Problem

2.

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors.]

A particle $P$ of mass $3\,\mathrm{kg}$ is moving on a smooth horizontal plane.

At time $t$ seconds, where $1\leqslant t\leqslant4$, the position vector, $\mathbf{r}$ metres, of $P$ relative to a fixed origin is given by $\mathbf{r}=4t^{\frac12}\mathbf{i}-3t\mathbf{j}$.

At the instant when $t=4$, $P$ receives an impulse $(6\mathbf{i}-3\mathbf{j})\,\mathrm{N\,s}$.

Find the speed of $P$ immediately after the impulse is received.