The motorist and the hill

[MarkFL]

Suppose a motorist is going over a hill...it is one mile up to the top and one mile from the top to the bottom on the other side. On the way up the hill, the motorist travels 30 miles per hour...how fast must the motorist go down the hill so that his/her average speed over the entire hill is 60 mile per hour?

 

[tango]

He would have to do the second mile in no time at all, i.e. an infinite speed.

 

[MarkFL]

He would have to do the second mile in no time at all, i.e. an infinite speed.

Exactly right...this question was posed by my Calc II prof. when I was a student, and the majority of the class immediately said 90 mph since the average of 90 and 30 is 60. But a couple of us realized the motorist wouldn't be traveling downhill for as long as the trip uphill took, so a weighted average would have to be used, if averaging is what you want to do.

I observed that in order to average 60 mph over 2 miles would take 2 minutes, but the motorist is already using 2 minutes to get to the top of the hill, leaving zero time for the trip down.

The lesson for that day was to think first and avoid common pitfalls.

 

[tango]

Exactly right...this question was posed by my Calc II prof. when I was a student, and the majority of the class immediately said 90 mph since the average of 90 and 30 is 60. But a couple of us realized the motorist wouldn't be traveling downhill for as long as the trip uphill took, so a weighted average would have to be used, if averaging is what you want to do.

I observed that in order to average 60 mph over 2 miles would take 2 minutes, but the motorist is already using 2 minutes to get to the top of the hill, leaving zero time for the trip down.

The lesson for that day was to think first and avoid common pitfalls.

I've done that kind of mental calculation many times, trying to figure out how going up and over a honking great hill affects my average fuel economy. Needless to say it's not as simple as the notion that going up the hill at 4mpg and down the hill at 98mpg means I averaged 51mpg over the trip.