# 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.

Mathematically it is infinite because your bird instantaneously changes direction and is zero length.
Factor in the fact that at each turn the bird must decelerate to zero, turn and accelerate back to w. When the period of time between the trains is less than the turning time and the distance between the trains is less than the birds length the bird is squashed.

The number of back and fore trips is infinite, but the time taken and the distance it covers is quite finite.

There are two ways to approach it, either using a geometric series and summing an infinite number of convergent terms or by realising that the bird moves at a constant speed the entire time and that you can calculate the time to collision simply from the train's motion.

This is a rewording of a question often attributed to Von Neumann.

I think your mathematic model fails because you did not take into account the size of the bird. At a certain point, when distance is equal to the bird size, the model must stop the iteration.