M2 June 2005 Q4
4. A darts player throws darts at a dart board which hangs vertically. The motion of a dart is modelled as that of a particle moving freely under gravity. The darts move in a vertical plane which is perpendicular to the plane of the dart board. A dart is thrown horizontally with speed 12.6 m s\(^{-1}\). It hits the board at a point which is 10 cm below the level from which it was thrown.
(a) Find the horizontal distance from the point where the dart was thrown to the dart board. (4)
The darts player moves his position. He now throws a dart from a point which is at a horizontal distance of 2.5 m from the board. He throws the dart at an angle of elevation \(\alpha\) to the horizontal, where \(\tan\alpha = \tfrac{7}{24}\). This dart hits the board at a point which is at the same level as the point from which it was thrown.
(b) Find the speed with which the dart is thrown. (6)
| Scheme | Marks |
|---|---|
| \(\rightarrow\ 12.6t = x\) | B1 |
| \(\downarrow\ 0.1 = 4.9t^2\) | B1 |
| \(\Rightarrow 0.1 = 4.9 \times \dfrac{x^2}{12.6^2}\) | M1 |
| \(\Rightarrow x = 1.8\) m | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\rightarrow\ u\cos\alpha.t = 2.5\) | M1 A1 |
| \(\uparrow\ u\sin\alpha.t = \tfrac{1}{2}gt^2\) | M1 A1 |
| \(u.\dfrac{24}{25}t = 2.5\) | |
| \(u.\dfrac{7}{25} = 4.9.\dfrac{2.5.25}{24u}\) | |
| \(u^2 = \dfrac{4.9 \times 2.5 \times 25^2}{7 \times 24}\) | |
| \(\Rightarrow u \approx 6.75\) or 6.8 m s\(^{-1}\) | M1 A1 |
| (6) | |
| (10 marks) |