A particle \(P\) starts from the point \(O\) and moves such that its position vector \(\mathbf { r } \mathbf { m }\) relative to \(O\) after \(t\) seconds is given by \(\mathbf { r } = a t ^ { 2 } \mathbf { i } + b t \mathbf { j }\).
60 seconds after \(P\) leaves \(O\) it is at the point \(Q\) with position vector \(( 90 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m }\).
Find the values of the constants \(a\) and \(b\).
Find the speed of \(P\) when it is at \(Q\).
Sketch the path followed by \(P\) for \(0 \leq t \leq 60\).
A lorry of mass 4200 kg can develop a maximum power of 84 kW . On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at \(20 \mathrm {~ms} ^ { - 1 }\) the resisting force has magnitude 2400 N . Find the maximum speed of the lorry when it is
travelling on a horizontal road,
climbing a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\).