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 velocity and the escape velocity. With a circular orbit, escape velocity is times the orbital speed. Thus the total escape velocity formula_29 when leaving one body orbiting a second and seeking to escape them both is, under simplified assumptions: where formula_31 for circular orbits. The last two columns will depend precisely where in orbit escape velocity is reached, as the orbits are not exactly circular (particularly Mercury and Pluto). Let "G" be the gravitational constant and let "M" be the mass of the earth (or other gravitating body) and "m" be the mass of the escaping body or projectile. At