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I drove my car along a stretch of road $500$m long. My car can accelerate uniformly from $0$ to $60$km h$^{-1}$ in $10$ seconds. Its maximum speed is $100$km h$^{-1}$. Here are some questions concerning speed-time diagrams:
1. I looked at my speedometer three times on the journey and read the speeds $10$ km h$^{-1}$, then $50$ km h$^{-1}$ then $10$ 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 $500$m of road, I looked at my speedometer twice: on one occasion it registered $50$ km h$^{-1}$, which was my maximum speed for the journey, and on another occasion is registered $10$ km h$^{-1}$, which was also my minimum speed for the journey.
What was the least and greatest possible times I spent on this section of road?
3. On the next $500$m section of road I alternately accelerate to $50$ km h$^{-1}$ and decelerate down to $10$ km h$^{-1}$.
What is the largest number of times I can record a speed of $10$ km h$^{-1}$?
4. On the final $500$m section of road, before I am arrested for dangerous driving, I alternately accelerate to $50$ km h$^{-1}$ and decelerate down to $10$ km h$^{-1}$.
What speed must I start the section of road to finish at $10$ km h$^{-1}$?
Don't forget that I start one section of road at the same speed that I finish the previous section.