3.4 Resultant motion

For the observer on the ground, the person walks longer distance. As the time interval involved in tworeference systems are same, the speed (v’) with which person appears to move in ground’s reference is greater.

Identifying velocities

Identification of velocities and assigning them the right subscript are critical to analyze situation. Weconsider the earlier example of a person moving on a moving train. Let us refer person with “A” and train with “B”. Then, the meaning of different notations are :

• ${\mathbf{v}}_{AB}$ : velocity of person (“A”) with respect to train (“B”)
• ${\mathbf{v}}_A$ : velocity of the person ("A") with respect to ground
• ${\mathbf{v}}_B$ : velocity of train ("B") with respect to ground

From the explanation given earlier, we can say that :

$$\begin{array}{l}{v}_{AB}=v\\ v_A=v\text{'}\\ v_B=u\end{array}$$

The second and third assignments are evident as they are measurements with respect to ground. What isnotable here is that the velocity of the person ("A") with respect to train ("B") is actually equal to the velocity of theperson on the ground "v" (when train is stationary with respect to ground). This velocity is his inherent ability to walk on earth, but as explained, it is alsohis relative velocity with respect to a medium, when the medium (train) is moving. After all, why should biological capacity of the person change on the train?

Similar is the situation in the case of a boat sailing in a stream of certain width. The inherent mechanical speedin the still water is equal to the relative velocity of the boat with respect to the moving stream. Thus, we conclude that relative velocity of a body with respect to a moving medium is equal to its velocity in the still medium.

$$\begin{array}{l}{\mathbf{v}}_{AB}=\mathrm{velocity\; of\; the\; body\; ("A")\; with\; respect\; to\; moving\; medium\; ("B")}\\ =\mathrm{velocity\; of\; the\; body\; ("A")\; in\; the\; still\; medium}\end{array}$$

This is the key aspect of learning for the motion in a medium. Rest is simply assigning values into the equation ofrelative velocity and evaluating the same as before.

$$\begin{array}{l}\Rightarrow {\mathbf{v}}_{AB}={\mathbf{v}}_A-{\mathbf{v}}_B\end{array}$$

A closer look at the equation of relative velocity says it all. Remember, we interpreted relative velocity as the velocity of the body when reference body is stationary. Extending the interpretation to the case in hand, we can say that relative velocity of the body "A" is the velocity when the reference medium is stationary i.e. still. Thus, there is no contradiction in the equivalence of two meanings expressed in the equation above. Let us now check our understanding and try to identify velocities in one of the questions considered earlier.

Problem 1 : An aircraft flies with a wind velocity of 200 km/hr blowing from south. If the relative velocity of aircraft with respect to wind is 1000 km/hr, then find the direction in which aircraft should fly such that it reaches a destination in north – east direction.

Can we answer the questions raised earlier - What does this relative velocity of aircraft with respect to wind mean? Yes, the answer is that the relative velocity of aircraft with respect to wind is same as velocity of aircraft in still air.

Resultant velocity and relative velocity are equivalent concepts

The concepts of resultant and relative velocities are equivalent. Rearranging the equation of relative velocity, we have :

$$\begin{array}{l}\Rightarrow {\mathbf{v}}_A={\mathbf{v}}_{AB}+{\mathbf{v}}_B\end{array}$$

This means that resultant velocity of person ( ${\mathbf{v}}_A$ ) is equal to the resultant of velocity of body in still medium ( ${\mathbf{v}}_{\mathrm{AB}}$ ) and velocity of the medium ( ${\mathbf{v}}_B$ ).

Example