Animated Solution for Physics - Gravitation: Planet A has mass $M$ and radius $R$. Planet B has half the mass and half the radius of planet A. If the escape velocities from the planets A and B are $v_A$ and $v_B$ respectively, then $\frac{v_A}{v_B}=\frac{n}{4}$. The value of $n$ is

A Tale of Two Planets

Imagine you are standing on the surface of a massive planet, looking up at the stars. To leave this planet forever and never be pulled back by its gravity, you need to be launched with a very specific minimum speed. This magical speed is known as the escape velocity.

In this problem, we are introduced to two different worlds: Planet A and Planet B. Planet A is our reference world, possessing a mass $M$ and a radius $R$. Planet B, on the other hand, is a scaled-down version. It has exactly half the mass, $\frac{M}{2}$, and half the radius, $\frac{R}{2}$, of Planet A.

To understand how hard it is to escape these planets, we need our trusty mathematical tool. The escape velocity $v_e$ from the surface of any spherical body is given by the beautiful equation:

Here, $G$ is the universal gravitational constant, $M$ is the mass of the planet, and $R$ is its radius. Notice how the escape velocity depends on the ratio of the mass to the radius. This is the crucial insight that will crack the problem wide open.

Let's apply our master equation to both planets. For Planet A, the escape velocity $v_A$ is straightforwardly:

Now, let's carefully construct the expression for Planet B. We must substitute its specific mass and radius into the formula:

Look closely at the expression for $v_B$. We have a factor of $2$ dividing the mass in the numerator, and a factor of $2$ dividing the radius in the denominator. Mathematically, these two factors perfectly cancel each other out!

This is a profound physical result. Even though Planet B is smaller and less massive, the ratio of its mass to its radius is exactly the same as Planet A's. Consequently, the escape velocity for Planet B is identical to that of Planet A.

Since $v_A$ and $v_B$ are mathematically identical, their ratio is simply $1$:

The problem states that this ratio is equal to $\frac{n}{4}$. By equating our finding to the given expression, we can easily solve for $n$:

And there we have it! By understanding the proportional relationship within the escape velocity formula, we effortlessly navigated through the problem to find our final answer.