Relative Motion: Suppose two cars are moving at the same velocity. Even though they are moving with respect to an observer on the road, they are not moving with respect to each other. A passenger in one car will see the other car at the same distance. Thus, with respect to time, the other car is not having any velocity. We say that relative motion of one car with respect to other car is zero. We take up this problem in a general way.
Supposing an axes system is moving with respect to other axes system, what is the relationship between velocity in two system?
X-Y is a fixed reference frame and x-y is a moving reference frame. To begin with, consider the x-y axes only translate with respect to X-Y, but do not rotate. If A is any particle. The position vector of A as measured relative to the frame x-y is , where subscript A/B means "A relative to B" or "A with respect to B". The position of A with respect to X-Y,
rA = rB r A/B
Thus, the absolute velocity of a particle is the vector sum of the velocity of a particle with respect to a translating frame of reference and the velocity of the frame.
The absolute acceleration of a particle is the sum of the acceleration of the particle with respect to the translating frame and the acceleration of the frame.
Note that have zero derivatives with respect to time as these are constant vectors. Their direction and magnitude both remain constant with respect to time. They only translate.
A translating reference frame which has no acceleration is known as inertial frame.