Question

The escape velocity of a body on a planet 'A' is 12 kms^–1. The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is ______.

Options

• 12 kms^-1

• 24 kms^-1

• 36 kms^-1

• 6 kms^-1

Answer 1

The escape velocity of a body on a planet 'A' is 12 kms^–1. The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is 12 kms^-1.

Explanation:

Escape velocity of a body on a planet is given by:

V_e = `sqrt(2"gR")` or V_e = `sqrt(2"GM")/"R"`

where g ⇒ acceleration due to gravity at the surface

m ⇒ mass of planet and R = Radius of planet

Assuming plant to be solid sphere uniform, density ρ

Mass = Density × Volume

M = ρ × `4/3` πR³

The expression for escape velocity can be written as

V_e = `sqrt (2  rho xx -  pi"R")/"R"` 

= `sqrt(4/3"G"rho"R")`

V_e = for planet A = 12 km/s.

Let it’s Radius R_A and density ρ

For planet B ⇒ `("V"_e)_"B" ⇒ "R"_"B"`

= `("R"_A")/2`, ρ_B = 4 ρ_A

`(("V"_"e")_"A")/(("V"_"e")_"B") = sqrt((8pi  "G"rho_"A""R"_"A"^2)/3)/((8pi  "G"rho_"B""R"_"B"^2)/3)`

= `sqrt(rho_"A"/rho_"B" ("R"_"A"/("R"_"B"))^2`

= `sqrt(1/4xx4)`

= 1

`("V"_"e")_"A"` = `("V"_"e")_"B"` = 12 km/s