Relative velocity in one dimension

The measurements, describing motion, are subject to the state of motion of the frame of reference with respect to which measurements are made. Our day to day perception of motion is generally earth’s view – a view common to all bodies at rest with respect to earth. However, we encounter occasions when there is perceptible change to our earth’s view. One such occasion is traveling on the city trains. We find that it takes lot longer to overtake another train on a parallel track. Also, we see two people talking while driving separate cars in the parallel lane, as if they were stationary to each other!. In terms of kinematics, as a matter of fact, they are actually stationary to each other - even though each of them are in motion with respect to ground.

In this module, we set ourselves to study motion from a perspective other than that of earth. Only condition we subject ourselves is that two references or two observers making the measurements of motion of an object, are moving at constant velocity (We shall learn afterward that two such reference systems moving with constant velocity is known as inertial frames, where Newton’s laws of motion are valid.).

The observers themselves are not accelerated. There is, however, no restriction on the motion of the object itself, which the observers are going to observe from different reference systems. The motion of the object can very well be accelerated. Further, we shall study relative motion for two categories of motion : (i) one dimension (in this module) and (ii) two dimensions (in another module). We shall skip three dimensional motion – though two dimensional study can easily be extended to three dimensional motion as well.

Relative motion in one dimension

We start here with relative motion in one dimension. It means that the individual motions of the object and observers are along a straight line with only two possible directions of motion.

Position of the point object

We consider two observers “A” and “B”. The observer “A” is at rest with earth, whereas observer “B” moves with a velocity $v_{BA}$ with respect to the observer “A”. The two observers watch the motion of the point like object “C”. The motions of “B” and “C” are along the same straight line.

The position of the object “C” as measured by the two observers “A” and “B” are $x_{CA}$ and $x_{CB}$ as shown in the figure. The observers are represented by their respective frame of reference in the figure.

Here,

$$\begin{array}{l}{x}_{CA}=x_{BA}+x_{CB}\end{array}$$

Velocity of the point object

We can obtain velocity of the object by differentiating its position with respect to time. As the measurements of position in two references are different, it is expected that velocities in two references are different, because one observer is at rest, whereas other observer is moving with constant velocity.