A lauching vehicle carrying an artificial satellite of mass `m` is set for launch on the surface of the earth of mass `M` and radius `R`. If the satellite intended to move in a circular orbit of radius `7R`, the minimum energy required to be spent by the launching vehicle on the satellite is

Explanation

Here `r=R+h=7R`
Orbital velocity `V_(0)=sqrt((GM)/r)`
`TE` of satellite in its orbit `=PE+KE`
`=-(GMm)/r+1/2mV_(0)^(2)`
`=-(GMm)/r+1/2m((GM)/r)`
`=-(GMm)/(7R)+1/2m((GM)/(7R))=-(GMm)/(14R)`
`TE` of satellite on the earth `=-(GMm)/R`
If the minimum energy to launch the satellite in to its orbit is `E_(min)`, then,
`-(GMm)/R+E_(min)=-(GMm)/(14R)rArr E_(min)=(13GMm)/(14R)`