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  • What is the escape velocity of an object? Derive an expression for escape velocity.

What is the escape velocity of an object? Derive an expression for escape velocity.

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Escape velocity is defined as the minimum velocity an object must have in order to escape from the planet's gravitational pull.

  •  Escape velocity ( in terms of the radius of the earth)

To escape a body from earth's surface means to displace it from the surface of the earth to infinity.

The work done to displace a body from the surface of the earth (r = R) to infinity ( r =\infty  ) is 

W=\int_{R}^{\infty}\frac{GMm}{x^2}dx=\frac{GMm}{R}

So if we provide kinetic energy equal to W to the body at the surface of the earth then it will be able to escape from the earth's gravitational pull.

So KE=\frac{GMm}{R}

And Kinetic energy can be written as KE= \frac{1}{2}mV_e^2

Where V_e is the required escape velocity

By comparing we get

\frac{1}{2}mV_e^2=\frac{GMm}{R}\\ \Rightarrow V_e=\sqrt{\frac{2GM}{R}}

Using GM=gR^2

We get V_{e}=\sqrt{2gR}

V_{e} \rightarrow Escape velocity

R \rightarrowRadius of earth

And using g=\frac{4}{3}\pi \rho \, GR

V_{e}=R\sqrt{\frac{8}{3}\pi G\rho }

For the earth

 V_{e}=11.2Km/s

Posted by

avinash.dongre

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