Answer
$v=0.99986c$
Work Step by Step
Dilated time is equal to $$\Delta t'=\gamma \Delta t$$ If 1.0000 second corresponds to 1.0000 minute on Earth, the gamma value must be $\frac{60s}{1s}=60.$ Use the formula for gamma $$\gamma=\frac{1}{\sqrt{1-\beta^2}}$$ where beta is $v=\beta c$, to solve for $\beta$. $$\gamma^2=\frac{1}{1-\beta^2}$$ $$1-\beta^2=\gamma^{-2}$$ $$\beta=\sqrt{1-\gamma^{-2}}$$ Substituting the known value of $\gamma=60.000$ yields a beta value of $$\beta=\sqrt{1-60.000^{-2}}=0.99986$$ Therefore, using the definition of beta, the velocity is equal to $v=0.99986c$.
