Animated Solution for Physics - Gravitation: The radius in kilometre to which the present radius of Earth ($R=6400\text{ km}$) to be compressed so that the escape velocity is increased 10 times is ......... .

Escaping a Compressed Earth

Imagine standing on the surface of the Earth, looking up at the night sky. To completely break free from our planet's gravitational grip and travel into deep space, a spacecraft must be launched with a very specific minimum speed. This magical speed is known as the escape velocity.

For our current Earth, with a radius of $6400\text{ km}$, this escape velocity is roughly $11.2\text{ km/s}$. But what if we could play god and squeeze the entire Earth into a much smaller sphere without losing any of its mass? How would that affect the escape velocity? Let's dive into the physics of this fascinating thought experiment.

The escape velocity $v_e$ from the surface of a spherical body is derived from the principle of conservation of mechanical energy. To just escape the gravitational field, the total mechanical energy of the object at the surface must be at least zero.

Solving for $v_e$, we get the master equation:

Here, $G$ is the universal gravitational constant, $M$ is the mass of the planet, and $R$ is its radius.

Notice something crucial here: the problem states that the Earth is compressed. This means the mass $M$ remains absolutely constant. Therefore, the escape velocity depends solely on the radius $R$. Specifically, it is inversely proportional to the square root of the radius:

We are given that the new escape velocity ($v_{e2}$) is $10$ times the initial escape velocity ($v_{e1}$).

Using our proportionality relation, we can set up a ratio between the two states:

Substituting $v_{e2}=10v_{e1}$ into the equation:

To get rid of the square root, we simply square both sides of the equation:

Now, we rearrange the formula to solve for the new radius $R_2$:

We know the initial radius of the Earth is $R_1=6400\text{ km}$. Substituting this value in:

A Density Nightmare

Take a moment to appreciate what this means. The entire mass of the Earth—every mountain, ocean, and continent—is now packed into a sphere with a radius of just $64\text{ km}$ (roughly the size of a large city).

Because density is mass divided by volume ($\rho =\frac{M}{\frac{4}{3}\pi R^3}$), decreasing the radius by a factor of $100$ increases the density by a staggering factor of $100^3$, or one million times! While this is a purely theoretical exercise, it beautifully mirrors the real-world physics of collapsing stars forming ultra-dense neutron stars.