What is ratio of forces between …

Suppose two balls A and B, of masses m_A and m_B are travelling in the same direction along a straight line at different velocities u_A and u_B. And there are no other external unbalanced forces acting on them. Let u_A > u_B and the two balls collide with each other. During collision which lasts for a time t, the ball A exerts a force F_AB on ball B and the ball B exerts a force F_BA  on ball A.

Suppose v_A  and v_B are the velocities of the two balls A and B after the collision, respectively.

The momenta of ball A before and after the collision are m_A u_A and m_A v_A. The rate of change of its momentum (or F_AB, action) during the collision will be

m_A (v_A - u_A)/t

Similarly, the rate of change of momentum of ball B (= F_BA or reaction) during the collision will be

m_B (v_B - u_B)/t.

According to the third law of motion, the force F_AB  exerted by ball A on ball B (action) and the force F_BA  exerted by the ball B on ball A (reaction) must be equal and opposite to each other. Therefore,

F_AB = – F_BA

or

m_A (v_A - u_A)/t = -m_B (v_B - u_B)/t. This gives,

m_A u_A + m_B u_B = m_A u_A + m_B v_B

where m_A and m_B are masses of the bodies, u₁ and u₂ are initial velocities of the bodies and v₁ and v₂ are final velocities of the bodies.