During a lunar mission, it is necessary to make a midcourse correction of 22.6 m/s in the speed of the spacecraft, which is moving at 388 m/s. The exhaust speed of the rocket engine is 1230 m/s. What fraction of the initial mass of the spacecraft must be discarded as exhaust?

It is important to note that the negative sign in v_rel highlights the fact that the relative velocity of the exhaust gases is in the opposite direction relative to the direction of the vector highlighting the change in rockets velocity with magnitude ?v.
Therefore, the fraction of initial mass of the spacecraft that must be discarded as exhaust is 0.98.