Estimate whether it takes more energy to get a satellite upto `1600 km` above the earth than to put in orbit there. Earth's radius is `6400 km`. Does your answer remain same for height `3200 km` or for height `4800 km`?

Answer

`=1.5`

Explanation

Energy required to raise a satellite to height `h`,
`E_(1)=-(GMm)/((R+h))-(-(GMm)/R)`
`=GMm(1/R-1/(R+h))=(GMmh)/(R(R+h))`
Energy required to put the satellite in the orbit at that the height `=KE` of the satellite
`E_(2)=1/2mv^(2)=1/2m((GM)/r)=(GMm)/(2r)=(GMm)/(2(R+h))`
`(E_(1))/(E_(2))=(2(R+h))/(R(R+h))=(2h)/R=(2xx(1600km))/((6400km))=1/2=0.5`
Clearly the answer depends on height if h=3200 km
`(E_(1))/(E_(2))=(2h)/R=(2xx(3200km))/((6400km))=1.10`
For height `h=4800 km`
`(E_(1))/(E_(2))=(2xx(4800km))/((6400km))=3/2=1.5`