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Train $A$ is $x$ feet long and is going east at $r_1$ mph. On a parallel track going west is train $B$ going at $r_2$ mph. f the trains take $y$ seconds to pass each other completely, how many feet long is train $B$ ?

To approach this problem I've thought first a simpler case where I had train $A$ standing still.

In this case I realize (I think) that if $B$ takes $w$ seconds to pass completely $A$ I've that $B$ measures $r_2\cdot w -x $ feet.

While this case is clear I have some difficulty when I allow $A$ to have a rate $r_1$.

Intuitively I think that the solution in second case is found by adding the two rates of the trains and to multiply by the time it takes the trains to pass completely each other.

Thus if I take into consideration the data I am given in the problem I would have that $B$ measures $(r_1+r_2)\cdot y -x $, where $x$ is the length of train $A$.

However if this is correct I can't see the rigorous way to show that I must add the velocities beside the fact that of course if two trains are going at opposite directions the time it takes the two trains to pass each other is clearly less than the time it would take if one train would be still.

So my question is: what is a rigorous way to show that I have to add the velocities of the two trains? And if my idea is completely wrong then how should I have solved the problem ?

Edit: I am new at M.SE so forgive me if the question is not properly asked. I've tried to make the problem the most general possible so anyone else can benefit from it.

Idontknow
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    Consider things from the point of view of someone in train A. It's exactly as if train A were standing still and train B were approaching at a rate $r_1+r_2$. So your analysis is correct. – rogerl Nov 28 '15 at 13:30

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Hint: RELATIVE VELOCITY .See the $total time = sum of both trains(distance)/speed$ . Therefore $y=\frac{w+x}{r_1+r_2}$. So $w=y(r_1+r_2)-x$. They are going in opposite direction so $V_{AB}=V_{A}+V_{B}$.$w$ for more info you can go through the physics concept of relative velocity. is the length of trainB. Hope its clear.

Archis Welankar
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