A particle of mass \(m\) is projected vertically upwards, at time \(t = 0\), with speed \(U\). The particle is subject to air resistance of magnitude \(\frac { m g v ^ { 2 } } { k ^ { 2 } }\), where \(v\) is the speed of the particle at time \(t\) and \(k\) is a positive constant.
Show that the particle reaches its greatest height above the point of projection at time
$$\frac { k } { g } \tan ^ { - 1 } \left( \frac { U } { k } \right)$$
Find the greatest height above the point of projection attained by the particle.