A parcel of mass 2.5 kg is moving in a straight line on a smooth horizontal floor. Initially the parcel is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The parcel is brought to rest in a distance of 20 m by a constant horizontal force of magnitude \(R\) newtons. Modelling the parcel as a particle, find
the kinetic energy lost by the parcel in coming to rest,
the value of \(R\).
At time \(t\) seconds \(( t \geqslant 0 )\), a particle \(P\) has position vector \(\mathbf { p }\) metres, with respect to a fixed origin \(O\), where
$$\mathbf { p } = \left( 3 t ^ { 2 } - 6 t + 4 \right) \mathbf { i } + \left( 3 t ^ { 3 } - 4 t \right) \mathbf { j } .$$
Find
the velocity of \(P\) at time \(t\) seconds,
the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf { i }\).
When \(t = 1\), the particle \(P\) receives an impulse of \(( 2 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N } \mathrm { s }\). Given that the mass of \(P\) is 0.5 kg ,
find the velocity of \(P\) immediately after the impulse.