A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4m\), has the same radius as \(A\) and is moving on the table with velocity \(u\).
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The sphere \(A\) collides directly with the sphere \(B\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).
- Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision. [6 marks]
- The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined. [2 marks]
- Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{5}\).
The sphere \(B\) collides with \(A\) again after rebounding from the wall.
Show that \(e < b\), where \(b\) is a constant to be determined. [3 marks]
- Given that \(e = \frac{4}{7}\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall. [3 marks]