A satellite of mass m is in an elliptical orbit around the earth of mass `M(M gt gt m)` the speed of the satellite at its nearest point ot the earth (perigee) is `sqrt((6GM)/(5R))` where R= its closest distance to the earth it is desired to transfer this satellite into a circular orbit around the earth of radius equal its largest distance from the earth. Find the increase in its speed to be imparted at the apogee (farthest point on the elliptical orbit).

Answer

`sqrt((GM)/(R))[sqrt((2)/(3))-sqrt((8)/(15))]`

Explanation

Let r= distance of apogee
COAM: `sqrt((6GM)/(5R))R=vr` (i)
COME: `-(GMm)/(R)+(1)/(2)m((6GM)/(5R))`
`=-(GMm)/(r)+(1)/(2)mv^(2)` ..(ii)
`impliesr=(3R)/(2)` and `v=sqrt((8GM)/(15R))`
Orbital speed at r
`=sqrt((GM)/(r))=sqrt((2)/(3)(GM)/(R))`
`therefore `increase in speed
`=sqrt((GM)/(R))[sqrt((2)/(3))-sqrt((8)/(15))]`