Riddle regarding mean value theorem

Question

A man is photographed at a tollbooth at 12:00, and then arrives another tollbooth, 250 miles down the road, at 2:00. A cop pulls him over and gives him a traffic ticket for driving 125 mph.

His defense lawyer claims in court, "You can't prove that there was ever any particular moment when my client was actually driving 125 mph..."

Hi guys I came across this riddle and I was wondering how should I approach this while I know it might have something to do with the mean value theorem? :)

Answer 1

By the mean value theorem the driver must have been driving at $\displaystyle \frac {250-0}{2:00-12:00}=125\text{mph}$ at some point. If you join the two points on a distance-time graph using a line or curve the slope will have to be at least 125 at some point between them.

This law was attempted to be enforced in the courts of Boston, MA, USA but didn’t go through.

Answer 2

Assume integer speeds, because I’ve never heard of police giving rational tickets. If the client had been driving at 124 mph for two hours, they could not have reached the second toll point 250 miles away 2 hours later. Thus they had to exceed 124 mph at some point, hitting 125.