A man needs to go through a train tunnel to reach the other side. He starts running at 10 mph through the tunnel in an effort to reach his destination as soon as possible. When he is ¼ th of the way through the tunnel, he hears the whistle of the train behind him. Assuming the tunnel is not big enough for him and the train, he has to get out of the tunnel in order to survive. We know the following: Given 1. If he runs back, he will make it out of the tunnel by a whisker. Given 2. If he continues running forward, he will still make it out through the other end by a whisker. What is the speed of the train?
Solution: Let S = Speed of train, s = Speed of man = 10 mph, L = Length of the tunnel, D = Distance of train to start of tunnel. Distance of man from start of tunnel = L/4
By Given 1, if he runs back, he will make it out of the tunnel by a whisker. Hence, D/S = L/(4s)
By Given 2, if he continues running forward, he will still make out through the other end by a whisker. Hence,
(D + L)/S = 3L/(4s)
Combining the two equations:
S/2 = s, S = 2s = 20 mph !
More elegant solution: (submitted by Mr. Gerard Delaney of Goa, India)
We know he is 1/4 of the way into the tunnel when he hears the whistle. And he will just escape if he runs back towards the train (Given 1). This means that if he keeps running after he hears the train, he will be 1/2 way into the tunnel when the train enters the tunnel. And (Given 2) we know that the time he takes to run the remaining half of the tunnel will be the same as the time the train takes to travel the whole tunnel.
Hence the speed of the train is twice his speed or 20 mph.