Question

Let the period of revolution of a Planet at a distance R from a star be T prove that if it was at a distance of 2 R from the star period of revolution will be √8 T.

Answer 1

Explanation:

Here is your valuable proof.

Answer 2

Answer:

The period of revolution of the planet will be $\sf\:\sqrt{8}\:T$

Step-by-step-explanation:

We have given that,

$\bullet\sf\:Distance\:of\:planet\:from\:star\:=\:R\\\\\\\bullet\sf\:Period\:of\:revolution\:of\:planet\:=\:T$

We have to prove that,

If the planet was at a distance of 2R from star, its period of revolution will be $\sf\:\sqrt{8}\:T$.

According to Kepler's third law of planetary motion,

$\pink{\sf\:\dfrac{T^2}{R^3}\:=\:k}\sf\:\:\:-\:-\:(\:1\:)$

Where, k is the constant of proportionality.

Now,

Let x be the period of revolution of planet when it was at distance of 2R from the star.

Again by Kepler's third law, we get,

$\sf\:\dfrac{x^2}{(\:2R\:)^3}\:=\:k\:\:\:-\:-\:(\:2\:)$

From equations ( 1 ) & ( 2 ), we get,

$\sf\:\dfrac{x^2}{(\:2R\:)^3}\:=\:\dfrac{T^2}{R^3}\\\\\\\implies\sf\:\dfrac{x^2}{8R^3}\:=\:\dfrac{T^2}{R^3}\\\\\\\implies\sf\:x^2\:=\:\dfrac{T^2\:\times\:8\:\cancel{R^3}}{\cancel{R^3}}\\\\\\\implies\sf\:x^2\:=\:8\:T^2\\\\\\\implies\boxed{\red{\sf\:x\:=\:\sqrt{8}\:T}}$

Hence proved!