Faced with a trip of 100 miles and the need to arrive at my destination by 3:30pm, I deliberately chose to leave home at 1:30. Experience has taught me that while driving, I can comfortably average 50 miles per hour from start to finish. In answer to Question 1, the following calculation revealed the time needed to complete the journey.


100 miles

=2 hours
50 miles/hour

Your average driving speed may be faster than mine. (Don't forget to include pit stops!) Type in your average speed to check how long it would take for you to drive from Corvallis to Portland.

Driving Time for a 100 Mile Trip
Enter your average driving speed
miles per hour
Check your driving time

hour(s) and minute(s)

For example, you should find that at an average driving speed of 60 miles per hour, one could spend up to 20 minutes with a police officer and still complete the trip in two hours.

These calculations involve three quantities: elapsed time, distance traveled, and average speed. The following formula captures the relationship between these variables.

Average Speed
distance traveled
average speed=
elapsed time

Distance traveled is a function of elapsed time. Measuring these quantities with respect to my instant of departure at 1:30pm on that November afternoon, we set

t = time elapsed since departure at 1:30pm


s(t) = distance traveled since departure

The function s is called the distance function. In order to make numerical calculations with this function, we must specify units: Time will be measured in seconds and distance traveled will be measured in feet. (The reason for this choice will become clear in a moment.) With these choices made, average speed will be given in feet per second.

Average speed can be expressed in terms of the distance function. Consider a time interval that begins at time t0 and ends at time t1. For such a time interval, the definition of average speed can be expressed in terms of the distance function as follows.

Average Speed in Terms of the Distance Function
s(t1) - s(t0)
average speed=
t1 - t0


The expression on the right hand side of this equation is called a difference quotient for the function s. The numerator of this difference quotient represents the change in the odometer readings from time t0 to time t1, and so is equal to the distance traveled during this time interval. The denominator is equal to the elapsed time between t0 and t1. The difference quotient itself therefore computes the average speed for this time interval.

A difference quotient for the distance function determines average speed for a suitable time interval. Average speed is the key to Question 1. When it comes to Question 2, a closer look at the distance function will reveal how the police officer was able to clock my speed at 38 miles per hour. So get out your stopwatch and let's consider how the police use radar technology to catch speeders!



© CalculusQuestTM
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson