Mass driver constrained by a limited amount of onboard spacecraft power. Thrust and momentum from exhaust, per unit mass expelled, scales up linearly with its velocity (momentum = mv), yet kinetic energy and energy input requirements scale up faster with velocity squared (kinetic energy = mv). Too low exhaust velocity would excessively increase propellant mass needed under the rocket equation, with too high a fraction of energy going into accelerating propellant not used yet. Higher exhaust velocity has both benefit and tradeoff, increasing propellant usage efficiency (more momentum per unit mass of propellant expelled) but decreasing thrust and the current rate of
Escape velocity "r" from the center of mass of the planet, whose mass is "M". Its initial speed is equal to its escape velocity, formula_6. At its final state, "f", it will be an infinite distance away from the planet, and its speed will be negligibly small and assumed to be 0. Kinetic energy "K" and gravitational potential energy "U" are the only types of energy that we will deal with, so by the conservation of energy, "K" = 0 because final velocity is zero, and "U" = 0 because its final distance is infinity, so where μ is the standard gravitational