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

Let `r_(P), r_(Q)` are the radii of planets `P` and `Q` respectively. Then
`4pi r_(p)^(2) = A, 4pi r_(Q)^(2) = 4A , r_(Q) = 2r_(p)`
For planet `R`,
`(4)/(3)pi r_(p)^(3) rho + (4)/(3) pi (2r_(p))^(3) = (4)/(3) pi (r_(R))^(3) rho`
or `r_(R) = 9^(1//3) r_(p)`
Escape velocity, `upsilon = sqrt((2GM)/(r )) = sqrt((2G(4)/(3) pi r^(3) rho)/(r ))`
`= r sqrt((8pi G rho)/(3))`
i.e., `upsilon prop r`
`:. upsilon_(P) : upsilon_(Q) : upsilon_(R ) = r_(p) : 2r_(p) : 9^(1//3) r_(p) = 1 : 2: 9^(1//3)`
`:. upsilon_(R) gt upsilon_(Q) gt upsilon_(P)` and `upsilon_(P)//upsilon_(Q) = 1//2`
Thus option (b) and (d) are true.