Four Ball Collision

Four equal radii balls of masses 1kg, 2kg, 3kg, and 4kg as shown,

each moving with velocity \(1 \text{ m/sec}\) towards the same point \(P\), collide at exactly the same instant. The points of contact form vertices of a square, the center of which is point \(P\). When the balls fly apart after the collision, they move at velocities \(v_1, v_2, v_3, v_4\).

If \({v_1}^{2}+{v_2}^{2}+{v_3}^{2}+{v_4}^{2}\) can be expressed as \(\dfrac{A}{B}\), where \(A,B\) are coprime positive integers, then what is \(A+B\)?

Assume collisions are ideal, i.e., perfectly elastic and frictionless.

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