A2 June 2019 Q1

EdexcelCurrent spec8 marksCollisions: 1 Sphere Direct

1.

Figure 1: plan of two parallel walls W2 (left) and W1 (right), 3 m apart; the particle at O is d m from W1 and is projected towards W1 with speed u m/s
Figure 1

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, \(0 \lt d \leqslant 3\), from \(W_1\)

At time \(t = 0\), the particle is projected from \(O\) towards \(W_1\) with speed \(u\ \text{m s}^{-1}\) in a direction perpendicular to the walls.

The coefficient of restitution between the particle and each wall is \(\dfrac{2}{3}\)

The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.

(a) Show that \(T = \dfrac{45 - 5d}{4u}\) (6)

The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary.

(b) Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer. (2)
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