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
Velocity-addition formula appear there must be inverted here: This matrix rotates the a pure time-axis vector to , and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation. If a fly is moving with four-velocity in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is , Dividing by the time component and substituting for the components of the four-vectors and in terms of the components of the three-vectors and gives the relativistic composition law as The form of the relativistic composition law can be