Question 239900: Let?s say that you are driving on a straight route to a set
destination, and you can drive at any speed you like. You
stop for a few minutes but when you arrive at the halfway
point, you discover that you have averaged only 20 miles
per hour. So you decide to forego any more stops and drive
fast enough to average 40 miles per hour for the entire trip. If you keep a steady speed, how fast should you drive? Answer by solver91311(17077) (Show Source):
You can't do it even if you went the speed of light. Let's look at a specific example first.
Let's say the whole trip was 40 miles. Then to average 40 mph over the entire trip, you would have to make the entire trip in 1 hour. But since you only managed 20 mph for the first 20 miles of the trip (halfway), it took the entire hour to get to the halfway point. No matter how fast you go, you can't complete the last half of the trip in zero time.
In general, Let represent the time to travel halfway, and let represent the time for the entire trip.
If is the distance traveled at 20 mph, then is the entire distance. The first half of the trip can be described as:
And the entire trip can be described as:
Which means that
Impossible because if we let be the time for the second half, we know from the fact that must be true so if , then , and that simply cannot be.