A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. Its initial speed is \(u \mathrm{~m} \mathrm{~s}^{-1}\) and its angle of projection is \(\sin ^{-1}\left(\frac{4}{5}\right)\) above the horizontal. At time 8 s after projection, \(P\) is at the point \(A\). At time 32 s after projection, \(P\) is at the point \(B\). The direction of motion of \(P\) at \(B\) is perpendicular to its direction of motion at \(A\).

Find the value of \(u\).