Finding when moving in specific direction

2. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) is moving on a horizontal plane with acceleration \(\left[ \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + ( 6 t - 5 ) \mathbf { j } \right] \mathrm { m } \mathrm { s } ^ { - 2 }\). When \(t = 3\) the velocity of \(P\) is \(( 11 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the velocity of \(P\) at time \(t\) seconds,
2. the speed of \(P\) when it is moving parallel to the vector \(\mathbf { i }\).

At time \(t\) seconds \((t \geq 0)\), a particle \(P\) has position vector \(\mathbf{p}\) metres, with respect to a fixed origin \(O\), where $$\mathbf{p} = (3t^2 - 6t + 4)\mathbf{i} + (3t^3 - 4t)\mathbf{j}.$$ Find

1. the velocity of \(P\) at time \(t\) seconds, [2]
2. the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf{i}\). [3]

When \(t = 1\), the particle \(P\) receives an impulse of \((2\mathbf{i} - 6\mathbf{j})\) N s. Given that the mass of \(P\) is 0.5 kg,

1. find the velocity of \(P\) immediately after the impulse. [4]