A car starts from rest to cover a distance s. the coefficient of friction between the road and the tyres is μ. The minimum time in which the car can cover the distance is proportional to

The correct option is 1/μ.

Let us analyze the motion of the car step-by-step:

1. Frictional Force:
When the car starts from rest, the maximum force of friction between the road and the tyres provides the force required to accelerate the car. The maximum static frictional force is given by:
$f_{\max }=\mu N=\mu mg$

where:
• $\mu$ is the coefficient of friction between the road and the tyres.
• $m$ is the mass of the car.
• $g$ is the acceleration due to gravity.

2. Maximum Acceleration:
Using Newton's second law of motion, the maximum acceleration $a$ of the car is:
$a=\frac{f_{\max }}{m}=\mu g$

3. Proportionality of Time:
The time taken to cover a fixed distance is inversely related to the acceleration. Since the maximum acceleration is directly proportional to the coefficient of friction ($a\propto \mu$), the minimum time $t$ required to cover the distance is inversely proportional to $\mu$:
$t\propto \frac{1}{\mu }$