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# An artificial satellite (of mass m) of the earth (radius R and mass M) moves in an orbit whose radius in n times the radius of the earth. Assuming resistance to the motion to be proportional to e square velocity, that is, F=av^2 , find how long the satellite will take to fall onn to the earth.

25 Points
11 years ago

initial radiusof the orbit of satellite is nR

here the gravitation pull of the earth provides the necessary centripital force

hence GMm/r2 =mv2/r  where r is the radius of the orbit of satellite at any time and v is the velocity.

hence v=√GM/r

it is given that resistance to motion of satellite issgiven by F=av2

so F=mdv/dt

hence av2=mdv/dt

integrating with limits of t varying from o to T and that of v varying from √GM/nR to √GM/R

m(√nR/GM -√R/GM)=aT

T=m/a*√R/GM*(√n-1)

Good Luck