\begin{enumerate} \item A small ball \(A\) is moving with velocity \(( 7 \mathbf { i } + 12 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). It collides in mid-air with another ball \(B\), of mass 0.4 kg , moving with velocity \(( - \mathrm { i } + 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\). Immediately after the collision, \(A\) has velocity \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(B\) has velocity \(( 6 \cdot 5 \mathbf { i } + 13 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Calculate the mass of \(A\). \item A stick of mass 0.75 kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick
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and the floor is 0.6 . Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \item An engine of mass 20000 kg climbs a hill inclined at \(10 ^ { \circ }\) to the horizontal. The total nongravitational resistance to its motion has magnitude 35000 N and the maximum speed of the engine on the hill is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Find, in kW , the maximum rate at which the engine can work.
2. Find the maximum speed of the engine when it is travelling on a horizontal track against the same non-gravitational resistance as before. \item Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m }\) and \(( 12 \mathbf { i } + \mathbf { j } )\) m respectively, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \(( 2 t + 4 ) \mathbf { i } + \left( k t ^ { 2 } - 5 \right) \mathbf { j }\).