M1 January 2011 Q4
4. A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. The velocity of \(P\) is \((2\mathbf{i} - 5\mathbf{j})\) m s\(^{-1}\) at time \(t = 0\), and \((7\mathbf{i} + 10\mathbf{j})\) m s\(^{-1}\) at time \(t = 5\) s.
Find
(a) the speed of \(P\) at \(t = 0\), (2)
(b) the vector \(\mathbf{F}\) in the form \(a\mathbf{i} + b\mathbf{j}\), (5)
(c) the value of \(t\) when \(P\) is moving parallel to \(\mathbf{i}\). (4)
| Scheme | Marks |
|---|---|
| speed \(= \sqrt{2^2 + (-5)^2}\) | M1 |
| \(= \sqrt{29} = 5.4\) or better | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\left((7\mathbf{i} + 10\mathbf{j}) - (2\mathbf{i} - 5\mathbf{j})\right)/5\) | M1 A1 |
| \(= (5\mathbf{i} + 15\mathbf{j})/5 = \mathbf{i} + 3\mathbf{j}\) | A1 |
| \(\mathbf{F} = m\mathbf{a} = 2(\mathbf{i} + 3\mathbf{j}) = 2\mathbf{i} + 6\mathbf{j}\) | DM1 A1ft |
| (5) |
| Scheme | Marks |
|---|---|
| \(\mathbf{v} = \mathbf{u} + \mathbf{a}t = (2\mathbf{i} - 5\mathbf{j}) + (\mathbf{i} + 3\mathbf{j})t\) | M1 |
| \((-5 + 3t)\mathbf{j}\) | A1 |
| Parallel to \(\mathbf{i} \Rightarrow -5 + 3t = 0\) | M1 |
| \(t = 5/3\) | A1 |
| (4) | |
| (11 marks) |