M2 June 2005 Q3
3. A particle \(P\) moves in a horizontal plane. At time \(t\) seconds, the position vector of \(P\) is \(\mathbf{r}\) metres relative to a fixed origin \(O\), and \(\mathbf{r}\) is given by
\[\mathbf{r} = (18t - 4t^3)\mathbf{i} + ct^2\mathbf{j},\]where \(c\) is a positive constant. When \(t = 1.5\), the speed of \(P\) is 15 m s\(^{-1}\). Find
(a) the value of \(c\), (6)
(b) the acceleration of \(P\) when \(t = 1.5\). (3)
| Scheme | Marks |
|---|---|
| \(\mathbf{v} = (18 - 12t^2)\mathbf{i} + 2ct\mathbf{j}\) | M1 A1 A1 |
| \(t = \tfrac{3}{2}\): \(\mathbf{v} = -9\mathbf{i} + 3c\mathbf{j}\) | M1 |
| \(|\mathbf{v}| = 15 \Rightarrow 9^2 + (3c)^2 = 15^2\) | M1 |
| \(\Rightarrow (3c)^2 = 144\ \Rightarrow\ c = 4\) | A1 |
| (6) |
| Scheme | Marks |
|---|---|
| \(\mathbf{a} = -24t\mathbf{i} + 8\mathbf{j}\) | M1 |
| \(t = \tfrac{3}{2}\): \(\mathbf{a} = -36\mathbf{i} + 8\mathbf{j}\) | M1 A1ft |
| (3) | |
| (9 marks) |