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
Escape velocity alternative expression for the escape velocity formula_6 particularly useful at the surface on the body is: where "r" is the distance between the center of the body and the point at which escape velocity is being calculated and "g" is the gravitational acceleration at that distance (i.e., the surface gravity). For a body with a spherically-symmetric distribution of mass, the escape velocity formula_6 from the surface is proportional to the radius assuming constant density, and proportional to the square root of the average density ρ. where formula_14 The escape velocity "relative to the surface" of a rotating body depends on