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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.