Answer
(a) $v=0.64c$ (b) $L'=3.1m$
Work Step by Step
(a) The length contraction formula equals $$L'=\frac{L}{\gamma}$$ Solving for gamma yields $$\gamma=\frac{L}{L'}$$ Substituting known values of $L=5.0m$ and $L'=4.0m$ yields a gamma value of $$\gamma=\frac{5.0m}{4.0m}=1.3$$ Use the gamma equation to solve for $\beta$. $$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$ $$\gamma^2=\frac{1}{1-\beta^2}$$ $$1-\beta^2=\gamma^{-2}$$ $$\beta=\sqrt{1-\gamma^{-2}}$$ Substituting the known value of $\gamma=1.3$ yields a beta value of $$\beta=\sqrt{1-1.3^{-2}}=0.64$$ Therefore, since $v=\beta c$, the velocity must be $v=0.64c$. (b) The length contraction formula is equal to $$L'=\frac{L}{\gamma}$$ Substituting known values of $\gamma=1.3$ and $L=4.00m$ yields a contracted length of $$L'=\frac{4.0m}{1.3}=3.1m$$
