Example 2: Average Velocity

A car starts at rest and moves with velocity v = 3t². Find its average velocity between times t = 0 and t = 5. At what point of time is its velocity equal to the average velocity?

To find the value of t where v = u_ave, we put

Suppose a car drives from city A to city B and back, a distance of 120 miles each way. From A to B it travels at a speed of 30 mph, and on the return trip it travels at 60 mph. What is the average speed?

If we choose distance as the independent variable we get one answer, and if we choose time we get another.

Average speed with respect to time: The car takes 120/30 = 4 hours to go from A to B and 120/60 = 2 hours to return to A. The total trip takes 6 hours.

Average speed with respect to distance: The car goes 120 miles at 30mph and 120 miles at 60 mph, with a total distance of 240 miles. Therefore

From Figure 6.5.4 we see that the average with respect to time is smaller because most of the time was spent at the lower speed of 30 mph.

Figure 6.5.4

In general, if y is given both as a function of s and of t, y = f(s) = g(t) then there is one average of y with respect to s, and another with respect to t.