# STEP I 2017 Q.9 – trajectory through (d, d tan β)

I/2017/9 – A particle is projected at speed u from a point O on a horizontal plane. It passes through a fixed point P which is at a horizontal distance d from O and at a height d tan β above the plane, where d > 0 and β is an acute angle. The angle of projection α is chosen so that u is as small as possible.

(i) Show that $u^2 = gd \tan \alpha$ and $2 \alpha = \beta + 90 ^{\circ}$

(ii) At what angle to the horizontal is the particle travelling when it passes through P? Express your answer in terms of α in its simplest form.