Question

best answer will Mark as brainlist​

Answer 1

In physics, escape velocity is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a massive body, that is, to achieve an infinite distance from it. Escape velocity is a function of the mass of the body and distance to the center of mass of the body.

Answer 2

✫ Escape Velocity ✫

• The minimum speed required for an object to escape the gravitational field of a massive body is known as the escape velocity for that massive body.

• Escape Velocity is independent of the mass of the escaping object. It only depends on the Massive Astronomical Body.

• For example, the Escape Velocity for Earth is about 11.2 km/s. This means that if any object wants to escape the gravitational field of the Earth, it must be travelling at at least 11.2 kilometres per second.

• Thus, any person, a spacecraft, a ball, anything! If it is moving faster than 11.2 km/s, it can escape the gravity of Earth.

• However, the Escape Velocity of two different planets is different. For example, the Escape Velocity of Jupiter is about 59.5 km/sec. Thus, any thing wanting to escape the gravity of Jupiter must travel a lot faster than it needs to do on Earth.

The formula for Escape Velocity is:-

$\Large \boxed{\sf v_{esc}=\sqrt{\frac{2GM}{R}}}$

where

• G = Universal Gravitational Constant
• M = Mass of planet/astronomical body
• R = Radius of planet/astronomical body

Derivation of Escape Velocity:-

Let us consider a planet of Mass M and Radius R.

• An object of mass m wants to escape the gravity of this planet. Let us assume that it needs a velocity $\sf v_{esc}$ to be able to do so.

• By Energy Conservation, the sum of Potential and Kinetic Energies should be conserved.

• So let's consider the initial and final positions. Initially it is at the surface of the planet. There it's gravitational potential energy is $\sf -\frac{GMm}{R}$ and it is given a Kinetic Energy of $\sf\frac{1}{2}mv_{esc}^2$.

• [Gravitational Potential Energy is defined to be zero when the separation between objects is infinite. So, at a finite separation, like here it is R, the gravitational potential energy is negative. This is because gravity is an attractive force, so it is energetically favourable to be at a closer distance than a farther distance. So, gravitational potential energy at finite separation is lesser (and hence negative) than potential energy at infinite separation (which is defined as 0)]

• In the final position, it is at a very large distance from the planet. So, potential energy becomes $\sf\frac{GMm}{\infty} = 0$. And also let's assume that all the kinetic energy was used up in reaching this gravity free state. So, we can now get the Escape Velocity:

$\sf\displaystyle \textsf{K+U on surface = K+U very far away} \\\\\\ \implies K_i+U_i = K_f+U_f \\\\\\ \implies \frac{1}{2}mv_{esc}^2-\frac{GMm}{R}=0+0 \\\\\\ \implies \frac{1}{2}\cancel{m}v_{esc}^2=\frac{GM\cancel{m}}{R}\\\\\\ \implies v_{esc}^2=\frac{2GM}{R}\\\\\\ \implies v_{esc}=\sqrt{\frac{2GM}{R}}$

We can also write it in form of $\sf g=\frac{GM}{R^2}$

$\sf\displaystyle v_{esc}=\sqrt{\frac{2GM}{R}}\\\\\\ \implies v_{esc}=\sqrt{2\times\frac{GM}{R^2}\times R} \\\\\\ \implies v_{esc}=\sqrt{2gR} \\\\\\ \Large \boxed{\boxed{\sf v_{esc}=\sqrt{\frac{2GM}{R}}=\sqrt{2gR}}}$

$\rule{300}{1}$

✫ Extra Info ✫

The answer has concluded. Read on for interesting extra info!

1) Escape Velocities of Objects in Solar System

• By putting the values of Mass M and Radius R, we can calculate the escape velocities of different planets. Here's the data:

$\begin{tabular}{|c|l|}\cline{1-2}\sf \textbf{Object} & \sf\textbf{Escape Velocity (km/s)} \\\cline{1-2} \sf Mercury & \sf 4.3 \\ \sf Venus & \sf 10.4 \\\sf Earth & \sf 11.2 \\ \sf Moon & \sf 2.4 \\\sf Mars & \sf 5.0 \\\sf Jupiter & \sf 59.5\\\sf Saturn & \sf 35.5 \\\sf Uranus & \sf 21.3 \\\sf Neptune & \sf 23.5 \\\sf Pluto & \sf 1.3 \\\cline{1-2}\end{tabular}$

Jupiter has the highest Escape Velocity, a whopping 59.5 km/s!!

2) Black Holes and Schwarzschild Radius

• A black hole is an object with such a strong gravitational field whose escape velocity is greater than or equal to the speed of light!

• And since nothing can travel faster than the speed of light in vacuum, anything that goes inside a black hole can never come out!

• We can calculate the critical radius beyond which if the object gets smaller it will become a Black Hole. This critical radius is called the Schwarzschild Radius.

Let c = velocity of light in vacuum. Then,

$\sf\displaystyle c = v_{esc} = \sqrt{\frac{2GM}{R}} \\\\\\ \implies c^2=\frac{2GM}{R_S} \\\\\\ \implies \boxed{R_S = \frac{2GM}{c^2}}$

The Schwarzschild Radius of the Earth is around 8.9 mm!!! Thus, if the Earth would be compressed into a radius smaller than 8.9 millimetres, it would become a Black Hole!

• We live in a strange, but wonderful Universe!