The friction coefficient between an athelete's shoes and the ground is $0.90$. Suppose a superman wears these shoes and races for $50 \mathrm{~m}$. There is no upper limit on his capacity of running at high speeds. (a) Find the minimum time that he will have to take in completingthe $50 \mathrm{~m}$ starting from rest. (b) Suppose he takes exactly this minimum time to complete the $50 \mathrm{~m}$, what minimum time will he take to stop?
(a) frictional force exerted by ground will be in forward direction.
$f f=m a$
$\mu N=m a$
$\mu m g=m a$
$a=\mu g=0.9 \times 10=9 \mathrm{~m} / \mathrm{s}^{2}$
$u=0 ; a=9 \frac{m}{s^{2}} ; s=50 \mathrm{~m}$
$s=u t+\frac{1}{2} a t^{2}$
$50=0+\frac{1}{2}(9)(t)^{2}$
$t=\frac{10}{3}$ sec
(b) While stopping, frictional force will act in opposite direction of motion.
$0-f f=m a$
$-\mu N=m a$
$-\mu m g=m a$
$a=-\mu g=-0.9 \times 10=-9 \mathrm{~m} / \mathrm{s}^{2}$
After covering $50 \mathrm{~m}$, the velocity of athelete is
$v=u+a t$
$=0+9 \times \frac{10}{3}$
$v=30 \mathrm{~m} / \mathrm{s}$
Now,
$u=30 \frac{m}{s} ; v=0 ; a=-9 m / s^{2}$
$v=u+a t$
$t=3.33 \mathrm{sec}$