Question 414: Gravitation

Question

The radius of a planet is twice the radius of earth. Both have almost equal average mass densities. VP and VE 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

VP = 1.5 VE

Answer

VP = 2 VE

Answer

VE = 3 VP

Answer

VE = 1.5 VP

Gravitation

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

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