Question

a a Let the period of a planet at a distance R from star be to prove that it it was at a distance of 2R from the star, its period of revolution will be √8 T​

Answer 1

Answer:

Prove that if it was at a distance of 2R from the star, it's period of revolution will be √8T. Hence,

Answer 2

According to Kepler’s third law

$T^2∝a^3$

T is time period of planet and a is semi major axis of its orbit.

So, according to question

$T^2=kR^3--------(1)$

When distance is 2R then

$T'2 =k(2R)^3$

$T'2 =8R^3k - - - - - - - (2)$

Taking ratios of (1) and (2)

$\frac{T {}^{2} }{T' {}^{2} } = \frac{kR {}^{3} }{8R {}^{3}k }$

$\frac{T {}^{2} }{T' {}^{2} } = \frac{1}{8 {} }$

$T' {}^{2} = 8T{}^{2}$

$T' = \sqrt{8} T$