Q13 (b) (iv)

Particle $A$ is projected from the origin with initial speed $v\ \mathrm{m\,s^{-1}}$ at an angle $\theta$ with the horizontal plane. At the same time, particle $B$ is projected horizontally with initial speed $u\ \mathrm{m\,s^{-1}}$ from a point that is $H$ metres above the origin, as shown in the diagram.

The position vector of particle $A$, $t$ seconds after it is projected, is given by

$$\mathbf r_A(t)=\begin{pmatrix}vt\cos\theta\\ vt\sin\theta-\frac12gt^2\end{pmatrix}.$$

(Do NOT prove this.)

The position vector of particle $B$, $t$ seconds after it is projected, is given by

$$\mathbf r_B(t)=\begin{pmatrix}ut\\ H-\frac12gt^2\end{pmatrix}.$$

(Do NOT prove this.)

The angle $\theta$ is chosen so that $\tan\theta=2$.

The two particles collide.

Prior to the collision, the trajectory of particle $A$ was a parabola. (Do NOT prove this.)

(iv) Find the height of the vertex of that parabola above the horizontal plane. Give your answer in terms of $H$.