Animated Solution for Physics - Gravitation: A satellite is revolving in a circular orbit at a height $h$ from the earth surface such that $h<<R$, where $R$ is the radius of the earth. Assuming that the effect of earth's atmosphere can be neglected the minimum increase in the speed required so that the satellite could escape from the gravitational field of earth is

Imagine you are a mission commander at a space agency. You have a satellite peacefully orbiting the Earth at a relatively low altitude, $h$. Suddenly, you get a new directive: the satellite needs to leave Earth's orbit entirely and venture into deep space. How much extra speed do you need to give it? This classic physics problem is a beautiful exercise in understanding orbital mechanics and making smart mathematical approximations.

First, let's look at the satellite's current state. It is revolving in a circular orbit at a height $h$ above the Earth's surface. The distance from the center of the Earth is $r=R+h$, where $R$ is the radius of the Earth.

To maintain this circular orbit, the satellite must be traveling at a specific speed, known as the orbital velocity ($v_o$). The gravitational force provides the necessary centripetal force, leading to the formula:

Here, $G$ is the universal gravitational constant, and $M$ is the mass of the Earth.

Now, the problem gives us a crucial piece of information: $h\ll R$. This means the height of the satellite is negligible compared to the massive radius of the Earth. We can safely approximate $R+h\approx R$. Substituting this into our orbital velocity equation gives:

We also know the relationship between the acceleration due to gravity at the surface ($g$) and the gravitational constant: $g=\frac{GM}{R^2}$, which means $GM=g{R}^2$. Substituting this in, we get a very neat expression for the orbital velocity:

Now, what is our goal? We want the satellite to escape Earth's gravitational field. To do this, it needs to reach the escape velocity ($v_e$) from its current position. The escape velocity from a distance $r$ from the center of a massive body is given by:

For our satellite at height $h$, the distance is $r=R+h$. So, the required escape velocity is:

Once again, we apply our powerful approximation, $R+h\approx R$:

Using the same substitution $GM=g{R}^2$, we find:

Notice a fascinating relationship here: the escape velocity is exactly $\sqrt{2}$ times the orbital velocity for a circular orbit at the same distance!

The question asks for the minimum increase in speed required. This is simply the difference between the speed the satellite needs to escape ($v_e$) and the speed it already has ($v_o$). Let's call this required boost $\Delta v$.

Substitute the elegant expressions we derived:

To make this look cleaner, we can factor out the common term, $\sqrt{gR}$:

And there we have it! By understanding the fundamental formulas for orbital and escape velocities and applying a practical approximation, we've determined the exact speed boost required to send our satellite on an interstellar journey.