Answer
(a) $1.22\times 10^{-15}s$
(b) $8\times 10^6$orbits
Work Step by Step
(a) We know that
$v_2=\frac{2\pi ke^2}{2h}$
We plug in the known values to obtain:
$v_2=\frac{2(3.14)(9\times 10^4)(1.6\times 10^{-19})^2}{2(6.63\times 10^{-34})}$
$v_2=10.91\times 10^5m/s$
Now $t=\frac{(2)(3.14)(5.29\times 10^{-11})}{10.91\times 10^5}$
$v_2=1.22\times 10^{-15}s$
(b) We can determine the required number of orbits as follow:
$n=\frac{10^{-8}}{1.22\times 10^{-15}}$
$\implies n=8\times 10^6$orbits