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A comet (assumed to be in an elliptical orbit around the sun) is at a distance of $0.4 \mathrm{AU}$ from the sun at the per ihelion. If the time period of the comet is 125 years, what is the aphelion distance ? AU : Astronomical Unit.
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The correct answer is:
$49.6 \mathrm{AU}$
$$
\therefore r=\frac{0.4+y}{2}
$$
also $\left(\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}\right)^{2}=\left(\frac{\mathrm{r}_{1}}{\mathrm{r}_{2}}\right)^{3}$ by Kepler's law of time- periods $\left(\mathrm{T}_{1}, \mathrm{r}_{1}\right.$ are taken for earth)
$$
\therefore\left(\frac{1 \mathrm{y}}{125 \mathrm{y}}\right)^{2}=\left(\frac{1 \mathrm{AU}}{\left(\frac{0.4+\mathrm{y}}{2}\right) \mathrm{AU}}\right)^{3}
$$
ving we get $\mathrm{y}=49.6 \mathrm{AU}$
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