First slide

Gravitaional potential energy

Question

The radius of a planet is twice the radius of earth. Both have almost equal average mass-densities. $V_{P} a n d V_{E}$ are escape velocities of the planet and the earth, respectively, then

Moderate

A

$V_{P} = 1.5 V_{E}$
B

$V_{P} = 2 V_{E}$
C

$V_{E} = 3 V_{P}$
D

$V_{E} = 1.5 V_{P}$

Solution

$Here, R_{P} = 2 R_{E}, ρ_{E} = ρ_{P} E s c a p e v e l o c i t y o f t h e e a r t h, V_{E} = \sqrt{\frac{2 G M_{E}}{R_{E}}} = \sqrt{\frac{2 G}{R_{E}} (\frac{4}{3} π R_{E}^{3} ρ_{E})} = R_{E} \sqrt{\frac{8}{3} π G ρ_{E}} . . . . (i) E s c a p e v e l o c i t y o f p l a n e t, V_{P} = \sqrt{\frac{2 G M_{P}}{R_{P}}} = \sqrt{\frac{2 G}{R_{P}} (\frac{4}{3} π R_{P}^{3} ρ_{P})} = R_{P} \sqrt{\frac{8}{3} π G ρ_{P}} . . . . . . . (i i) D i v i d e (i) b y (i i), w e g e t \frac{V_{E}}{V_{P}} = \frac{R_{E}}{R_{P}} \sqrt{\frac{ρ_{E}}{ρ_{P}}} \frac{V_{E}}{V_{P}} = \frac{R_{E}}{2 R_{E}} \sqrt{\frac{ρ_{E}}{ρ_{E}}} = \frac{1}{2} or V_{P} = 2 V_{E}$

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