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 start, it's period of revolution will be /8T

Answer 1

According to Kepler's law , T²\proptoa³
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,
T' = \sqrt{8}T
Hence, time period will be \sqrt{8}T , when distance between planet and star will be 2R