At time \(t = 0\) a ball is projected vertically upwards from a point \(O\) and rises to a maximum height of 40 m above \(O\). The ball is modelled as a particle moving freely under gravity.
- Show that the speed of projection is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the times, in seconds, when the ball is 33.6 m above \(O\).
- Particle \(P\) has mass 3 kg and particle \(Q\) has mass 2 kg . The particles are moving in opposite directions on a smooth horizontal plane when they collide directly. Immediately before the collision, \(P\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, both particles move in the same direction and the difference in their speeds is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Find the speed of each particle after the collision.
- Find the magnitude of the impulse exerted on \(P\) by \(Q\).
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\caption{Figure 1}
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A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N . The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1.
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
Given that the particle is on the point of sliding down the plane,
(i) show that the magnitude of the normal reaction between the particle and the plane is 20 N ,
(ii) find the value of \(W\).