Two spherical planets P and Q have the same uniform density `rho,` masses `M_p and M_Q` and surface areas A and 4A respectively. A spherical planet R also has uniform density `rho` and its mass is `(M_P + M_Q).` The escape velocities from the plantes P,Q and R are `V_P V_Q and V_R` respectively. Then

Options

A. `V_(Q)gtV_(R)gtV_(P)`
B. `V_(R)gtV_(Q)gtV_(P)`
C. `V_(R)//V_(P)=3`
D. `V_(P)//V_(Q)=1/2`

Answer

B, D

Explanation

`V_(P)=sqrt((2GM)/(R)),V_(Q)=sqrt((2G(8M))/(2R))=2V_(P)`
`V_(R)=sqrt((2G(9M))/(9^(1//3)R))=9^(1//3)V_(P)`