A Car Trip...and a certain Train

Suppose you are in your car, in the middle of a long drive, while on vacation. Your car has an odometer, and you always make note of its readings when you are at rest stops. At 2pm you are at one rest stop. At 4pm you are at another rest stop.

(a) Suppose you want to know your average speed between 2pm and 4pm. What information would use, and, in as precise a way as you can, explain how exactly you would use it:

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

(b) Describe how, in general, you can find your "average speed, between [starting time] and [ending time]. Use full sentences, but also be mathematically precise about which quantities you would use, and how you would use them.

Use S for your starting time [the time at which you would be at the first rest stop] and E for your ending time [the time at which you arrive at the second rest stop]. Yes, it's ok to subtract, for example, 4:20pm "minus" 3:30pm" would be "50 minutes" Your explanation:

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

(c) Write down a formula for the average speed between a starting time (S) and an ending time (E) if your odometer at the first time equaled D1 and at the second time (at the "end") it equaled D2 (since you are driving forward, D2 will be a bigger number than D1).

Formula for average speed, using the above constants and notation:





(d) Now give a more general definition of average speed using "elapsed time" -- that is, how much time passed between the "first instant (point in time) we are interested in" and the "second instant" -- and using "elapsed distance" (this would correspond to the distance between the two rest stops in our earlier example).

Average Speed = (__________________) ÷ (__________________).

(e) Explain how "average speed" is different from "what my speedometer reads" (the latter gives the instantaneous speed at any one point in time when we care to look at it). And, intuitively, why are these two quantities different?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

(f) Explain how "average speed" readings can approximate your car's instantaneous speed (i.e. its "Speedometer reading" speed). One way to think about it: Suppose that you were driving your car while a person in the back seat held a stopwatch in their hands and could tell the person in the passenger seat to record the odometer at any given time; whenever the person in the back shouted "Now!" The person in the back would ask for the odometer reading at exactly 3:00pm and zero seconds. Then they'd ask for a reading 60 seconds later, and then the "average speed between 3:00pm and 3:01pm" could be found. Now imagine the same thing happening, only with a wait of only 30 second between the two odometer readings. Would the "average speed" computed now be likely to be closer to the speedometer readings during those 30 seconds? Why? Imagine waiting only 5 seconds between "starting odometer reading" and ending. Would the "average speed" from that (that is, the car's "average speed" between 3:00pm exactly, and 5 seconds past 3pm) come even closer to the speedometer reading? Why?

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________

______________________________________________________________________