Two particles \(P\) and \(Q\) are moving in opposite directions along the same horizontal straight line. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(k m\). The particles collide directly. Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(2 u\). As a result of the collision, the direction of motion of each particle is reversed and the speed of each particle is halved.
Find the value of \(k\).
(4)
Find, in terms of \(m\) and \(u\) only, the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.
(2)
A block \(A\) of mass 9 kg is released from rest from a point \(P\) which is a height \(h\) metres above horizontal soft ground. The block falls and strikes another block \(B\) of mass 1.5 kg which is on the ground vertically below \(P\). The speed of \(A\) immediately before it strikes \(B\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The blocks are modelled as particles.
Find the value of \(h\).
(2)
Immediately after the impact the blocks move downwards together with the same speed and both come to rest after sinking a vertical distance of 12 cm into the ground. Assuming that the resistance offered by the ground has constant magnitude \(R\) newtons,
find the value of \(R\).
(8)
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