Question

A rod moves lengthwise with a constant velocity $v$ relative to the inertial reference frame $K$. At what value of $v$ will the length of the rod in this frame be $\eta =0.5\%$ less than its proper length?

Answer 1

From the formula for length contraction
$(l_0-l_0\sqrt{1-\frac{v^2}{c^2}})=\eta l_0$
So, $1-\frac{v^2}{c^2}={(1-\eta )}^2$ or $v=c\sqrt{\eta (2-\eta )}$

$⟹v=c\sqrt{0.005(2-0.005)}$

$⟹v=0.09988c$