1. \hspace{0pt} [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 \(( 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the speed of the ball,
2. the position vector of the ball after \(t\) seconds. The point \(B\) on the field has position vector \(( 10 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m }\).
3. Find the time when the ball is due north of \(B\). At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that he intercepts the ball,
4. find the value of \(v\).
5. State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic.