The ratio of escape velocity at earth to the escape velocity at a planet whose radius and mean density are twice as the one of the earth is:
Step 1: Given
Given, the escape velocity of Earth is and the escape velocity of the planet is .
Let be the density of Earth and be the density of the planet.
We are given,
Let be the radius of Earth and be the radius of the planet.
We are given,
Step 2: Formulas used
We know that the density of an object is given as,
where is the mass and is the volume.
We know that the volume of a sphere is given as,
where is its radius.
We know that the escape velocity is given as,
where is the mass of the planet, is its radius and is the gravitational constant.
Step 3: Express mass in terms of density
The mass of the earth in terms of its density can be written as,
Similarly, the mass of the planet is,
Step 4: Find the ratio of escape velocities
The ratio of the escape velocity of the earth to the planet is,
Therefore, the ratio of the escape velocity of the earth to the escape speed of the planet is .