Answer

Verified

34.8k+ views

**Hint:**From the question we can see that the radius and mean density of the planet is given. We can use it to calculate escape velocity of both earth and the planet.

Formula Used:

${v_e} = \sqrt {\dfrac{{2G{M_e}}}{{{R_e}}}} $\[\]

**Complete step by step answer:**

As the name suggests, escape velocity is an initial velocity at which, a body when thrown will leave the gravitational field of earth and never come back and its formula is

${v_e} = \sqrt {\dfrac{{2G{M_e}}}{{{R_e}}}} $\[\] where G is the gravitational constant, ${M_e},{R_e}$ are the mass and radius of earth

Since density is given, we can write mass as the product of volume and density

Hence $M = V\rho = \dfrac{{4\pi {R^3}}}{3}\rho $ (the shape of earth and planet is spherical)

V is the volume and $\rho $ is the density, now put it in the escape velocity equation,

$v = \sqrt {\dfrac{{2G \times 4\pi {R^3} \times \rho }}{{3R}}} = \sqrt {\dfrac{{2G \times 4\pi {R^2} \times \rho }}{3}} $

Escape velocity of earth${v_e} = \sqrt {\dfrac{{2G \times 4\pi {R_e}^2 \times {\rho _e}}}{3}} $

Similarly escape velocity of planet ${v_p} = \sqrt {\dfrac{{2G \times 4\pi {R_p}^2 \times {\rho _p}}}{3}} $

The ratio is \[\dfrac{{{v_e}}}{{{v_p}}} = \sqrt {\dfrac{{2G \times 4\pi {R_e}^2 \times {\rho _e}}}{3}} \Rightarrow \dfrac{{{v_e}}}{{{v_p}}} = \dfrac{{{R_e}}}{{{R_p}}}\sqrt {\dfrac{{{\rho _e}}}{{{\rho _p}}}} \]

Since it is given in the question that the radius and mean density of planet is two times to that of earth

Putting this value in above equation, it becomes

\[\dfrac{{{v_e}}}{{{v_p}}} = \dfrac{1}{2}\sqrt {\dfrac{1}{2}} \Rightarrow \dfrac{{{v_e}}}{{{v_p}}} = \dfrac{1}{{2\sqrt 2 }}\]

**Hence, the correct option is D**

**Additional information:**

Mathematically, gravitational constant is the force of attraction of two particles which are of unit mass and are kept at a distance of a unit. It is not affected by the presence of any other body or medium. It is the same in every condition. The SI unit of G is \[N{m^2}k{g^{ - 2}}\]

**Note:**

The escape velocity of smaller planets like mars, is less and there is no atmosphere and bigger planets like Jupiter, Saturn have very large escape velocity hence they have denser atmospheres in these planets.

Recently Updated Pages

To get a maximum current in an external resistance class 1 physics JEE_Main

f a body travels with constant acceleration which of class 1 physics JEE_Main

A hollow sphere of mass M and radius R is rotating class 1 physics JEE_Main

If the beams of electrons and protons move parallel class 1 physics JEE_Main

Two radioactive nuclei P and Q in a given sample decay class 1 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main

Other Pages

Oxidation state of S in H2S2O8 is A 6 B 7 C +8 D 0 class 12 chemistry JEE_Main

What is the volume of water that must be added to a class 11 chemistry JEE_Main

Which of the following sets of displacements might class 11 physics JEE_Main

What is the pH of 001 M solution of HCl a 1 b 10 c class 11 chemistry JEE_Main

1mol of ferric oxalate is oxidized by x mol of MnO4in class 11 chemistry JEE_Main

The mole fraction of the solute in a 1 molal aqueous class 11 chemistry JEE_Main