3. During a cricket match, the batsman hits the ball and begins running with constant velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to try and score a run. When the batsman is at the fixed origin \(O\), the ball is thrown by a member of the opposing team with velocity \(\left( { } ^ { - } 8 \mathbf { i } + 24 \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\) from the point with position vector \(( 30 \mathbf { i } - 60 \mathbf { j } ) \mathrm { m }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors. At time \(t\) seconds after the ball is thrown, the position vectors of the batsman and the ball are \(\mathbf { r }\) metres and s metres respectively. In a model of the situation, the ball is assumed to travel horizontally and air resistance is considered to be negligible.

1. Find expressions for \(\mathbf { r }\) and \(\mathbf { s }\) in terms of \(t\).
2. Show that the ball hits the batsman and find the position vector of the batsman when this occurs.
3. Write down two reasons why the assumptions used in these calculations are unlikely to provide a realistic model.
   (2 marks)