Answer
(a) $v = 0.9798~c$
(b) $E = 8.485\times 10^{-11}~J$
Work Step by Step
(a) Since the half-life is five times longer when moving at high speed, $\gamma = 5$
We can find the speed when $\gamma = 5$:
$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
$\sqrt{1-\frac{v^2}{c^2}} = \frac{1}{\gamma}$
$1-\frac{v^2}{c^2} = \frac{1}{\gamma^2}$
$\frac{v^2}{c^2} = 1-\frac{1}{\gamma^2}$
$v^2 = (1-\frac{1}{\gamma^2})~c^2$
$v = \sqrt{1-\frac{1}{\gamma^2}}~c$
$v = \sqrt{1-\frac{1}{5^2}}~c$
$v = \sqrt{0.96}~c$
$v = 0.9798~c$
(b) We can find the total energy:
$E = \gamma~mc^2$
$E = (5)(207)(9.109\times 10^{-31}~kg)(3.0\times 10^8~m/s)^2$
$E = 8.485\times 10^{-11}~J$
