A Little Road Trip

What does it mean when the speedometer on your car reads 35 miles per hour at some particular instant It does not mean, you will readily agree, that you have come exactly 35 miles in the last hour or that you can expect to travel 35 miles in the next hour. You will almost certainly slow down and speed up during different parts of your trip and your speedometer will respond accordingly. You can certainly calculate your average speed for the entire journey by dividing the number of miles travelled by the number of hours, but your speedometer readings will have been sometimes slower, sometimes faster, than this average value. 

Let the variable r represent the distance you have travelled from your starting point, and let t represent the elapsed time. Fig. 6.1 is a plot of your progress, distance travelled as function of time, as represented by the function r t). Your stops for a red lights show up as horizontal segments, where t continues to increase but r stands still. 

Suppose, at the beginning of your trip, your watch reads ta and your odometer reads r, while at the end your watch and odometer read tt, and rt, respectively. Your average speed—we will call it v for velocity—for the whole trip is given by 

Plot of road trip. Speedometer reading gives 

You might have noted that the later part of your trip, after odometer reading Tc at time tc, was somewhat faster than the earlier part. You could thereby calculate your speeds for the separate legs of the trip 

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