Animated Solution for Physics - Gravitation: The escape velocity of a body depends upon mass as

Imagine you are standing on the surface of a planet, holding a ball. If you throw it upwards, it comes back down due to gravity. If you throw it harder, it goes higher before returning. But what if you could throw it so incredibly fast that it never comes back? It just keeps going, escaping the planet's gravitational grip forever. This magical minimum speed is known as the escape velocity.

To understand what this speed depends on, we need to look at the mathematics of gravity. By using the principle of conservation of mechanical energy, we equate the total energy of the object at the planet's surface to its total energy at infinity (which is zero).

When we solve for the velocity, we get the famous equation for escape velocity:

Alternatively, using the relation $g=\frac{GM}{R^2}$, we can also write it as:

Now, let's play a game of observation. Look closely at the variables inside the square root.

We have $G$, the universal gravitational constant. We have $M$, the mass of the planet. And we have $R$, the radius of the planet.

But wait, where is $m$, the mass of the object we are throwing?

It's not there! The mass of the object completely cancels out during the derivation. This is a profound realization. The universe doesn't care if you are trying to throw a tiny grain of sand, a bowling ball, or a massive interstellar spaceship. The speed required to escape the planet's gravity is exactly the same for all of them.

Because the escape velocity is completely independent of the object's mass $m$, we can express this mathematical relationship by saying that $v_e$ is proportional to $m^0$.

Why? Because any non-zero number raised to the power of zero is exactly $1$. Multiplying by $1$ changes nothing, perfectly representing the concept of independence. Therefore, the correct option is indeed $m^0$.