Answer
$v=0.81c$
Work Step by Step
Relativistic length is equal to $$L'=\frac{L}{\gamma}$$ Solving for $\gamma$ yields $$\gamma=\frac{L}{L'}$$ Substituting known values of $L=250m$ and $L'=150m$ yields a gamma value of $$\gamma=\frac{250m}{150m}=1.7$$ Use the gamma equation $$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$ where $v=\beta c$, to solve for speed parameter $\beta$. $$\gamma^2=\frac{1}{1-\beta^2}$$ $$1-\beta^2=\gamma^{-2}$$ $$\beta=\sqrt{1-\gamma^{-2}}$$ Substituting the known value of $\gamma=1.7$ yields a beta value of $$\beta=\sqrt{1-1.7^{-2}}=0.81$$ Therefore, the speed of the ship must be $v=0.81c$.
