Question 398: Gravitation

Question

The radius of a planet is twice the radius of earth. Both have almost equal average mass densities. V
P
and V
E
are escape velocities of the planet and the earth, respectively, then

Accepted Answer

Here, R
P
= 2R
E
, $${\rho _E} = {\rho _P}$$
Escape velocity of the earth,
$${V_E} = \sqrt {{{2G{M_E}} \over {{R_E}}}}  = \sqrt {{{2G} \over {{R_E}}}\left( {{4 \over 3}\pi R_E^3{\rho _E}} \right)} $$
$$ = {R_E}\sqrt {{8 \over 3}\pi G{\rho _E}} $$   ...(i)
Escape velocity of the planet
$${V_P} = \sqrt {{{2G{M_P}} \over {{R_P}}}}  = \sqrt {{{2G} \over {{R_P}}}\left( {{4 \over 3}\pi R_P^3{\rho _P}} \right)} $$
$$ = {R_P}\sqrt {{8 \over 3}\pi G{\rho _P}} $$   ...(ii)
Divide (i) by (ii), we get
$${{{V_E}} \over {{V_P}}} = {{{R_E}} \over {{R_P}}}\sqrt {{{{\rho _E}} \over {{\rho _P}}}}  = {{{R_E}} \over {2{R_E}}}\sqrt {{{{\rho _E}} \over {{\rho _E}}}}  = {1 \over 2}$$
$$ \Rightarrow {V_P} = 2{V_E}$$

Answer

V
P
= 1.5 V
E

Answer

V
P
= 2 V
E

Answer

V
E
= 3 V
P

Answer

V
E
= 1.5 V
P

Gravitation

The radius of a planet is twice the radius of earth. Both have almost equal average mass densities. V P and V E are escape velocities of the planet and the earth, respectively, then

Choose the correct answer: