# The bird's got some guts

• Rainbow

#### Rainbow

Here’s a question that I’m stuck with.
Two trains initially separated by distance L are heading towards each other on the same track each with speed v, and a bird flies from train A towards B with constant speed w>v reaches train B and immediately comes back to A with same speed and continues to do so till it sandwiches between the two. Find out the number of trips and time taken before it sandwiches.
I solved it mathematically and got the answer as infinity, which I find hard to accept. I think this is due to the wrong mathematical approach. I mean, at some point of time the velocities of both the trains and the bird change to zero. So, I think we would have to account for this sudden change of variables in our equations. But, the question is how.

'The answer'? You were asked two questions. There certainly will be an infinite number of trips (this is a perfectly valid thing in a model like this that has nothing to do with reality). But the time taken certainly isn't infinite. Indeed, the time taken can be deduced without even thinking about the bird.

This goes to a series with an infinite number of terms, or trips, but finite value, or time taken.

This remembers me of an anecdote about the famous mathematician John von Neumann:

When this problem was posed to John von Neumann, he immediately replied,
"150 miles."

"It is very strange," said the poser, "but nearly everyone tries to sum the
infinite series."

"What do you mean, strange?" asked Von Neumann. "That's how I did it!"

How long does it take for the trains to touch each other?

How long does it take for the trains to touch each other?

As long as v is not equal to 0, wouldn't that actually be "...for the trains to crash into each other"? Poor bird.

Two trains initially separated by distance L are heading towards each other on the same track each with speed v
So they are closing on one another at speed 2v. It will take time L/2v (in whatever units are appropriate) for the two trains to "touch" (more correctly, crash). Since the bird flies at speed w, in that time it will have flown distance Lw/2v.