M2 January 2010 Q4
4.
The points \(A\), \(B\) and \(C\) lie in a horizontal plane. A batsman strikes a ball of mass 0.25 kg. Immediately before being struck, the ball is moving along the horizontal line \(AB\) with speed 30 m s\(^{-1}\). Immediately after being struck, the ball moves along the horizontal line \(BC\) with speed 40 m s\(^{-1}\). The line \(BC\) makes an angle of 60\(^\circ\) with the original direction of motion \(AB\), as shown in Figure 1.
Find, to 3 significant figures,
(i) the magnitude of the impulse given to the ball,
(ii) the size of the angle that the direction of this impulse makes with the original direction of motion \(AB\). (8)
| Scheme | Marks |
|---|---|
| \(I\uparrow\ = 0.25 \times 40\sin 60 = 5\sqrt{3}\quad (8.66)\) one component | M1 |
| \(I\leftarrow\ = 0.25(-20 + 30) = 2.5\) both | A1 |
| \(|I| = \sqrt{75 + 6.25} = 9.01\ \text{(Ns)}\) | M1 A1 |
| (4) |
Alternative to 4(i)
| Use of \(\ \mathbf{I} = m(\mathbf{v} - \mathbf{u})\) | M1 |
| \(30^2 + 40^2 - 2 \times 30 \times 40\cos 60^\circ\quad (= 1300)\) | M1 A1 |
| \(I = 0.25\sqrt{1300} = 9.01\) N s (3 s.f.) | A1 |
2nd Alternative to 4(i)
| \(\mathbf{u} = 30\mathbf{i},\quad \mathbf{v} = 40\cos 60\mathbf{i} + 40\sin 60\mathbf{j} = 20\mathbf{i} + 20\sqrt{3}\mathbf{j}\) | |
| \(I = \dfrac{1}{4}(-10\mathbf{i} + 20\sqrt{3}\mathbf{j}) = -2.5\mathbf{i} + 5\sqrt{3}\mathbf{j}\) | M1 A1 etc |
| Scheme | Marks |
|---|---|
| \(\dfrac{\sin\theta}{40} = \dfrac{\sin 60^\circ}{\sqrt{1300}}\) | |
| \(\theta = 106^\circ\) (3 s.f.) | M1 A1 |
| or \(\ \tan\theta = \pm\dfrac{5\sqrt{3}}{2.5}\) oee \(\theta = 106^\circ\) | M1 A1 |
| (4) | |
| (8 marks) |