Question

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?

Answer 1

To reach the 50 m distance in minimum time, the superman has to move with maximum possible acceleration.
Suppose the maximum acceleration required is 'a'.
∴ ma − μR = 0 ⇒ ma = μ mg
⇒ a = μg = 0.9 × 10 = 9 m/s²


(a) As per the question, the initial velocity,
u = 0, t = ?
a = 9 m/s², s = 50 m

From the equation of motion,
$s=ut+\left(\frac{1}{2}\right)a{t}^2$
$$\begin{gathered} 50=0+\left(\frac{1}{2}\right)9t^2 \\ \Rightarrow t=\frac{10}{3}\mathrm{s} \end{gathered}$$

(b) After covering 50 m, the velocity of the athelete is
v = u + at
$$\begin{gathered} =0+9\times \left(\frac{10}{3}\right)\mathrm{m}/\mathrm{s} \\ =30\mathrm{m}/\mathrm{s} \end{gathered}$$
The superman has to stop in minimum time. So, 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
$$\begin{gathered} \Rightarrow t=\frac{v_1-u_1}{a} \\ =\frac{0-30}{-a} \\ =\frac{-30}{-a}=\frac{10}{3}\mathrm{s} \end{gathered}$$