# Relative motion

Relative motion is the study of the motion of an object with regard to some other moving object. Thus, the motion is not calculated with reference to the earth but is the velocity of the object in reference to the other moving object as if it were in a static state. Normally the reference system used is the Cartesian plane (or a three-dimensional Cartesian coordinate system) or a polar coordinate system (because depending on the case it may be more useful). The laws of physics which apply when you are at rest on the earth also apply when you are in any reference frame which is moving at a constant velocity with respect to the earth. The importance of relative motions concerns the correct application of the laws of kinematics and dynamics with respect to the reference system studied.

From the point of view of physics, to say that a body has a certain speed value or is located in a particular position is incomplete information if we do not specify compared to what we measured the speed or that position. All motions are relative motions with respect to a fixed reference system: that is, we must always specify the reference system used to describe the physical phenomena we are studying. Depending on the reference system chosen based on the needs and convenience of the model we want to study, the position and speed change: the motion is therefore always relative. Different observers in different reference systems describe what they see differently.

The velocity of the moving objects with respect to other moving or stationary object is called “relative velocity” and this motion is called “relative motion”. The motion may have a different appearance as viewed from a different reference frame, but this can be explained by including the relative velocity, relative speed, or relative acceleration (which is the change in velocity divided by the change in time) of the reference frame in the description of the motion.

Even the trajectory of a body changes in relative motions and according to the reference system from which its motion is observed. For example, when it snows and we look at the flakes as they fall from the window, we see them fall vertically; however, if we find ourselves traveling by car, the flakes go towards the windscreen with a certain inclination. If a person traveling in a train wagon throws a ball upwards and then picks it up again, for him the ball will have made a simple vertical motion in free fall, but for an observer who is stationary on the platform, the ball will have a parabolic movement because the ball, in addition to moving vertically, also moved horizontally with the same speed as the train.

## Relative motion in one dimension

In one-dimensional motion (in the classical or non-relativistic, or the Newtonian approximation) objects move in a straight line at speeds much less than the speed of light. So there are only two possible cases:

• objects are moving in the same direction;
• objects are moving in the opposite direction.

## Relative motion in two dimensions

The same concept will be applicable in two-dimensional motion. If you have to find the velocity of A with respect to B, assume that B is at rest and give the velocity of B to A in the opposite direction. Let us consider two objects A and B which are moving with velocities $$V_a$$ and $$V_b$$ with respect to some common frame of reference, say, with respect to the ground or the earth. We have to find the velocity of A with respect to B, so assume that B is at rest and give the velocity of B to A in the opposite direction.

$V_{ab} = v_a – v_b$

Similarly, for the velocity of object B with respect to A, assume that A is at rest and give the velocity of A to B in the opposite direction.

$V_{ba} = v_b – v_a$