M1 June 2005 Q8

8. [In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal vectors due east and north respectively.]

At time \(t = 0\), a football player kicks a ball from the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j})\) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity \((5\mathbf{i} + 8\mathbf{j})\) m s\(^{-1}\). Find

The point \(B\) on the field has position vector \((10\mathbf{i} + 7\mathbf{j})\) m.

At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v\) m s\(^{-1}\). Given that he intercepts the ball,