M3 June 2010 Q6
6. At time \(t = 0\), a particle \(P\) is at the origin \(O\) moving with speed 2 m s\(^{-1}\) along the \(x\)-axis in the positive \(x\)-direction. At time \(t\) seconds \((t > 0)\), the acceleration of \(P\) has magnitude \(\dfrac{3}{(t + 1)^2}\) m s\(^{-2}\) and is directed towards \(O\).
(a) Show that at time \(t\) seconds the velocity of \(P\) is \(\left(\dfrac{3}{t + 1} - 1\right)\) m s\(^{-1}\). (5)
(b) Find, to 3 significant figures, the distance of \(P\) from \(O\) when \(P\) is instantaneously at rest. (7)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}^2x}{\mathrm{d}t^2} = -\dfrac{3}{(t + 1)^2}\) | M1 |
| \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = \displaystyle\int -3(t + 1)^{-2}\,\mathrm{d}t\) | |
| \(= 3(t + 1)^{-1}\ (+c)\) | M1 A1 |
| \(t = 0,\ \ v = 2 \quad 2 = 3 + c \quad c = -1\) | M1 |
| \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{3}{t + 1} - 1\) * | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(x = \displaystyle\int\left(\dfrac{3}{t + 1} - 1\right)\mathrm{d}t\) | M1 |
| \(= 3\ln(t + 1) - t \quad (+c^{\prime})\) | A1 |
| \(t = 0,\ \ x = 0 \quad \Rightarrow c^{\prime} = 0\) \(x = 3\ln(t + 1) - t\) | B1 |
| \(v = 0 \Rightarrow \dfrac{3}{t + 1} = 1\) | M1 |
| \(t = 2\) | A1 |
| \(x = 3\ln 3 - 2\) | M1 |
| \(= 1.295\ldots\) \(= 1.30\) m (Allow 1.3) | A1 |
| (7) | |
| (12 marks) |