Question

prove that the ratio of maximum to minimun velocity of a planet in its circular orbit is equal to the ratio of the radio of the orbit.

Answer 1

the planet will have maximum velocity when it is at minimum distance from the sun and vice-versa.

Net torque on the planet about the sun is zero, coz force (Gravitational) acts along the line joining the sun and the planet.

so, Angular momentum of the planet is conserved.
i.e. L = constant.
=> L = m x V x R= const.

since, m = constant.

so, if "R" is maximum then "V" will be minimum and vice-versa.

Again,

=> m x V1 x R1 = m x V2 x R2
=> V1 / V2 = R2 / R1................. proved .