What is the relation between the velocity of the same object, as seen from two different frames of reference? That is, how is v_A for the object seen from viewpoint A related to v_B, the velocity of the object seen from viewpoint B?

We can instantly see that r_B, the position vector locating the object from coordinate system B, and r_A, the position vector locating the object from coordinate system A, are related by

If we take the time derivative, we instantly find the answer we are looking for:

If we take the time derivative again and the velocity u of A seen from B is a constant, then we get

which is sometimes called “the Newtonian Principle of Relativity.”

However if reference frame A is accelerating as seen from reference frame B, then since the derivative of u is not zero, but rather something like a_AB, we find