Example 4: Driving Speeds

Click here to open an associated Mathcad worksheet:

Problem. As you leave Boston on the Massachusetts Turnpike a machine spits out a card with the time and place of your entry encoded on a magnetic strip. You throw the card on the floor of your car and drive to the other end of the Commonwealth. There, a ticket-taker in a tollbooth swipes the card through another machine, frowns, and asks you to pull over. You are being ticketed for exceeding the  65 mph  speed limit somewhere on the Turnpike, even though no one actually saw you speeding. Should you fight the ticket?

Solution. Miles per hour is a rate, given by  Dx/Dt, where  Dx  is the distance between Boston and the tollbooth (change in position) and  Dt  is the length of your trip (change in time) – both pieces of information encoded on the card.

Your only defense would be to argue that this is an average rate, averaged over your entire trip, and that the actual rate on your speedometer never exceeded  65 mph. Unfortunately, this is impossible. Why?

You'd better pay the ticket and slow down.