4. A boy starts at the corner \(O\) of a rectangular playing field and runs across the field with constant velocity vector \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions of two perpendicular sides of the field. After 40 seconds, at the point \(P\) in the field, he changes speed and direction so that his new velocity vector is \(( 2 \cdot 4 \mathbf { i } - 1 \cdot 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and maintains this velocity until he reaches the point \(Q\), where \(P Q = 75 \mathrm {~m}\).
Calculate (a) the distance \(O P\),
(b) the time taken to travel from \(P\) to \(Q\),
(c) the position vector of \(Q\) relative to \(O\). Another boy travels directly from \(O\) to \(Q\) with constant velocity \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), leaving \(O\) and reaching \(Q\) at the same times as the first boy.
(d) Find the values of the constants \(a\) and \(b\). \section*{MECHANICS 1 (A)TEST PAPER 2 Page 2}