The friction coefficient between an athlete's shoes and the ground...

The maximum acceleration of the superhero is provided by the friction between the shoes and the ground.

a_\max = \mu g = 0.90 \times 10 = 9 \text{ m/s}^2

To complete the $50 \text{ m}$ race in the minimum possible time, the superhero must accelerate at this maximum rate for the entire distance.

Using the equation of motion $v^2 = u^2 + 2as$, the velocity $v$ at the end of the $50 \text{ m}$ mark is:

$v^2 = 0 + 2 \times 9 \times 50 = 900$

$v = 30 \text{ m/s}$

To come to a stop in the minimum time after completing the race, the superhero must decelerate at the maximum possible rate, which is also $a_{\text{dec}} = \mu g = 9 \text{ m/s}^2$.

Using the equation $v_f = v - a_{\text{dec}}t$ for deceleration:

$0 = 30 - 9t$

$t = \dfrac{30}{9} = \dfrac{10}{3} \text{ s}$