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 2R from the star, its period of revolution will be â8T.

Answer 1

According to Kepler's law , $\bold{T^2 \propto a^3}$
Here T denotes the time period and a denotes the distance between two objects

Here, when distance between planet and star is R then time period is T
means T² = kR³ [ here k is proportionality constant ]----(1)

Again, Let the time period is T' when distance between planet and star is 2R
T'² = k(2R)³ = k.8R³ -----(2)

Dividing equation (1) by (2)
T²/T'² = kR³/k8R³ = 1/8
T'² = 8T²
square root both sides,
$\bold{T' = \sqrt{8}T}$
Hence, time period will be √8T , when distance between planet and star will be 2R

Answer 2

Explanation:

According to Kepler's law , \bold{T^2 \propto a^3}T

2

∝a

3

Here T denotes the time period and a denotes the distance between two objects

Here, when distance between planet and star is R then time period is T

means T² = kR³ [ here k is proportionality constant ]----(1)

Again, Let the time period is T' when distance between planet and star is 2R

T'² = k(2R)³ = k.8R³ -----(2)

Dividing equation (1) by (2)

T²/T'² = kR³/k8R³ = 1/8

T'² = 8T²

square root both sides,

\bold{T' = \sqrt{8}T}T

′

=

8

T

Hence, time period will be √8T , when distance between planet and star will be 2R