Average speeds are treacherous. Sometimes they cannot be made up for—and by the time you notice, all the time is already gone.
The Missing Half
Riddle statement
You travel from city A to city B.
You cover the first half of the distance at 30 km/h.
You want your average speed for the whole trip to be 60 km/h.
How fast must you travel over the second half?
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Solution
Suppose the total distance is $D$.
For the average speed over the whole trip to be 60 km/h, the total time would have to be:
But the first half of the trip has length $D/2$, and you cover it at 30 km/h. So that first half alone already takes:
In other words, the first half has used up all the time available for the entire trip.
To achieve an average speed of 60 km/h, you would have to cover the second half in zero time.
Answer: it is impossible. The second half would require infinite speed.