Imagine that you are in a spacecraft orbiting around the earth in a circle of radius `7000 km` (from the centre of the earth). If you decrease the magnitude of mechanical energy of the spacecraft `-` earth system by `10%` by firing the rockets, then what is the greatest height you can take your spacecraft above the surface of the earth? [`R_(e) = 6400 km`]

Options

A. `6400 km`
B. `540km`
C. `2140 km`
D. `3000 km`

Answer

C

Explanation

As we decrease the magnitude of mechanical energy of the spacecraft earth system, it means we are increasing the energy of the spacecraft earth system as the total energy of the bounded system is negative.
As we change the energy, the circular orbit of the spacecraft will become elliptical. Let a bet the semi major axis of the new elliptical orbit.
`E_("final")=(GM_(m))/(2a)`
`E_("initial")=(GM_(m))/(2r)`, where` r=7000 km`
`=(GM_(m))/(2r)=-(GM_(m))/(2a)`
`a=r/0.9=1.11r`
`r_("max")=2a` [where `r_("max")]` is the distance corresponding to aphelion]
`=2.2r-r=1.22r`
Required greatest height `h-r_("max")-R_(e)=2140km`