma = (1)/(0.9865)=1.0137`

Now let's re-arrange eq.(1), and we get

`v=c \sqrt{1- (1)/(\gamma^2) }`
and if we use
`\gamma=1.0137` as we found before, and
`c=3 \cdot 10^8 m/s`, we find the corresponding velocity:

`v=(3 \cdot 10^8 m/s) \sqrt{1- (1)/((1.0137)^2) } = 4.9 \cdot 10^7 m/s`