M2 June 2006 Q3
3. A cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity of the ball is \((-30\mathbf{i})\) m s\(^{-1}\). Immediately after being struck, the velocity of the ball is \((16\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\).
(a) Find the magnitude of the impulse exerted on the ball by the bat. (4)
In the subsequent motion, the position vector of the ball is \(\mathbf{r}\) metres at time \(t\) seconds. In a model of the situation, it is assumed that \(\mathbf{r} = [16t\mathbf{i} + (20t - 5t^2)\mathbf{j}]\). Using this model,
(b) find the speed of the ball when \(t = 3\). (4)
| Scheme | Marks |
|---|---|
| \(I = \pm 0.5(16\mathbf{i} + 20\mathbf{j} - (-30\mathbf{i}))\) | M1 |
| \(= \pm(23\mathbf{i} + 10\mathbf{j})\) indep | M1 |
| magn \(= \sqrt{(23^2 + 10^2)} \approx \underline{25.1\ \text{Ns}}\) indep | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\mathbf{v} = 16\mathbf{i} + (20 - 10t)\mathbf{j}\) | M1 |
| \(t = 3\ \Rightarrow\ \mathbf{v} = 16\mathbf{i} - 10\mathbf{j}\) indep | M1 |
| \(v = \sqrt{(16^2 + 10^2)} \approx \underline{18.9\ \text{m s}^{-1}}\) indep | M1 A1 |
| (4) | |
| (8 marks) |