1.
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\caption{Figure 1}
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Figure 1 represents the plan of part of a smooth horizontal floor, where \(W _ { 1 }\) and \(W _ { 2 }\) are two fixed parallel vertical walls. The walls are 3 metres apart.
A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leqslant 3\), from \(W _ { 1 }\)
At time \(t = 0\), the particle is projected from \(O\) towards \(W _ { 1 }\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls.
The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\)
The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.
- Show that \(T = \frac { 45 - 5 d } { 4 u }\)
The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary.
- Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer.