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
Four-velocity the 3d position vector, written as a column vector The components of the velocity formula_2 (tangent to the curve) at any point on the world line are Each component is simply written In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions "x"("τ"), where μ is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates. The zeroth component is defined as the time coordinate multiplied by "c", Each function depends on one parameter "τ" called