The radius of the planet is double that of the earth, but their average densities are same. V p and V E are the escape velocities of planet and earth respectively. If V E V p = x , the value

Question

The radius of the planet is double that of the earth, but their average densities are same. V p ​ and V E ​ are the escape velocities of planet and earth respectively. If V E ​ V p ​ ​ = x , the value of ' x ' is

TestHub · Accepted Answer

The escape velocity of a planet is given by $V = \frac{2 GM}{R}$ .

Substituting mass $M = \frac{4}{3} π R^{3} ρ$ , we get:

$V = \frac{2 G}{R} \times \frac{4}{3} π R^{3} ρ = R \frac{8}{3} π Gρ$

Since the average density $ρ$ is the same for both the planet and the earth, the escape velocity is directly proportional to the radius of the planet.

$V \propto R$

Given that the radius of the planet is double that of the earth ( $R_{p} = 2 R_{E}$ ), the ratio of their escape velocities is:

$\frac{V _{p}}{V _{E}} = \frac{R _{p}}{R _{E}} = \frac{2 R _{E}}{R _{E}} = 2$

Thus, $x = 2$ .

Answer: 2

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