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

Answer

B, D

Explanation

`{:((P)/(rho),(Q)/(rho),(R )/(rho)),(M,8M,9M),(A,4A,),(R,2R,3^(2/3)R):}`
`V_(P) = sqrt((GM)/(R ))`
`V_(Q) = sqrt((G8M)/(2R)) = 2sqrt((GM)/(R )) = 8^(1//3) sqrt((GM)/(R ))`
`therefore (V_(P))/(V_(Q)) = (1)/(2)`
`V_(R ) = sqrt((G(9M))/(3^(2//3)R)) = sqrt(3^(4//3)(GM)/(R )) = 3^(2/3) sqrt((GM)/(R ))= 9^(1//3) sqrt((GM)/(R ))`
`therefore V_(R )gt V_(Q)gt V_(P)`