Question

The friction coefficient between an athelete's shoes and the ground is 0.90. Suppose a superman wears these shoses and races for 50m. 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 completing the 50 m starting from rst. (b) Suppose he takes exactly this minimum time to complete the 50 m, what minimum time will he take to stop?

Answer 1

To reach minimum time he has to move with maximum possible accleration.
Let the maximum acceleration is 'a'
\therefore ma - \mu R = 0 \Rightarrow ma = \mu mg\\
\Rightarrow a = \mu g = 0.9 \times 10 = 9 m/s^2 \\

(a) Initial velocity,
u = 0 t = ?\\
$a=9m/s^2$ S = 50 m
From the equation $S=ut+\left(\frac{1}{2}\right)a{t}^{2}50=0+\left(\frac{1}{2}\right)9t^2\Rightarrow t=\frac{10}{3}sec$
(b) After covering 50 m, velocity of the athelete is given by
v = u + at
$=0+9\times \left(\frac{10}{3}\right)m/s$
he has to stop in minimum time. So deceleration is $-a=-9m/s^2$ (max)
R = mg
$ma=\mu R$
(maximum frictional force)
$\Rightarrow a=\mu g$
$=9m/s^2$(Deceleration)
u_1 = 30 m/s, v= 0 \\
$\Rightarrow t=\frac{v_1-v_1}{a}=\frac{0-30}{-a}=\frac{-30}{a}=\frac{10}{3}sec$