1 A ball of mass 0.2 kg is hit directly by a bat. Just before the impact, the ball is travelling horizontally with speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Just after the impact, the ball is travelling horizontally with speed \(32 \mathrm {~ms} ^ { - 1 }\) in the opposite direction.

1. Find the magnitude of the impulse exerted on the ball.
2. At time \(t\) seconds after the ball first comes into contact with the bat, the force exerted by the bat on the ball is \(k \left( 0.9 t - 10 t ^ { 2 } \right)\) newtons, where \(k\) is a constant and \(0 \leqslant t \leqslant 0.09\). The bat stays in contact with the ball for 0.09 seconds. Find the value of \(k\).