Deriving (Literally) the Acceleration of the Center of Mass

Starting with the equation expressing the velocity of the center of mass in terms of the masses and velocities of the particles in the system::

Take the derivative of both sides of this equation with respect to time, t:

Now, all of the d(v)/dt's, (the rate the velocity changes), are accelerations, a. Therefore:

which is the desired result.

last update February 5, 2010 by JL Stanbrough