There are two trains a and b. which are 30 kms apart.
both the trains are moving towards each other with
velocity of train a is 40 kms / hr
velocity od train b is 60 kms / hr
there is a crow which moves in to and fro motion between the two trains with the speed of 80 kms/hr.
determine the total distance covered by the crow before it gets crushed between the two trains.

-- If one train covers 40 km per hour and the other one covers 60 km per hour,
then they reduce the gap between them at the rate of 100 km per hour.

-- Since they begin 30 km apart, they meet and crush the crow in 30/100 = 0.3 hour.

-- The crow's speed is 80 km per hour.  In 0.3 hour, he covers 0.3 x 80 = 24 km.

Note: 40 km/hr and 60 km/hr are speeds, not velocities.

Lets consider the trains first. The train A is traveling at 40 km/ hour and the train B is going 60 kms/ hour
So they are approaching each other at (60+40) kms/ hour = 100 kms/ hour.
Distance between them = 30kms
So Distance = Rate * Time
Time = Distance / Rate
= (30 km) / (100 kms/ hour)
= 3/10 hour.
=18min

The fly spends the same amount of time traveling as the trains. It goes 80 kms/ hour, so in the 18 min the trains take to collide, the fly will go 24kms.  

But The solution above ignores the shape of the fly's path. To picture this shape, think of the fly as a point made out of rubber. It's bouncing between the trains at a very high speed. As the trains get closer and closer the bounces get shorter and shorter, until they are microscopic. Even then, if you had a strong enough magnifying glass, you could still see the bounces getting shorter. No matter how much you magnify, there will always be a tinier bounce that you can't see.  

You can analyze this path by combining the bounces into a series of round trips, from the first train to the second and back again. It turns out that the length of each trip is a fraction of the trip before. No matter how many times you multiply by a fraction, you will never reach zero. The fly makes an infinite number of round trips, each one smaller than the last.