M2 June 2011 Q3
3. A ball of mass 0.5 kg is moving with velocity \(12\mathbf{i}\) m s\(^{-1}\) when it is struck by a bat. The impulse received by the ball is \((-4\mathbf{i} + 7\mathbf{j})\) N s. By modelling the ball as a particle, find
(a) the speed of the ball immediately after the impact, (4)
(b) the angle, in degrees, between the velocity of the ball immediately after the impact and the vector \(\mathbf{i}\), (2)
(c) the kinetic energy gained by the ball as a result of the impact. (2)
| Scheme | Marks |
|---|---|
| \(\mathbf{I} = m\mathbf{v} - m\mathbf{u}\) | M1 |
| \(-4\mathbf{i} + 7\mathbf{j} = 0.5(\mathbf{v} - 12\mathbf{i})\) | |
| \(4\mathbf{i} + 14\mathbf{j} = \mathbf{v}\) | A1 |
| Speed \(= \sqrt{16 + 196} = \sqrt{212}\) m s\(^{-1}\) (14.6 or better) | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\tan\theta = \dfrac{7}{2}\) | M1 |
| \(\theta = 74.0\ldots\) | |
| \(\theta = 74^\circ\) | A1ft |
| (2) |
| Scheme | Marks |
|---|---|
| Gain in K.E. \(= \dfrac{1}{2} \times 0.5\left(212 - 12^2\right),\quad = 17\) J | M1 A1 |
| (2) | |
| (8 marks) |