Answer
(a) $t = 7.69~s$
The minimum coefficient of friction is $1.38$
(b) $t = ~9.20s$
Work Step by Step
(a) We can find the acceleration:
$v_f^2 = v_0^2+2ad$
$a = \frac{v_f^2-v_0^2}{2d}$
$a = \frac{(104~m/s)^2-0}{(2)(400.0~m)}$
$a = 13.52~m/s^2$
We can find the time to complete the race:
$v_f = v_0+at$
$t = \frac{v_f-v_0}{a}$
$t = \frac{104~m/s-0}{13.52~m/s^2}$
$t = 7.69~s$
We can find the minimum coefficient of static friction:
$mg~\mu_s = ma$
$\mu_s = \frac{a}{g}$
$\mu_s = \frac{13.52~m/s^2}{9.80~m/s^2}$
$\mu_s = 1.38$
(b) We can find the new acceleration:
$a = 0.70~a_0 = (0.70)(13.52~m/s^2) = 9.46~m/s^2$
We can find the time to complete the race:
$d = \frac{1}{2}at^2$
$t = \sqrt{\frac{2d}{a}}$
$t = \sqrt{\frac{(2)(400.0~m)}{9.46~m/s^2}}$
$t = ~9.20s$
