Question

what is the meaning of escape velocity?​

Answer 1

HEY MATE

HERE IS YOUR ANSWER

The velocity required to over come the earths gravity and not fall back on the surface on the earth is known as escape velocity

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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 $\sf g=\frac{GM}{R^2}$

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

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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!