A particle \(P\) of mass 2 kg is moving with velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse. Immediately after the impulse is applied, \(P\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the magnitude of the impulse.
Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied.
A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where
$$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$
When \(t = 0 , P\) is at the origin \(O\).
Find the acceleration of \(P\) when \(t = 3\)
Find the total distance travelled by \(P\) in the first 3 seconds of its motion.