**Relative Motion**

**The frame of Reference:**

Any set of planes or curves, with respect to which position or motion of other points is defined, is known as a Reference Frame. For example, if we consider a moving bus as a frame of reference, the velocities of two persons sitting inside the bus, with respect to each other is zero, but the velocity of these persons with respect to a man walking on the road has some value.

Consider the following figure in which two individuals A and B are sitting inside a bus, which is moving with a velocity V km/hr and a person, C is standing at the bus stop waiting for a bus.

If we ask an individual “A” about what is the velocity of individual “B”? Individual A’s answer will be individual “B” is at rest. But if the same question is asked to individual “C”, his answer will be individual “”B is moving with the velocity V in the positive “X” direction. So it can be observed that before specifying the velocity, we need to specify in which frame we are. In simple terms, we need to define a frame of reference.

Earth is considered as the frame of reference and calculations pertaining to position, or motion is made with the earth as the reference frame. In **n** dimensions, to adequately define a reference frame, n+1 reference points are sufficient. In dynamics, frames of reference used are known as coordinate systems. By defining a reference as the origin and the coordinate axes, the position of a point can be figured out. It is also necessary to define the positive and negative direction of the frame of reference.

The figure represents two frames of reference S and S’. The position of point P can be measured using the three coordinate axes, which intersect at an origin. The coordinates can be given as P (x,y,z) in S and by P (x’,y’,z’) in S’.

**Once we have chosen our reference frame, which is of two types:**

- Inertial Frame of Reference: An inertial frame of reference is a frame where Newton’s law hold true. That means a body will remain in its position in a uniform motion until and unless no other external force is acted upon it. An inertial frame at rest or moves with constant velocity with respect to an assumed inertial reference frame.
- Non-inertial Frame of Reference: A frame, which is accelerated with respect to the assumed inertial frame of reference is a non-inertial frame of reference. Newton’s laws will not hold true in these frames. Newton’s laws can be applied in such frames by using pseudo forces.

Relative motion is the motion of an object with respect to another object, which is at rest or in motion. Suppose a car A is moving with a velocity of \( \overrightarrow{v_a} \) and another car B is moving with a velocity of \( \overrightarrow{v_b} \). To find out velocity of car A with respect to B, we need to find out the relative velocity of A w.r.t B, i.e.,

\( \overrightarrow{v_{a|b}}~=~ \overrightarrow{v_a} ~-~\overrightarrow{v_b}\)