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.

tamperedLet 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.

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