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Circle, point, gravity and time!

Source: South Africa 1997

October 8, 2005
trigonometrygeometry solvedgeometry

Problem Statement

A circle and a point PP higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from PP to a point QQ on the circle. Given that the distance travelled from PP in time tt is equal to 12gt2sin⁡α\dfrac{1}{2}gt^2 \sin{\alpha}, where α\alpha is the angle of inclination of the line PQPQ to the horizontal, give a geometrical characterization of the point QQ for which the time taken from PP to QQ is a minimum.
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