Challenge Level

I drove my car along a stretch of road $500\textrm{ m}$ long. My car can accelerate uniformly from $0$ to $60\textrm{ km h}^{-1}$ in $10$ seconds. Its maximum speed is $100\textrm{ km h}^{-1}$.

1. I looked at my speedometer three times on the journey and read the speeds $10\textrm{ km h}^{-1}$, then $50\textrm{ km h}^{-1}$, then $10\textrm{ km h}^{-1}$. What was the least possible time to travel along the stretch of road? What was the greatest possible time?

2. On the next $500\textrm{ m}$ of road, I looked at my speedometer twice: on one occasion it registered $50\textrm{ km h}^{-1}$, which was my maximum speed for the journey, and on another occasion is registered $10\textrm{ km h}^{-1}$, which was also my minimum speed for the journey.

What were the least and greatest possible times I spent on this section of road?

3. On the next $500\textrm{ m}$ section of road I alternately accelerate to $50\textrm{ km h}^{-1}$ and decelerate down to $10\textrm{ km h}^{-1}$.

What is the largest number of times I can record a speed of $10\textrm{ km h}^{-1}$?

4. On the final $500\textrm{ m}$ section of road, before I am arrested for dangerous driving, I alternately accelerate to $50\textrm{ km h}^{-1}$ and decelerate down to $10\textrm{ km h}^{-1}$.

What speed must I start the section of road to finish at $10\textrm{ km h}^{-1}$?

Don't forget that I start one section of road at the same speed that I finish the previous section.