Question

using the law of conservation of energy obtain the expression for the escape velocity​

Answer 1

Answer:

this is the answer u meed

Answer 2

Answer:

The required expression for the escape velocity is

$\boxed{\red{\sf\:v_{esc}\:=\:\sqrt{\dfrac{2\:G\:M}{R}}}}$

Explanation:

We have to use the law of conservation of energy and obtain the expression for the escape velocity.

As the escape velocity term is related to air and atmosphere, we shouldn't consider the effects of air and other forces.

Let an object be of mass $\sf\:m$ is thrown vertically upwards from the surface of the earth.

Let the initial velocity of the object be the escape velocity i. e. $\sf\:v_{esc}$.

Case I:

The object is on the surface of the earth.

The total energy of the object is given by,

$\sf\:E_1\:=\:Kinetic\:energy\:+\:Potential\:energy\\\\\\\implies\sf\:E_1\:=\:\dfrac{1}{2}\:m\:v^2_{esc}\:+\:\bigg(\:-\:\dfrac{G\:m\:M}{R}\:\bigg)\:\:\:-\:-\:[\:Formulae\:]\\\\\\\sf\:Where\:,\\\\\\\longrightarrow\sf\:G\:=\:Universal\:Gravitational\:constant\\\\\\\longrightarrow\sf\:M\:=\:Mass\:of\:the\:earth\\\\\\\longrightarrow\sf\:m\:=\:Mass\:of\:the\:object\\\\\\\longrightarrow\sf\:R\:=\:Radius\:of\:the\:earth\\\\\\\therefore\:\boxed{\red{\sf\:E_1\:=\:\dfrac{1}{2}\:m\:v^2_{esc}\:+\:\bigg(\:-\:\dfrac{G\:m\:M}{R}\:\bigg)}}$

Case II:

The object is moving to the infinity and comes to rest there.

The total energy of the object is given by,

$\sf\:E_2\:=\:Kinetic\:energy\:+\:Potential\:energy\\\\\\\implies\sf\:E_2\:=\:\dfrac{1}{2}\:m\:(\:0\:)^2\:+\:\bigg(\:\-\:\dfrac{G\:m\:M}{\infty}\:\bigg)\\\\\\\implies\sf\:E_2\:=\:0\:+\:0\\\\\\\therefore\:\boxed{\red{\sf\:E_2\:=\:0}}$

Now,

According to the law of conservation of energy,

$\pink{\sf\:E_1\:=\:E_2}\\\\\\\implies\sf\:\dfrac{1}{2}\:m\:v^2_{esc}\:-\:\dfrac{G\:m\:M}{R}\:=\:0\\\\\\\implies\sf\:\dfrac{1}{2}\:v^2_{esc}\:-\:\dfrac{G\:M}{R}\:=\:0\:\:\:-\:-\:[\:Mass\:of\:object\:is\:not\:constant\:]\\\\\\\implies\sf\:\dfrac{1}{2}\:v^2_{esc}\:=\:\dfrac{G\:M}{R}\\\\\\\implies\sf\:v^2_{esc}\:=\:\dfrac{2\:G\:M}{R}\\\\\\\implies\boxed{\red{\sf\:v_{esc}\:=\:\sqrt{\dfrac{2\:G\:M}{R}}}}$

This is the required expression for the escape velocity.