Question

Escape velocity?derive expressions for this and suitable graph

Answer 1

Answer:

$\boxed{\huge{ \blue{ Escape \: Velocity}}}$

It is the velocity with which an object projected from the surface of a planet goes out of the gravitational pull of the planet (i.e reaches infinity ).

It is the minimum velocity with which infinity can be reached.

Derivation:

We have to apply Conservation Of Mechanical Energy such that final Kinetic and Potential energy will be zero.

∴ KE1 + PE1 = KE2 + PE2

$= > \dfrac{1}{2} m {v}^{2} - \dfrac{GmM}{ r} =0 + 0$

$= > \dfrac{1}{2} m {v}^{2} = \dfrac{GMm}{ r }$

$= > v = \sqrt{ \dfrac{2GM}{r} }$

So Escape velocity is :

$\boxed{ \large{ \blue{ v_{esc.} = \sqrt{ \dfrac{2GM}{r} }}}}$

This Equation gives us an important fact :

• Escape Velocity is not dependent on mass of object.
• It solely depends on mass of planet and radius of planet.

Considering density as constant , we get :

$\: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \huge{ \bold{ \blue{v_{esc} \: \propto \: r}}}}$

On this basis , see the graph attached ,

on x axis (radius ) and on y axis(v esc.)