M2 June 2013 (R) Q7
7.
A small ball is projected from a fixed point \(O\) so as to hit a target \(T\) which is at a horizontal distance \(9a\) from \(O\) and at a height \(6a\) above the level of \(O\). The ball is projected with speed \(\sqrt{(27ag)}\) at an angle \(\theta\) to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.
The two possible angles of projection are \(\theta_1\) and \(\theta_2\), where \(\theta_1 > \theta_2\).
The particle is projected at the larger angle \(\theta_1\).
| Scheme | Marks |
|---|---|
| \((\rightarrow)\sqrt{27ag}\cos\theta.\ t = 9a\) | M1 A1 |
| \((\uparrow)\sqrt{27ag}\sin\theta.\ t - \tfrac{1}{2}gt^2 = 6a\) | M1 A1 |
| \((\uparrow)\sqrt{27ag}\sin\theta.\ \dfrac{9a}{\sqrt{27ag}\cos\theta} - \tfrac{1}{2}g\left(\dfrac{9a}{\sqrt{27ag}\cos\theta}\right)^2 = 6a\) | DM1 |
| \(9a\tan\theta - \tfrac{1}{2}g.81a^2\dfrac{(1 + \tan^2\theta)}{27ag} = 6a\) | DM1 |
| \(\tan^2\theta - 6\tan\theta + 5 = 0\) | A1 |
| (7) |
Notes
M1 Horizontal motion. Condone trig confusion.
M1 Vertical motion. Condone sign errors and trig confusion.
DM1 Substitute for \(t\) (unsimplified). Dependent on both previous M marks
DM1 Express all trig terms in terms of tan. Dependent on preceding M.
| Scheme | Marks |
|---|---|
| \(\tan^2\theta - 6\tan\theta + 5 = 0\) | |
| \((\tan\theta - 1)(\tan\theta - 5) = 0\) | M1 |
| \(\tan\theta_2 = 1\) or \(\tan\theta_1 = 5\) | A1 A1 |
| (3) |
Notes
M1 Method to find one root of the quadratic
| Scheme | Marks |
|---|---|
| \(t = \dfrac{9a}{\sqrt{27ag}\cos\theta} = \dfrac{9a}{\sqrt{27ag}} \times \dfrac{\sqrt{26}}{1}\) | M1 A1ft |
| \(= \sqrt{\dfrac{81a^2.26}{27a}} = \sqrt{\dfrac{78a}{g}}\ \ \ \)*Answer given* | A1 |
| (3) |
Notes
M1 Use \(\tan\theta =\) their 5 to find t.
A1ft Correct unsimplified. Correct \(\cos\theta\) for their \(\tan\theta\)
A1 Given answer \(\rightarrow\)evidence of working is required
| Scheme | Marks |
|---|---|
| \(\tfrac{1}{2}m(27ag - v^2) = mg6a\) | M1 A1 |
| \(v = \sqrt{15ag}\) | A1 |
| (3) | |
| (16 marks) |
Notes
M1 Conservation of energy. Requires all 3 terms. Condone sign error
A1 Correct equation
Or (d)
| \(v^2 = \left(\sqrt{27ag}\cos\theta\right)^2 + \left(\sqrt{27ag}\sin\theta - g.\sqrt{\dfrac{78a}{g}}\right)^2\) | M1 |
| \(= \left(\dfrac{27ag}{26}\right) + \left(5\sqrt{\dfrac{27ag}{26}} - \sqrt{78ag}\right)^2\ \left(= ag\left(\dfrac{27}{26} + \dfrac{363}{26}\right)\right)\) | A1 |
| \(v = \sqrt{15ag}\) | A1 |
M1 Horizontal and vertical components and Pythagoras. Condone trig confusion.
A1 Correctly substituted