To find the minimum time of travel for a bus starting from rest and traveling a distance \(s\) along a straight horizontal road, we need to consider the forces acting on the bus and the principles of motion, specifically involving friction.
### Key Points:
3. **Equations of Motion:**
- To determine the time taken to travel distance \( s \) with constant acceleration \( a \), we use the equation of motion:
\[
s = \frac{1}{2} a t^2
\]
- Plugging in \( a = \mu_s g \):
\[
s = \frac{1}{2} (\mu_s g) t^2
\]
- Solving for \( t \):
\[
t^2 = \frac{2s}{\mu_s g}
\]
\[
t = \sqrt{\frac{2s}{\mu_s g}}
\]
### Conclusion:
The minimum time \( t \) to travel the distance \( s \) is:
\[
t \propto \sqrt{\frac{s}{\mu_s g}}
\]
Therefore, the minimum time of travel is proportional to \( \sqrt{\frac{s}{\mu_s}} \). This relationship shows that the minimum time is directly proportional to the square root of the distance traveled and inversely proportional to the square root of the static friction coefficient.
