A particle is projected from a point O with a velocity u at an angle α (upwards) to the horizontal. At a certain point P, it moves at right angles to its initial direction. It follows that

(d) vcos⁡(90−α)=ucos⁡α                v=ucot⁡α (c) tan⁡(−(90−α))=usin⁡α−gtucos⁡α⇒ t=ugsin⁡α (b) r=(ucos⁡α)ugsin⁡αi^+usin⁡αugsin⁡α−12gugsin⁡α2j^r=u2gcot⁡αi^+u2g−u22gcosec2⁡αj^r=u2gcot⁡αi^+u2g1−cosec2⁡α2j^OP=u2gcot⁡α2+u2g1−cosec2⁡α22 (a) tan⁡θ=u2g1−cosec2⁡α2u22cot⁡α