Question

The minimum velocity of projection of a body to send it to infinity from the surface of a planet is `(1)/(sqrt(6))` times that is required from the surface of the earth. The radius of the planet is `(1)/(36)` times the radius of the earth. The planet is surrounded by an atmosphere which contains monoatomic innert gas `(gamma = 5//3)` of constant density up to a height h(`h lt lt` radius of the planet). Find the velocity of sound on the surface of the planet.

Answer 1

Escape velocity from the surface of the planet
`v_(p)=sqrt(2g_(p)R_(p))`
Given `v_(p)=(v_(e ))/(sqrt(6)) = sqrt((2g_(e )R_(e ))/(6))`
`sqrt((g_(e )R_(e ))/(3)) = sqrt(2g_(p)R_(e )//36) rArr g_(p) =6g_(e )`
Pressure exerted by the atmospheric column of height h on the surface of the planet `P = rho g_(P) h`
Using equation of state `P = (rho RT)/(M)`
Hence speed of the sound `v = sqrt((yRT)/(M)) = sqrt(yg_(p)h) = sqrt(6y g_(e )h) = sqrt(10 g_(e )h)`