If the distance between earth and the sun were half of its present value, that how many number of days will be there in a year?

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Solution

From Kepler's Law of planetary motion, $T^{2} \propto R^{3}$

$\frac{T_{1}^{2}}{T_{2}^{2}} = \frac{R_{1}^{3}}{R_{2}^{3}}$

$\frac{T_{1}^{2}}{T_{2}^{2}} = \frac{8 R_{1}^{3}}{R_{1}^{3}}$

$\frac{T_{1}^{2}}{T_{2}^{2}} = 8$

${(\frac{T_{1}^{2}}{T_{2}^{2}})}^{3} = 8$

Asked for $1$ year

$T_{2} = \frac{365}{2 \sqrt{2}}$

$T_{2} = 129 d a y s$

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