Particle motion: 2D direction/bearing

7 A particle moves on a smooth horizontal surface with acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). Initially the velocity of the particle is \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression for the velocity of the particle at time \(t\) seconds.
2. Find the time when the particle is travelling in the \(\mathbf { i }\) direction.
3. Show that when \(t = 4\) the speed of the particle is \(20 \mathrm {~ms} ^ { - 1 }\).

7. A particle has an initial velocity of \(( \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and is accelerating uniformly in the direction \(( 2 \mathbf { i } + \mathbf { j } )\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. Given that the magnitude of the acceleration is \(3 \sqrt { } 5 \mathrm {~ms} ^ { - 2 }\),

1. show that, after \(t\) seconds, the velocity vector of the particle is $$[ ( 6 t + 1 ) \mathbf { i } + ( 3 t - 5 ) \mathbf { j } ] \mathrm { ms } ^ { - 1 }$$
2. Using your answer to part (a), or otherwise, find the value of \(t\) for which the speed of the particle is at its minimum.
   (5 marks)

1 The velocity of a model boat, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { 1 }\), is given by $$\mathbf { v } = \binom { 5 } { 10 } + t \binom { 6 } { 8 }$$ where \(t\) is the time in seconds and the vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are east and north respectively.

1. Show that when \(t = 2.5\) the boat is travelling south-east (i.e. on a bearing of \(135 ^ { \circ }\) ). Calculate its speed at this time. The boat is at a point O when \(t = 0\).
2. Calculate the bearing of the boat from O when \(t = 2.5\).