Collision then wall impact

Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) after the collision. [3]

Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan\theta = \frac{3}{4}\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{3}\). Immediately after \(B\) collides with the wall, the kinetic energy of \(B\) is \(\frac{5}{27}\) of the kinetic energy of \(B\).

1. Find the possible values of \(e\). [7]

Two objects, \(A\) of mass 18 kg and \(B\) of mass 7 kg, are moving in the same straight line on a smooth horizontal surface. Initially, they are moving with the same speed of \(4\text{ ms}^{-1}\) and in the same direction. Object \(B\) collides with a vertical wall which is perpendicular to its direction of motion and rebounds with a speed of \(3\text{ ms}^{-1}\). Subsequently, the two objects \(A\) and \(B\) collide directly. The coefficient of restitution between the two objects is \(\frac{5}{7}\).

1. Find the coefficient of restitution between \(B\) and the wall. [1]
2. Determine the speed of \(A\) and the speed of \(B\) immediately after the two objects collide. [7]
3. Calculate the impulse exerted by \(A\) on \(B\) due to the collision and clearly state its units. [2]
4. Find the loss in energy due to the collision between \(A\) and \(B\). [2]
5. State the direction of motion of \(A\) relative to the wall after the collision with \(B\). [1]