Sharon cycles up a hill at 15 km/h. When she reaches the top, she immediately turns around and comes down the same road at 30 km/h, stopping where she'd started. Is her average speed greater than, less than, or equal to 22.5 km/h? Explain
What will be an ideal response?
We define average speed as the distance traveled divided by the time required to
travel that distance. The average of the two speeds, 22/5 km/h here, is the same as the
average speed only if both segments take the same time.
The uphill distance is the same as the downhill distance. Therefore, the times are going to
be different. We can still use the relation
distance traveled = (average speed) x (time traveled)
though in the form
time traveled = (distance traveled)/(average speed).
Call the distance traveled from the bottom to the top of the hill D. The uphill time is D/(15
km/h). The downhill time is D/(30 km/h). It takes twice as long to go uphill as to go
downhill. The total time is D/(15 km/h) + D/(30 km/h), so we find we can rewrite the time
as D/(10 km/h)..
Now using the above relation written as
average speed = (distance traveled)/(time traveled),
we find now the total distance to be 2D, the total time has to be D/(10 km/h), as already
discussed, so the average speed is vav = (2D)/[D/(10 km/h)] = 20 km/h.
The average speed is less than the average of the speeds.
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