From Mechanics

From part a) we have

\displaystyle \mathbf{v}=(80+t\ )\mathbf{i}+4t\ \mathbf{j}.

and

\displaystyle \mathbf{r}=(80t+\frac{1}{2}{t}^{\ 2}\ )\mathbf{i}+2{t}^{\ 2}\ \mathbf{j}

We see the height is

Thus the time when the aeroplane is at a height \displaystyle 50 \text{ m} .

We see the height is \displaystyle 2{t}^{\ 2} for any time \displaystyle t.

Thus the time when the aeroplane is at a height \displaystyle 50 \text{ m} is given by

\displaystyle 2{{t}^{\ 2}}=50

or

\displaystyle t=5\ \text{s}

At this time the velocity is

\displaystyle \mathbf{v}=(80+5 )\mathbf{i}+4 \times 5\ \mathbf{j}=85\mathbf{i}+20\mathbf{j}

The speed is the magnitude of this velocity

\displaystyle \sqrt{{{85}^{2}}+{{20}^{2}}}=87\textrm{.}3\ \text{m}{{\text{s}}^{-1}}