Animated Solution for Physics - Gravitation: The kinetic energy needed to project a body of mass $m$ from the earth's surface (radius $R$) to infinity is

Imagine you are standing on the surface of the Earth, holding a small object of mass $m$. The Earth, a massive sphere of mass $M_e$ and radius $R$, exerts a relentless inward pull on this object. This pull creates a 'gravity well'—a deep pit of negative potential energy that binds the object to the planet.

To throw this object so hard that it never comes back, you must provide it with enough kinetic energy to climb completely out of this well. This journey to 'infinity' is the ultimate test of escaping a gravitational field.

To solve this, we turn to one of the most powerful tools in physics: the Conservation of Mechanical Energy. The universe keeps a strict balance sheet. The total energy of the object at the Earth's surface must equal its total energy when it reaches infinity.

At the surface, the object possesses a gravitational potential energy given by:

The negative sign is crucial; it signifies that the object is trapped. To free it, we must supply a positive kinetic energy, $\text{KE}$.

Now, what happens at infinity? For the minimum energy required to escape, the object should just barely reach infinity. This means it arrives there completely exhausted, with a final velocity of zero. Furthermore, at an infinite distance, the gravitational pull is zero, meaning the potential energy is also zero.

Therefore, the total final energy is:

Equating the initial and final energies:

This is the raw escape energy. But we are not done yet.

If you look at the options provided in the question, they are not expressed in terms of the universal gravitational constant $G$ or the Earth's mass $M_e$. Instead, they use the local acceleration due to gravity, $g$.

We must bridge the gap between the cosmic scale ($G,M_e$) and the local scale ($g$). Recall the fundamental relationship for gravity at the surface of a planet:

By rearranging this, we can express the massive term $G{M}_e$ in terms of familiar local variables:

Now, let's substitute this beautiful substitution back into our kinetic energy equation:

One $R$ from the numerator gracefully cancels with the $R$ in the denominator, leaving us with a remarkably simple and elegant result:

This tells us that the energy required to escape the Earth entirely is exactly equal to the work you would do lifting the object to a height equal to the Earth's radius, assuming gravity remained constant at its surface value. Physics is full of such beautiful symmetries!