Answer
$v=0.99c$
Work Step by Step
To find the ratio of relativistic momentum to classical momentum, make substitutions that $p=mv$ and $p_{rel}=\gamma mv$ to get the ratio $$=\frac{\gamma mv}{mv}=\gamma$$ This ratio is equal to $7.5$, so this leads to $$\gamma=7.5$$ Solving the gamma equation $$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$ for the speed parameter $\beta$ yields $$\gamma^2=\frac{1}{1-\beta^2}$$ $$1-\beta^2=\gamma^{-2}$$ $$\beta=\sqrt{1-\gamma^{-2}}$$ Substituting the known value of $\gamma=7.5$ yields a speed parameter beta of $$\beta=\sqrt{1-7.5^{-2}}=0.99$$ Since $v=\beta c$, the speed of the object must be $v=0.99c$.
