At noon, two boats \(P\) and \(Q\) have position vectors \(( \mathbf { i } + 7 \mathbf { j } ) \mathrm { km }\) and \(( 3 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\) respectively relative to an origin \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively. \(P\) is moving with constant velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and \(Q\) is moving with constant velocity \(( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
Find the position vector of each boat at time \(t\) hours after noon, giving your answers in the form \(\mathrm { f } ( t ) \mathrm { i } + \mathrm { g } ( t ) \mathrm { j }\), where \(\mathrm { f } ( t )\) and \(\mathrm { g } ( t )\) are linear functions of \(t\) to be found.
Find, in terms of \(t\), the distance between the boats \(t\) hours after noon.
Calculate the time when the boats are closest together and find the distance between them at this time.
A particle starts from rest and accelerates at a uniform rate over a distance of 12 m . It then travels at a constant speed of \(u \mathrm {~ms} ^ { - 1 }\) for a further 30 seconds. Finally it decelerates uniformly to rest at \(1.6 \mathrm {~ms} ^ { - 2 }\).
Sketch the velocity-time graph for this motion.
Show that the total time for which the particle is in motion is