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$$\begin{array}{l}\Rightarrow {v}_{C}={v}_{B}+{v}_{CB}\\ \Rightarrow {v}_{CB}={v}_{C}-{v}_{B}\end{array}$$
This is an important relation. This is the working relation for relative motion in one dimension. We shall be using this form of equation most of the time, while working with problems in relative motion. This equation can be used effectively to determine relative velocity of two moving objects with uniform velocities (C and B), when their velocities in Earth’s reference are known. Let us work out an exercise, using new notation and see the ease of working.
Problem : Two cars, initially 100 m distant apart, start moving towards each other with speeds 1 m/s and 2 m/s along a straight road. When would they meet ?
Solution : The relative velocity of two cars (say 1 and 2) is :
$$\begin{array}{l}{v}_{21}={v}_{2}-{v}_{1}\end{array}$$
Let us consider that the direction ${v}_{1}$ is the positive reference direction.
Here, ${v}_{1}$ = 1 m/s and ${v}_{2}$ = -2 m/s. Thus, relative velocity of two cars (of 2 w.r.t 1) is :
$$\begin{array}{l}\Rightarrow {v}_{21}=-2-1=-3\phantom{\rule{2pt}{0ex}}m/s\end{array}$$
This means that car "2" is approaching car "1" with a speed of -3 m/s along the straight road. Similarly, car "1" is approaching car "2" with a speed of 3 m/s along the straight road. Therefore, we can say that two cars are approaching at a speed of 3 m/s. Now, let the two cars meet after time “t” :
$$\begin{array}{l}t=\frac{\mathrm{Displacement}}{\mathrm{Relative\; velocity}}=\frac{100}{3}=33.3\phantom{\rule{2pt}{0ex}}s\end{array}$$
There is slight possibility of misunderstanding or confusion as a result of the order of subscript in the equation. However, if we observe the subscript in the equation, it is easy to formulate a rule as far as writing subscript in the equation for relative motion is concerned. For any two subscripts say “A” and “B”, the relative velocity of “A” (first subscript) with respect to “B” (second subscript) is equal to velocity of “A” (first subscript) subtracted by the velocity of “B” (second subscript) :
$$\begin{array}{l}{v}_{AB}={v}_{A}-{v}_{B}\end{array}$$
and the relative velocity of B (first subscript) with respect to A (second subscript) is equal to velocity of B (first subscript) subtracted by the velocity of A (second subscript):
$$\begin{array}{l}{v}_{BA}={v}_{B}-{v}_{A}\end{array}$$
An inspection of the equation of relative velocity points to an interesting feature of the equation. We need to emphasize that the equation of relative velocity is essentially a vector equation. In one dimensional motion, we have taken the liberty to write them as scalar equation :
$$\begin{array}{l}{v}_{BA}={v}_{B}-{v}_{A}\end{array}$$
Now, the equation comprises of two vector quantities ( ${v}_{B}$ and $-{v}_{A}$ ) on the right hand side of the equation. The vector “ $-{v}_{A}$ ” is actually the negative vector i.e. a vector equal in magnitude, but opposite in direction to “ ${v}_{A}$ ”. Thus, we can evaluate relative velocity as following :
This concept of rendering the reference object stationary is explained in the figure below. In order to determine relative velocity of car "B" with reference to car "A", we apply velocity vector of car "A" to both cars. The relative velocity of car "B" with respect to car "A" is equal to the resultant velocity of car "B".
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