An artificial satellite (mass m) of a planet (mass M) revolves in a circular orbit whose radius is n times the radius R of thhe planet in the process of motion the satellite experiences a slight resistance due to cosmic dust. Assuming the force of resistance on satellite to depend on velocity as `F=av^(2)` where 'a' is a constant caculate how long the satellite will stay in the space 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))`