M3 January 2005 Q4

Figure 4: box AB of length 4L with window PQ of width b at the B end, target C in the window — Figure 4

In a game at a fair, a small target \(C\) moves horizontally with simple harmonic motion between the points \(A\) and \(B\), where \(AB = 4L\). The target moves inside a box and takes 3 s to travel from \(A\) to \(B\). A player has to shoot at \(C\), but \(C\) is only visible to the player when it passes a window \(PQ\), where \(PQ = b\). The window is initially placed with \(Q\) at the point as shown in Figure 4. The target \(C\) takes 0.75 s to pass from \(Q\) to \(P\).

(a) Show that \(b = (2 - \sqrt{2})L\). (5)

(b) Find the speed of \(C\) as it passes \(P\). (2)

Figure 5: box AB of length 4L with window PQ of width b at the centre — Figure 5

For advanced players, the window \(PQ\) is moved to the centre of \(AB\) so that \(AP = QB\), as shown in Figure 5.

Mark scheme (a)

Scheme	Marks
\(6 = \dfrac{2\pi}{\omega} \Rightarrow \omega = \dfrac{\pi}{3}\)	M1
\(a = 2L\)	B1
\(x = 2L\cos\omega t\)	M1
\(2L - b = 2L\cos\left(\dfrac{\pi}{3} \cdot \dfrac{3}{4}\right)\)	A1ft
\(b = L(2 - \sqrt{2})\) *	A1 cso
	(5)

Scheme	Marks
\(\tfrac{1}{2}(2-\sqrt{2})L = 2L\sin\omega t\)	M1 A1
\(t = 0.1469\ldots\)
\(\therefore\) Total time \(= 2 \times 0.14\ldots\)
\(= 0.28\) (2 d.p.)	A1
	(3)
	(10 marks)

Scheme	Marks
\(\dot{x} = -2L\omega\sin\omega t\)
\(= -2L\dfrac{\pi}{3}\sin\dfrac{\pi}{4}\)	M1
Speed \(= \dfrac{\sqrt{2}L\pi}{3}\)	A1
	(2)