3.2 Relative velocity in two dimensions

The concept of relative motion in two or three dimensions is exactly same as discussed for the case of one dimension. The motion of an object is observed in two reference systems as before – the earth and a reference system, which moves with constant velocity with respect to earth. The only difference here is that the motion of the reference system and the object ,being observed, can take place in two dimensions. The condition that observations be carried out in inertial frames is still a requirement to the scope of our study of relative motion in two dimensions.

As a matter of fact, theoretical development of the equation of relative velocity is so much alike with one dimensional case that the treatment in this module may appear repetition of the text of earlier module. However, application of relative velocity concept in two dimensions is different in content and details, requiring a separate module to study the topic.

Relative motion in two dimensions

The important aspect of relative motion in two dimensions is that we can not denote vector attributes of motion like position, velocity and acceleration as signed scalars as in the case of one dimension. These attributes can now have any direction in two dimensional plane (say “xy” plane) and as such they should be denoted with either vector notations or component scalars with unit vectors.

Position of the point object

We consider two observers A and B. The observer “A” is at rest with respect to earth, whereas observer “B” moves with a constant velocity with respect to the observer on earth i.e. “A”. The two observers watch the motion of the point like object “C”. The motions of “B” and “C” are as shown along dotted curves in the figure below. Note that the path of observer "B" is a straight line as it is moving with constant velocity. However, there is no such restriction on the motion of object C, which can be accelerated as well.

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

Here,

$$\begin{array}{l}{\mathbf{r}}_{CA}={\mathbf{r}}_{BA}+{\mathbf{r}}_{CB}\end{array}$$

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,

$$\begin{array}{l}{\mathbf{v}}_{CA}=\frac{đ{\mathbf{r}}_{CA}}{đt}\end{array}$$

and

$$\begin{array}{l}{\mathbf{v}}_{CB}=\frac{đ{\mathbf{r}}_{CB}}{đt}\end{array}$$

The velocities of the moving object “C” ( ${\mathbf{v}}_{CA}$ and ${\mathbf{v}}_{CB}$ ) as measured in two reference systems are shown in the figure. Since the figure is drawn from the perspective of “A” i.e. the observer on the ground, the velocity ${\mathbf{v}}_{CA}$ of the object "C" with respect to "A" is tangent to the curved path.

Now, we can obtain relation between these two velocities, using the relation ${\mathbf{r}}_{CA}={\mathbf{r}}_{BA}+{\mathbf{r}}_{CB}$ and differentiating the terms of the equation with respect to time as :