Two planets A and B have the same material density. If the radius of A is twice that of B, then the ratio of escape velocity (vA/vB) is

Question

Two planets A and B have the same material density. If the radius of A is twice that of B, then the ratio of escape velocity (vA/vB) is (A) 2 (B) √2 (C) (1/√2) (D) (1/2)

Tardigrade Admin · Accepted Answer

Escape velocity is given by ve=√(2 G M/R)=√(2 G/R) × (4/3) π R3 ρ ⇒ ve =R √(8/3) π G ρ ∴ (vA/vB)=(RA/RB)=2

Q. Two planets $A$ and $B$ have the same material density. If the radius of $A$ is twice that of $B$ , then the ratio of escape velocity $\frac{v _{A}}{v _{B}}$ is

$2$

$2$

$\frac{1}{2}$

$\frac{1}{2}$

Escape velocity is given by
$v_{e} = \frac{2 GM}{R} = \frac{2 G}{R} \times \frac{4}{3} π R^{3} ρ$
$\Rightarrow v_{e} = R \frac{8}{3} π Gρ$
$∴ \frac{v _{A}}{v _{B}} = \frac{R _{A}}{R _{B}} = 2$

Q. Two planets A and B have the same material density. If the radius of A is twice that of B, then the ratio of escape velocity vB​vA​​ is

2

2​

2​1​

21​

Escape velocity is given by ve​=R2GM​​=R2G​×34​πR3ρ​ ⇒ve​=R38​πGρ​ ∴vB​vA​​=RB​RA​​=2

Q. Two planets $A$ and $B$ have the same material density. If the radius of $A$ is twice that of $B$ , then the ratio of escape velocity $\frac{v _{A}}{v _{B}}$ is

$2$

$2$

$\frac{1}{2}$

$\frac{1}{2}$

Escape velocity is given by
$v_{e} = \frac{2 GM}{R} = \frac{2 G}{R} \times \frac{4}{3} π R^{3} ρ$
$\Rightarrow v_{e} = R \frac{8}{3} π Gρ$
$∴ \frac{v _{A}}{v _{B}} = \frac{R _{A}}{R _{B}} = 2$