Finding the velocity change before colliding

[WorldOfPhysics]

Homework Statement

Two identical blocks of mass M are tied with a mass-less string of length L. If the string is pulled with a force F right at the midpoint of string L, the blocks will move towards each other. What will the velocity be when the blocks collide. The blocks start from rest and friction is not considered.

Homework Equations

Net Force = M*a

The Attempt at a Solution

The horizontal component T_x provides the horizontal acceleration to the block. and since both of them accelerate towards each other, 2T_x / M will be their horizontal acceleration towards each other.

But this horizontal acceleration is not a constant, but it must keep decreasing as the horizontal component keeps on decreasing as the blocks approach and theta moves towards 90 degrees.

how can we use calculus to model this situation and find out the change in velocity from the initial to final points.

 

[rcgldr]

this horizontal acceleration is not a constant, but it must keep decreasing as the horizontal component keeps on decreasing as the blocks approach and theta moves towards 90 degrees.

At the midpoint of the string, the horizontal components of tensions counter each other, so I'm thinking that 2 tension sin(phi) = force, or tension = force / (2 sin(phi)). Rate of downwards acceleration of each block (I think of them as point masses), is tension sin(phi) / mass = force / (2 mass). Rate of horizontal acceleration of each block = tension cos(phi) / mass = force cos(phi) / (2 sin(phi)) .

It's going on 3 days since this thread was started. I was a bit curious about a solution, but will wait for the OP to hopefully update this thread.