M2 June 2010 Q5
5. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane.]
A ball of mass 0.5 kg is moving with velocity \((10\mathbf{i} + 24\mathbf{j})\) m s\(^{-1}\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20\mathbf{i}\) m s\(^{-1}\).
Find
(a) the magnitude of the impulse of the bat on the ball, (4)
(b) the size of the angle between the vector \(\mathbf{i}\) and the impulse exerted by the bat on the ball, (2)
(c) the kinetic energy lost by the ball in the impact. (3)
| Scheme | Marks |
|---|---|
| \(\mathbf{I} = m\mathbf{v} - m\mathbf{u}\) | |
| \(= 0.5 \times 20\mathbf{i} - 0.5(10\mathbf{i} + 24\mathbf{j})\) | M1 |
| \(= 5\mathbf{i} - 12\mathbf{j}\) | A1 |
| \(|5\mathbf{i} - 12\mathbf{j}| = 13\) Ns | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\tan\theta = \dfrac{12}{5}\) | M1 |
| \(\theta = 67.38\) | |
| \(\theta = 67.4^\circ\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| K.E.lost \(= \frac{1}{2} \times 0.5\left(10^2 + 24^2\right) - \frac{1}{2} \times 0.5 \times 20^2\) | M1 A1 |
| \(= 69\) J | A1 |
| (3) | |
| (9 marks) |