A planet is revolving around the sun as shown in the figure. The radius vectors joining the sum and the planet at points A and B are `90 times 10^6 km and 60 times 10^6 km`, respectively. The ratio of velocities of the planet at the points A and B when its velocities makes angle `30^@` and `60^@` with major axis of the orbit is

Options

A. `3/(2sqrt3)`
B. `2/sqrt3`
C. `1/sqrt3`
D. `sqrt3/2`

Explanation

According to the law of conservation of angular momentum , angular momentum (J) of a planet is constant.
`impliesm u _A r_A sin theta= m u _ B r _B sin theta _B`
or `u_A/u_B=r_B/r_A (sin theta_B)/ (sin theta_A) `
Given `r_A=90 times 10^6 km, r_B=60 times 10^6 km`
`theta_A=30^@, theta_B=60^@`
or `u_A/u_B=(60 times10^6)/ (90 times10^6) times (sin60^@)/(sin30^@)`
`=2/3 times (sqrt3//2)/(1//2)`
or `u_A/u_B=2/sqrt3`
Hence, the ratio of velocities of the planet is `2sqrt3`.