The classical momentum is given by:
[tex]p_c=mv[/tex]
where m is the particle mass and v its velocity, while the relativistic momentum is given by:
[tex]p_r=\gamma mv[/tex]
where
[tex]\gamma = \frac{1}{ \sqrt{1- \frac{v^2}{c^2} } } [/tex] (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
[tex] \frac{p_r - p_c}{p_r} = 0.0135 [/tex]
which means
[tex]p_r = \frac{p_c}{0.9865} [/tex]
and since [tex]p_r = \gamma p_c[/tex], this also means that
[tex]\gamma = \frac{1}{0.9865}=1.0137 [/tex]
Now let's re-arrange eq.(1), and we get
[tex]v=c \sqrt{1- \frac{1}{\gamma^2} } [/tex]
and if we use [tex]\gamma=1.0137[/tex] as we found before, and [tex]c=3 \cdot 10^8 m/s[/tex], we find the corresponding velocity:
[tex]v=(3 \cdot 10^8 m/s) \sqrt{1- \frac{1}{(1.0137)^2} } = 4.9 \cdot 10^7 m/s[/tex]
