Escape Velocity and its Dependence on Radius and Density

Escape velocity is defined as the minimum velocity required by an object to escape the gravitational pull of a massive body. Mathematically, it is given by the formula:

v0 = sqrt(2GM/R)

where v0 is the escape velocity, G is the universal gravitational constant, M is the mass of the planet, and R is its radius.

Dependence on Radius

It is clear from the formula that the escape velocity is directly proportional to the square root of the mass of the planet and inversely proportional to the square root of its radius. Therefore, if we keep the mass of the planet constant and increase its radius, the escape velocity will decrease. This can be explained as follows:

As we increase the radius of the planet, the distance between the object and the center of the planet increases. This means that the gravitational force between the object and the planet decreases, since it is inversely proportional to the square of the distance between them. Therefore, the object needs a lower velocity to overcome the weaker gravitational pull and escape from the planet.

Dependence on Density

The density of a planet also affects its escape velocity. If we keep the mass and radius of the planet constant and change its density, the escape velocity will remain the same. This can be explained as follows:

The mass of the planet is directly proportional to its volume, which is given by the formula:

V = (4/3)πR^3

Since density is defined as mass per unit volume, we can write:

ρ = M/V = 3M/(4πR^3)

Therefore, if we increase the density of the planet, its mass will increase proportionately, but its radius will remain the same. This means that the escape velocity will also increase proportionately, since it is directly proportional to the square root of the mass and inversely proportional to the square root of the radius.

Answer to the Question

Now let's apply these concepts to answer the given question. We are given that the density of the planet remains the same, but its radius becomes twice the original value. Therefore, we can write:

R' = 2R

Since the mass of the planet remains the same, we can use the formula for escape velocity to find the new value:

v0' = sqrt(2GM/R')

= sqrt(2GM/(2R))

= sqrt(GM/R)

= sqrt(v0^2)

= v0

Therefore, the escape velocity from a planet having twice the radius but same density will be the same as the original escape velocity, which is given by option (b). However, it is important to note that if the density of the planet had also changed, the escape velocity would have been different, as explained above.