M1 June 2008 Q3
3. A particle \(P\) of mass 0.4 kg moves under the action of a single constant force \(\mathbf{F}\) newtons. The acceleration of \(P\) is \((6\mathbf{i} + 8\mathbf{j})\) m s\(^{-2}\). Find
(a) the angle between the acceleration and \(\mathbf{i}\), (2)
(b) the magnitude of \(\mathbf{F}\). (3)
At time \(t\) seconds the velocity of \(P\) is \(\mathbf{v}\) m s\(^{-1}\). Given that when \(t = 0\), \(\mathbf{v} = 9\mathbf{i} - 10\mathbf{j}\),
(c) find the velocity of \(P\) when \(t = 5\). (3)
| Scheme | Marks |
|---|---|
| \(\tan\theta = \dfrac{8}{6}\) | M1 |
| \(\theta \approx 53^\circ\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{F} = 0.4(6\mathbf{i} + 8\mathbf{j})\ \ (= 2.4\mathbf{i} + 3.2\mathbf{j})\) | M1 |
| \(|\mathbf{F}| = \sqrt{\left(2.4^2 + 3.2^2\right)} = 4\) | M1 A1 |
| (3) |
Notes
The method marks can be gained in either order.
| Scheme | Marks |
|---|---|
| \(\mathbf{v} = 9\mathbf{i} - 10\mathbf{j} + 5(6\mathbf{i} + 8\mathbf{j})\) | M1 A1 |
| \(= 39\mathbf{i} + 30\mathbf{j}\ \left(\text{ms}^{-1}\right)\) | A1 |
| (3) | |
| (8 marks) |