M3 January 2011 Q4
4. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its displacement, \(x\) metres, from the origin \(O\) is given by \(x = 5\sin\left(\tfrac{1}{3}\pi t\right)\).
(a) Prove that \(P\) is moving with simple harmonic motion. (3)
(b) Find the period and the amplitude of the motion. (2)
(c) Find the maximum speed of \(P\). (2)
The points \(A\) and \(B\) on the positive \(x\)-axis are such that \(OA = 2\) m and \(OB = 3\) m.
(d) Find the time taken by \(P\) to travel directly from \(A\) to \(B\). (4)
| Scheme | Marks |
|---|---|
| \(x = 5\sin\left(\dfrac{\pi t}{3}\right)\) \(\dot{x} = 5 \times \dfrac{\pi}{3}\cos\left(\dfrac{\pi t}{3}\right)\) \(\ddot{x} = -5 \times \left(\dfrac{\pi}{3}\right)^2\sin\left(\dfrac{\pi t}{3}\right)\) | M1A1 |
| \(\ddot{x} = -\dfrac{\pi^2}{9}x \qquad (\therefore \text{S.H.M.})\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| period \(= \dfrac{2\pi}{\frac{\pi}{3}} = 6\) | B1 |
| amplitude = 5 | B1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\ldots = 5 \times \dfrac{\pi}{3}\cos\left(\dfrac{\pi t}{3}\right) \qquad\) or \(\quad \left|v_{\max}\right| = a\omega\) | M1 |
| \(\max.\ v = \dfrac{5\pi}{3}\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| At \(A\quad x = 2 \qquad 2 = 5\sin\left(\dfrac{\pi t}{3}\right)\) | M1 |
| \(\sin\dfrac{\pi}{3}t = 0.4\) \(t_A = \dfrac{3}{\pi} \times \sin^{-1} 0.4\) | A1 |
| At \(B\quad x = 3 \qquad t_B = \dfrac{3}{\pi} \times \sin^{-1} 0.6\) time \(A \to B \quad = \dfrac{3}{\pi} \times \sin^{-1} 0.6 - \dfrac{3}{\pi} \times \sin^{-1} 0.4\) | A1 |
| \(= 0.2215\ldots = 0.22\) s accept awrt 0.22 | A1 |
| (4) | |
| (11 marks) |