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{\bf Question}

On a level plane the sound of a rifle and that of the bullet
striking the target are heard at the same instant.  What is the
locus of the hearer!

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{\bf Answer}

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${}$

Consider the wavefront $\left\{\begin{array}{ll} {\rm if\ } w=v &
{\rm locus\ is\ the\ half\ line\ }\epsilon T \\ {\rm if\ } w<v &
{\rm no\ real\ locus\ exists} \end{array} \right.$

The time taken from the rifle shot to reach $H$ is $\ds
\frac{RH}{v}$ where $v$ is the velocity of sound.

The time taken for the sound to reach $H$ via $T$ is $\ds
\frac{RT}{W} + \frac{TH}{v} = \frac{RH}{v}$ $$RH - TH =
\frac{RT.v}{W} = {\rm constant}$$ So thre locus of H is a branch
of a hyperbola

if $W>v$ (constant = $\lq \lq 2a " < RT$ = distance between foci)

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