Average speed of a round trip

Question

You drive up a hill and back down the same road. Going up, you average 30 mph. You want your average speed for the entire round trip to be 60 mph.

How fast must you drive on the way back down?

The tempting answer is 90 mph, but be careful.

Accepted Answer

The answer is that it's impossible, no finite speed works. Why "90 mph" is wrong. Average speed is total distance ÷ total time, which is a time-weighted quantity, not the simple average of the two leg speeds. Because you spend more time on the slow uphill leg, that slow speed dominates. Do the arithmetic. Let the hill be d miles each way, so the round trip is 2d miles. - Uphill at 30 mph takes time \dfrac{d}{30} hours. - To average 60 mph over 2d miles, the total trip must take \dfrac{2d}{60} = \dfrac{d}{30} hours. But you have already used \dfrac{d}{30} hours just going up! That leaves zero …

Average speed of a round trip

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