Past exam of the physics B course of the University of Cambridge / 2021 / Paper 1 / Section B / B6 / f / Solution

At transfer apoapsis, angular momentum conservation givesThe desired circular Kepler orbit has speed $v_c=\sqrt{GM/(3r_0)}=v_0/\sqrt{3}$. Thusof the original fuel supply is used in the second burn. This is about $21.8\%$ of the fuel remaining after the first burn. The total transfer consumes $f_1+f_2\simeq 39.4\%$ of the original supply, leaving about $60.6\%$. The two positive impulse magnitudes are added for the fuel budget, even though their directions at opposite ends of the orbit differ.