Question

The speed is needed for a gas molecule to escape from a planet is ___​

Answer 1

To find:

Speed of gas molecules needed to escape from a planet.

Calculation:

Let mass of planet be M , mass of gas be m , radius of planet be r and Universal Gravitational Constant be G ;

So, the gas molecules need to have such a kinetic energy such that it can escape earth's Gravitational field and theoretically reach infinity.

$\rm{ \therefore \: KE1 + PE1 = KE2 + PE2}$

$\rm{ = > \dfrac{1}{2} m {(v_{esc} )}^{2} - \dfrac{GMm}{r} = 0 + 0}$

$\rm{ = > \dfrac{1}{2} m {(v_{esc} )}^{2} - \dfrac{GMm}{r} = 0}$

$\rm{ = > \dfrac{1}{2} m {(v_{esc} )}^{2} = \dfrac{GMm}{r} }$

$\rm{ = > \dfrac{1}{2} {(v_{esc} )}^{2} = \dfrac{GM}{r} }$

$\rm{ = > {(v_{esc} )}^{2} = \dfrac{2GM}{r} }$

$\boxed{ \rm{ = > v_{esc} = \sqrt{\dfrac{2GM}{r}} }}$

So, for any gas to escape a planet , this Velocity is required.

• Note that escape velocity doesn't depend on mass of gas.

• For example , in case of Earth the value comes as 11.2 km/sec.

Hope It Helps.