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 center.
`impliesF=-(Gma)/(r )`
The work done by the resistance force,
`=dW=F dx=[Fv.dt]=[(GMz)/(r )sqrt((GM)/(r ))dt]`
`=[((GM)^(3//2)a)/(r^(3//2))dt]`............`(1)`
`implies` The loss of energy of the satellite `=dE`
`:. (dE)/(dr)=(d)/(dr)[-(GMm)/(2r )]=(GMm)/(2r^(2))`
`impliesdE=(GMm)/(2r^(2))dr`............`(2)`
Since `dE=-dW` (work energy theorem)
`-(GMm)/(2r^(2))dr=((GM)^(3//2)a)/(r^(3//2))dt`
`impliest=-(m)/(2asqrt(GM))int_(nR)^(R )(dr)/(sqrt(r ))`
`impliest=(msqrt(R )(sqrt(n)-1))/(asqrt(GM))`
`=(sqrt(n)-)(m)/(asqrt(gR))`