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. `R_R gt R_Q gt R_P`
D. `(v_P)/(v_Q) = 1/2`

Answer

A

Explanation

Escape velocity,
` v_e = sqrt((2GM)/( R) ) = sqrt ((2Grho 4/3 pi R^3)/(R )) = R sqrt((8pi G rho)/(3)) therefore v_e prop R` ...(i)
(Since `rho` is same for all planets.)
Let `R_P` and `R_Q` be radii of planets P and Q respectively. Then
Surface area of P, `4pi R_P^2 = A`
Surface area of Q, `4pi R_Q^2 = 4A therefore R_P/R_Q = 1/2 " or " R_Q = 2R_P ` .....(ii)
The spherical planet R has mass,`M_R = M_P + M_Q`
` therefore 4/3 pi R_R^3 rho = 4/3 pi R_P^3 rho + 4/3 pi R_Q^3 rho " or " R_R^3 = R_P^3 + R_Q^3`
or ` R_R^3 = R_P^3 + (2R_P)^3` (Using (ii))
or `R_R = (9)^(1//3) R_P therefore R_R gt R_Q gt R_P`
From equation (i), `v_R gt v_Q gt v_P` and from equation (ii), `(v_P)/(v_Q) = 1/2`