A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction, where \(v = 3 t ^ { 2 } - 4 t + 3\). When \(t = 0 , P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) is moving with minimum velocity.
(8)
Two particles, \(P\), of mass \(2 m\), and \(Q\), of mass \(m\), are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide. Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between the particles is \(e\), where \(e < 1\). Find, in terms of \(u\) and \(e\),
the speed of \(P\) immediately after the collision,
the speed of \(Q\) immediately after the collision.
A particle of mass 0.5 kg is projected vertically upwards from ground level with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It comes to instantaneous rest at a height of 10 m above the ground. As the particle moves it is subject to air resistance of constant magnitude \(R\) newtons. Using the work-energy principle, or otherwise, find the value of \(R\).
(6)
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The points \(A , B\) and \(C\) lie in a horizontal plane. A batsman strikes a ball of mass 0.25 kg . Immediately before being struck, the ball is moving along the horizontal line \(A B\) with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after being struck, the ball moves along the horizontal line \(B C\) with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The line \(B C\) makes an angle of \(60 ^ { \circ }\) with the original direction of motion \(A B\), as shown in Figure 1.
Find, to 3 significant figures,
the magnitude of the impulse given to the ball,
the size of the angle that the direction of this impulse makes with the original direction of motion \(A B\).