Question

How do you calculate escape velocity from the gravitational force?

Answer 1

Definition: The smallest speed at which a moving object, like a rocket, can leave the gravitational field of a celestial body, like the earth, and travel into space.

Step 1:Given that

Consider a planet with a perfect sphere shape, mass$M,$ and radius$R$. Now, imagine projecting a body of mass m from point $A$ on the planet's surface. Below is a picture for better illustration:

1. In the diagram, a line is traced from$O$, the planet's center, to $A\left(OA\right),$ and it is then drawn farther away from the surface. Two additional points are taken as $P$ and $Q$ in that extended line, which is separated from the center$O$by$xanddx$, respectively.
2. Let's now consider the bare minimum velocity be $v_e$ needed for a body to leave the planet.

As a result, Kinetic Energy will be

$\Rightarrow KineticEnergy\left(K.E.\right)=\frac{1}{2}m{v}_e^2$

Step 2: Calculate the work done

The body will be x distance from the planet's center at point P, and the gravitational attraction between the object and the planet will be as follows:

$\Rightarrow \mathrm{F}=\frac{\mathrm{GMm}}{{\mathrm{x}}^2}........\left(equation1\right)$

The labor required to move the body from $PtoQ$ against gravitational pull is

$$\begin{gathered} \Rightarrow \mathrm{dW}=\mathrm{Fdx} \\ \Rightarrow \mathrm{dW}=\frac{\mathrm{GMm}}{{\mathrm{x}}^2}\mathrm{dx} \end{gathered}$$

By integrating the equation for work done within the bounds $\mathrm{x}=\mathrm{R}\mathrm{to}\mathrm{x}=\infty$ , it is now simple to compute the work required to lift a body from the planet's surface to infinity.

$$\begin{gathered} \Rightarrow W={\int }_R^{\infty }dW \\ \Rightarrow W={\int }_R^{\infty }\frac{GMm}{x^2}dx \end{gathered}$$

Step 3: Further integrate the above expression

$$\begin{gathered} \Rightarrow \mathrm{W}=\mathrm{GMm}{\int }_{\mathrm{R}}^{\infty }{\mathrm{x}}^{-2}\mathrm{dx} \\ \Rightarrow \mathrm{W}=\mathrm{GMm}{\left[\frac{{\mathrm{x}}^{-1}}{-1}\right]}_{\mathrm{R}}^{\infty } \\ \Rightarrow \mathrm{W}=-\mathrm{GMm}{\left[\frac{1}{\mathrm{x}}\right]}_{\mathrm{R}}^{\infty } \\ \Rightarrow \mathrm{W}=-\mathrm{GMm}\left[\frac{1}{\infty }-\frac{1}{\mathrm{R}}\right] \end{gathered}$$

As a result, the work done will be:

$\Rightarrow \mathrm{W}=\frac{\mathrm{GMm}}{\mathrm{R}}$

Step 4: Calculate the escape velocity

Now, for a body to leave the planet's surface, its kinetic energy must match the amount of work required to defy gravity as it travels from the surface to infinity. So,

$\Rightarrow \mathrm{K}.\mathrm{E}.=\mathrm{W}$

Put the value of Kinetic energy and work done, and the following equation is obtained:

$\Rightarrow \frac{1}{2}{\mathrm{mv}}_{\mathrm{e}}^2=\frac{\mathrm{GMm}}{\mathrm{R}}$

The escape velocity can be simply derived from this equation and is given by:

$\Rightarrow v_e^2=\frac{2GMm}{Rm}$

Take square root on both sides

$\Rightarrow {\mathrm{v}}_{\mathrm{e}}=\sqrt{\frac{2\mathrm{GM}}{\mathrm{R}}}$

Put the value of $\mathrm{g}=\mathrm{GM}$ in the above equation the value of escape velocity becomes

$\Rightarrow {\mathrm{v}}_{\mathrm{e}}=\sqrt{2\mathrm{gR}}$

According to this equation, the escape velocity is solely influenced by the planet's mass and radius, not by the mass of the body.

Hence, the above equation is the formula for calculating the escape velocity.