Loss of energy in collision

3 A ball of mass 1.6 kg is released from rest at a point 5 m above horizontal ground. When the ball hits the ground it instantaneously loses 8 J of kinetic energy and starts to move upwards.

1. Use an energy method to find the greatest height that the ball reaches after hitting the ground.
2. Find the total time taken, from the initial release of the ball until it reaches this greatest height.

2. A particle \(P\) of mass 3 kg moves such that at time \(t\) seconds its position vector, \(\mathbf { r }\) metres, relative to a fixed origin, \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - 3 t \right) \mathbf { i } + \frac { 1 } { 6 } t ^ { 3 } \mathbf { j }$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.

1. Find the velocity of \(P\) when \(t = 0\).
2. Find the kinetic energy lost by \(P\) in the interval \(0 \leq t \leq 2\).

1. Three particles \(A , B\) and \(C\) are at rest on a smooth horizontal plane. The particles lie along a straight line with \(B\) between \(A\) and \(C\).

Particle \(B\) has mass \(4 m\) and particle \(C\) has mass \(k m\), where \(k\) is a positive constant. Particle \(B\) is projected with speed \(u\) along the plane towards \(C\) and they collide directly. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 1 } { 4 }\)

1. Find the range of values of \(k\) for which there would be no further collisions. The magnitude of the impulse on \(B\) in the collision between \(B\) and \(C\) is \(3 m u\)
2. Find the value of \(k\).

A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a\sqrt{3}\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt{\left(\frac{37ga}{5}\right)}\).

1. Show that \(S\) strikes the wall with speed \(\sqrt{\left(\frac{27ga}{5}\right)}\). [4] Given that the loss in kinetic energy due to the impact with the wall is \(\frac{3mga}{5}\),
2. find the coefficient of restitution between \(S\) and the wall. [7]