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]

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 be the semi-major axis of this new elliptical orbit. Efinal =−GMm2aEinitial =-GMm2r, where r=7000 kmGiven, -0.9GMm2r=−GMm2aa=r0.9=1.11rrmax=2a-r [where rmax is the distance corresponding to aphelion]       =2.22r−r=1.22rRequired greatest height, h=rmax−Re=2140 km