A pilot is performing at an air show. The position of her aeroplane at time `t` relative to a fixed origin `O` is given by
`underset~r_text(A) (t) = (450 - 150sin((pit)/6))underset~i + (400 - 200cos((pit)/6))underset~j`,
where `underset~i` is a unit vector in a horizontal direction, `underset~j` is a unit vector vertically up, displacement components are measured in metres and time `t` is measured in seconds where `t >= 0`.
- Find the maximum speed of the aeroplane. Give your answer in `text(ms)^(−1)`. (3 marks)
- i. Use `underset~r_text(A)(t)` to show that the cartesian equation of the path of the aeroplane is given by
- `((x - 450)^2)/(22\ 500) + ((y - 400)^2)/(40\ 000) = 1`. (2 marks)
- ii. Sketch the path of the aeroplane on the axes provided below. Label the position of the aeroplane when `t = 0`, using coordinates, and use an arrow to show the direction of motion of the aeroplane. (3 marks)
A friend of the pilot launches an experimental jet-powered drone to take photographs of the air show. The position of the drone at time `t` relative to the fixed origin is given by `underset~r_text(D)(t) = (30t)underset~i + (−t^2 + 40t)underset~j`, where `t` is in seconds and `0 <= t <= 40, underset~i` is a unit vector in the same horizontal direction, `underset~j` is a unit vector vertically up, and displacement components are measured in metres.
- Sketch the path of the drone on the axes provided in part b.ii. Using coordinates, label the points where the path of the drone crosses the path of the aeroplane, correct to the nearest metre. (3 marks)
- Determine whether the drone will make contact with the aeroplane. Give reasons for your answer. (3 marks)