Torque the vector cross product, p is the particle's linear momentum, and r is the displacement vector from the origin (the origin is assumed to be a fixed location anywhere in space). The time-derivative of this is: This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definition of force formula_17 (whether or not mass is constant) and the definition of velocity formula_18 The cross product of momentum formula_20 with its associated velocity formula_21 is zero because velocity and momentum are parallel, so the second term vanishes. By definition, torque τ
Mass driver then use the mass driver to accelerate pieces of matter of almost any sort, boosting itself in the opposite direction. At the smallest scale of reaction mass, this type of drive is called an ion drive. No absolute theoretical limit is known for the size, acceleration or muzzle energy of linear motors. However, practical engineering constraints apply for such as the power-to-mass ratio, waste heat dissipation, and the energy intake able to be supplied and handled. Exhaust velocity is best neither too low nor too high. There is a mission-dependent limited optimal exhaust velocity and specific impulse for any thruster