The most efficient way to transfer a spacecraft from an initial circular orbit at r1 to a larger circular orbit at r2 is to insert it into an intermediate elliptical orbit with radius r1 at perigee and r2 at apogee. the following equation relates the semi-major axis a, the total energy of the system e and the potential energy u(r) = −gmm/r ≡ −k/r for an elliptical orbit of the spacecraft of mass m about the earth of mass m:

r_1 + r_2 = 2a = \frac{k}{|e|}

a. derive the relation between the velocity v and the radius r for a circular orbit.

b. determine the velocity increase required to inject the spacecraft into the elliptical orbit as specified by r1 and r2. let v_1 be the velocity in the initial circular orbit and v_p be the velocity at perigee after the first boost so that $\delta$v = v_p - v_1 .

c. determine the velocity increment required to insert the spacecraft into the second circular orbit when it reaches apogee at r = r2. in this case let v_2 be the velocity in the final orbit and v_a be the velocity at apogee so that $\delta$v = v_2 - v_a .