9231 P32 - Nov 2023 - Q6 - 9 marks
6931
A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.
(a) Derive the equation of the trajectory of \(P\) in the form
\(y=x\tan\alpha-\frac{gx^2}{2u^2}\sec^2\alpha.\)
During its flight, \(P\) must clear an obstacle of height \(h\) metres that is at a horizontal distance of \(32\) metres from the point of projection.
When \(u=40\sqrt2\,\text{m s}^{-1}\), \(P\) just clears the obstacle. When \(u=40\,\text{m s}^{-1}\), \(P\) only achieves \(80\%\) of the height required to clear the obstacle.
(b) Find the two possible values of \(h\).
Solutions and mark schemes for 9231 are temporarily available to admins only.