- A particle \(P\) of mass 3 kg moves in a straight line on a smooth horizontal plane. When the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resultant force acting on \(P\) is a resistance to motion of magnitude \(2 v \mathrm {~N}\). Find the distance moved by \(P\) while slowing down from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(5)
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\caption{Figure 1}
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Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac { 4 } { 3 }\) and \(\tan \beta = \frac { 12 } { 5 }\) as shown in Fig. 1.
- Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision.
The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).
- Find, to one decimal place, the speed of each sphere after the collision.
(9)