An artificial satellite of mass `m` of a planet of mass `M`, revolves in a circular orbit whose radius is a times the radius `R` of the planet. In the process of motion, the satellite experiences a slight resistance due to cosmic dust. Assuming resistance force on the satellite depends on velocity as `F= av^(2)` where a is constant, calculate the time the satellite will stay in orbit before it falls onto the planet's surface.

Explanation

Air resistance `F=-av^(2)` where orbital velocity `v=sqrt((GM)/(r))`
`r=` the distance of the satellite from planet's centre `impliesF=-(Gma)/(r)`
the work by the resistance force dW=Fdx=Fvdt`=(Gma)/(r)sqrt((GM)/(r))dt=((GM)^(3//2))/(r^(3//2))dt` ..(i)
The loss of energy of the satellite `=dEtherefore(dE)/(dr)=(d)/(dr)[-(GMm)/(2r)]=(GMm)/(2r^(2))impliesdE=(GMm)/(2r^(2))dr` ...(ii)
since dE=`-dW` (work enerrgy theorem) `-(GMm)/(2r^(2))dr=((GM)^(3//2))/(r^(3//2))dt`
`impliest=-(m)/(2asqrt(GM))int_(nR)^(R)(dr)/(sqrt(r))=(msqrt(R)(sqrt(n)-1))/(asqrt(GM))=(sqrt(n)-1)(m)/(asqrt(gR))`