Ask Uncle Colin: A Pair Of Birds

Dear Uncle Colin,

I have a little problem. You see, there’s this bird, A, in its nest at time $t=0$ - the nest is at $(20,-17)$ - and it travels with a velocity of $-6\mathbf{i}+7\mathbf{j}$ (in the appropriate units). But there’s another bird, B, whose nest is at $(-8,9)$ and who travels at $p\mathbf{i}+2p\mathbf{j}$. Then when $t=4$, it turns out that B is (get this) south-west of bird A! For\ some reason, I have to figure out the speed of bird B. How would you do it?

- Help Evaluating Right Ornithology Numbers

Hi, HERON, and thanks for your message!

Ah, a classical two-bird problem with vectors. Luckily, we have quite a lot of information and it should fall apart fairly quickly.

A picture

Before doing anything else, I’d draw a picture of what’s going on.

I did that. I also plotted it in Desmos:

Where are the birds?

We can write the position vectors of the birds as ${\mathbf{r}}_{\mathbf{A}}=(20-6t)\mathbf{i}+(-17+7t)\mathbf{j}$ and ${\mathbf{r}}_{\mathbf{B}}=(-8+pt)\mathbf{i}+(9+2pt)\mathbf{j}$.

Better than that, we’re only really interested in the situation when $t=4$, so we can substitute that in: ${\mathbf{r}}_{\mathbf{A}}=-4\mathbf{i}+11\mathbf{j}$, which corresponds to the picture, and ${\mathbf{r}}_{\mathbf{B}}=(-8+4p)\mathbf{i}+(9+8p)\mathbf{j}$.

Vector $\overrightarrow{BA}$

If B is southwest of A, vector $\overrightarrow{BA}$ is in the northwesterly direction: it’s a multiple of $\mathbf{i}+\mathbf{j}$.

But we can work out $\overrightarrow{BA}={\mathbf{r}}_{\mathbf{A}}-{\mathbf{r}}_{\mathbf{B}}$, which is $((-4)-(-8+4p))\mathbf{i}+(11-(9+8p))\mathbf{j}$.

More clearly, that’s $(4-4p)\mathbf{i}+(2-8p)\mathbf{j}$, which needs to be in the form $k\mathbf{i}+k\mathbf{j}$ - or rather, the components must be equal.

So $4-4p=2-8p$, giving $2=-4p$ and $p=-\frac{1}{2}$.

Finally the speed

B’s speed is $\sqrt{p^2+(2p^2)}$, or $|p|\sqrt{5}$, which is $\frac{1}{2}\sqrt{5}$ of those appropriate units.

Hope that helps!

- Uncle Colin