Two Trains and a Persistent Fly
The problem
Two railway stations are kilometres apart. Train A leaves the first station at km/h towards the second; simultaneously train B leaves the second station at km/h towards the first. A fly starts on the front of train A and flies at a steady km/h back and forth between the two trains, turning around instantly each time, until the trains meet. How far does the fly travel in total?
- The trains travel on the same straight track towards each other.
- The fly never slows down and its turns take no time.
Take a minute
Reveal hint 1
Summing the infinitely many legs of the fly's journey is possible but painful. Is there something you already know that makes it unnecessary?
Reveal hint 2
The fly flies for exactly as long as it takes the trains to meet.
Solution
Answer kilometres.
Derivation
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Compute the time until the trains meet.
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Recognise that the fly is airborne the whole time.
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Multiply by its speed.
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Compare with the hard route.
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Notice how far it is.