Consider a spacecraft in an elliptical orbit around the earth At the lowest point or perigee of its orbit it is `300km` above the earth's surface at the highest point or apogee it is `3000km` above the earth surface
(a) What is the period of the spacecraft's orbit
(b) using conservation of angular momentum find the ratio of spacecraft's speed at perigee to its speed at apogee
(c) Using conservation of energy find the speed at perigee and the speed at apogee
(d) it is derised to have the spacecraft escape from the earth completrly if the spacecraft's rockets are fired at perigee by how much would the speed have to be increased to achieve this? What if the rockets were fired at apogee Which point in the orbit is the most efficient to use ?

Answer

(a) `T = (4pi)/(Rsqrtg)a^(3//2) = 7.16 xx 10^(3)sec`
(b) `(v_(1))/(v_(2)) = (94)/(67) = 1.4`
(c) `V_(p) = 896 xx 10^(2) sqrt((94)/(67 xx 161))m//sec`
`8.35 xx 10^(3)m//s`
`V_(a) = 896 xx 10^(@) sqrt((67)/(94 xx 161)) m//sec = 5.95 xx 10^(3)m//s`
(d) `DeltaV = 14 xx 10^(2) sqrt67 -V_(p) = 3.09 xx 10^(3)` m//s perigee .