The friction coefficient between an athelete’s shoes and the ground is 0.90. Suppose a superman wears these shoes and races for 50 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 completing the 50 m starting from rest. (b) Suppose he takes exactly this minimum time to complete the 50 m, what minimum time will he take to stop?
The superman has to move with maximum possible acceleration, to reach the given distance in minimum time,
Let us consider the maximum acceleration is ‘a‘.
∴ ma − μR = 0
⇒ ma = μ mg
⇒ a = μg = 0.9 × 10 = 9 m/s2
(a) First figure shows the first case
As per the question, the initial velocity,
initial velocity u = 0, t = ?
acceleration a = 9 m/s2,
distance s = 50 m
From the equation of motion,
s=ut + 
at2
50=0+(1/2)9t2
⇒t=10/3 s
(b) Second figure represents the second case
After covering 50 m, the velocity of the athlete is
v = u + at
=0+9×103 m/s
=30 m/s
He has to stop in minimum time. Hence, the deceleration,
a =−9 m/s2 (max)
R = mg
ma = μR (maximum frictional force)
ma = μmg
⇒ a = μg
= 9 m/s2 (deceleration)
u1 = 30 m/s, v = 0
⇒ t=10/3 sec