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
Angular velocity two-dimensional case above, one may define the angular velocity vector as: where "θ" is the angle between r and v. In terms of the cross product, this is: From this, one can recover the tangential velocity as: If a point rotates with angular velocity formula_28 in a coordinate frame formula_29 which itself rotates with an angular velocity formula_30 with respect to an external frame formula_31, we can define formula_32 to be the composite angular velocity vector of the point with respect to formula_31. This operation coincides with usual addition of vectors, and it gives angular velocity the algebraic structure of