# Tunnel Escape

Antonio, and his twin brother, Eugenio, decide to do a stupid thing. They enter a narrow train tunnel and start walking inside towards the other end.

Image: Antonio Martinetti

Once they have walked a quarter of the way through the tunnel they hear a train approaching them from behind (the direction they came from). They don’t know how far the train is away, but they know the tunnel is too narrow to safely shelter in so they must exit, and quickly. Antonio decides to run back to the entrance they started from and,

*just*, manages to get out safely.Eugenio decides to run to the exit at the far end of the tunnel and also,

*just*, manages makes it out safely.If both boys run at the same (constant) speed of 12 mph, and the train is also travelling at constant speed, what is the speed of the train?

*What is the speed of the train?*

## Diagram

Let’s define the length of the tunnel to be

*t*miles, and the distance the train is away from the start of the tunnel to be*d*miles. The speed of the train is*s*miles per hour.We know each of the twins can run 12 mph.

Antonio heads back to the start of the tunnel, so has to travel a distance of

*t/4*miles. As his speed is 12 mph, the time this takes him is*t/48*hours. We know he just makes it, so the train, travels to be at the entrance at the same time. The time the train makes it is*d/s*. Equating these we see that*ts=48d*.Eugenio heads the opposite direction. He has to travel a distance of

*3t/4*, at the same speed as his brother, 12 mph. His trip takes him*3t/48*. He also, just, makes it out alive, so the train will be at the exit of the tunnel after the same time. To get to the exit, the train has had to travel*d+t*miles, so the time it takes it is*(d+t)/s*, and this should be the same time as Eugenio’s run. Equating these we get*3ts=48(d+t)*.Substituting the result

*ts=48d*into*3ts=48(d+t)*we get the result that*2d=t*. The tunnel is twice as long as the train is away, and the train is travelling at 24 mph.The train is travelling at 24 mph, twice as fast as the boys can run.

*The train is travelling at 24 mph, twice as fast as the boys can run.*

## Logical Approach

There’s another way to think about this that does not require simultaneous equations, just a bit of common-sense logic.

We know that Antonio starts ¼ of the way into the tunnel, and if he runs towards the entrance, just makes it. He has travelled ¼ of the length of the tunnel in the time it takes the train to get to the entrance. If, instead, he’d have run the

*other way*, this ¼ length run would have placed him exactly halfway through the tunnel.The train is now at the start of the tunnel, and the boy is at the midway point of the tunnel. For the boy and the train to be at the exit at the same time, it’s pretty clear the train has to travel exactly twice as quick as the boy can run (twice the distance in the same time).

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## Dumb Ways to Die

If it’s not obvious, playing around train tracks is very dangerous.

Sing along with me …