Collision with mass ratio parameter

4 Two smooth spheres \(A\) and \(B\), of equal radii, have masses 0.1 kg and \(m \mathrm {~kg}\) respectively. They are moving towards each other in a straight line on a smooth horizontal table and collide directly. Immediately before collision the speed of \(A\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Assume that in the collision \(A\) does not change direction. The speeds of \(A\) and \(B\) after the collision are \(v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Express \(m\) in terms of \(v _ { A }\) and \(v _ { B }\), and hence show that \(m < 0.25\).
2. Assume instead that \(m = 0.2\) and that the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find the magnitude of the impulse acting on \(A\) in the collision.

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 2 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a smooth horizontal surface, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the greatest possible value of \(m\). It is given that \(m = 1\).
2. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
3. Find the kinetic energy lost due to the collision.

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.24 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a horizontal table, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse exerted by \(A\) on \(B\).
2. Show that \(m \leqslant 0.08\). It is given that \(m = 0.06\).
3. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
4. Find the speeds of \(A\) and \(B\) immediately after the collision.