A particle \(P\) is projected with speed \(u\,\text{m s}^{-1}\) at an angle \(\tan^{-1}2\) above the horizontal from a point \(O\) on a horizontal plane. It moves freely under gravity.

When \(P\) has travelled \(56\) m horizontally from \(O\), it is at a vertical height \(H\) m above the plane. When \(P\) has travelled \(84\) m horizontally from \(O\), it is at a vertical height \(\dfrac12H\) m above the plane.

Find, in either order, the value of \(u\) and the value of \(H\).