# Puzzle #20 – HP2: The Rogue Bludger

In a well-intentioned attempt to hospitalise Harry, Dobby the house-elf enchants a bludger such that it will pursue him relentlessly throughout the upcoming Quidditch match.

Let Harry’s position be $\bold{H}(t)$, and that of the pursuing bludger be $\bold{B}(t)$. The bludger’s velocity is at all times parallel to the vector joining the bludger to Harry, and we will assume for simplicity that it flies at constant speed $\beta$. Its instantaneous velocity may then be expressed as

$\displaystyle \frac{d}{dt}\bold{B}=\beta\left(\frac{\bold{H}-\bold{B}}{|\bold{H}-\bold{B}|}\right),$

where $|\bold{r}|$ represents the magnitude of vector $\bold{r}$.

Suppose that, to evade the bludger, Harry simply flies in a circle of radius $H$ at constant speed $H\omega=\eta$ in the $xy$ plane. His position may then be described by

$\displaystyle \bold{H}(t)=H\binom{\cos\omega t}{\sin\omega t},$

where we have taken $\bold{H}(0)=(H,0)$ without loss of generality.

The bludger’s equation of motion is practically insoluble for arbitrary $t$. That being said, we can make a pretty good guess as to what the bludger’s steady-state motion will be, if such a state exists; the bludger will also move in a circle, trailing Harry by some fixed distance.

Let’s use the ansatz

$\displaystyle \bold{B}(t)=B\binom{\cos\left(\omega t-\varphi\right)}{\sin\left(\omega t-\varphi\right)}$

The phase $\varphi$ is included to account for the fact that the bludger’s motion will lag Harry’s slightly.

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To solve for $B$ and $\varphi$, proceed as follows:

1) Write out the components of the bludger’s equation of motion, and take their ratio to obtain an expression for $\frac{dB_y}{dB_x}$.

2) Given that the bludger moves in a circle, find an expression for $\frac{dB_y}{dB_x}$ in terms of $B_y$ and $B_x$.

3) Equate your two expressions, and use our ansätze for $\bold{H}$ and $\bold{B}$ to generate an equation containing only the variable $t$.

4) Rearrange your equation to find an expression for the angle $\varphi$ in terms of the radii $H$ and $B$, and hence in terms of $\eta$ and $\beta$. Write down an expression for $B$ in terms of the same variables.

5) If $\beta=\frac{4}{5}\eta$, by what distance are Harry and the rogue bludger separated? What happens if $\beta>\eta$?

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Hints:

For part (2), differentiate the Cartesian equation of a circle; doing so implicitly will give the desired result fastest.

For part (4), you may (or may not) want to make use of the compound angle formulae $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$ and $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$. If we have made a reasonable guess, $t$ will drop out of the equation entirely.