# Relative velocity in one dimension  (Page 3/4)

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$\begin{array}{l}⇒{v}_{C}={v}_{B}+{v}_{CB}\\ ⇒{v}_{CB}={v}_{C}-{v}_{B}\end{array}$

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 :

$\begin{array}{l}⇒{v}_{21}=-2-1=-3\phantom{\rule{2pt}{0ex}}m/s\end{array}$

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” :

$\begin{array}{l}t=\frac{\mathrm{Displacement}}{\mathrm{Relative velocity}}=\frac{100}{3}=33.3\phantom{\rule{2pt}{0ex}}s\end{array}$

## Order of subscript

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):

$\begin{array}{l}{v}_{BA}={v}_{B}-{v}_{A}\end{array}$

## Evaluating relative velocity by making reference object stationary

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 :

• Apply velocity of the reference object (say object "A") to both objects and render the reference object at rest.
• The resultant velocity of the other object ("B") is equal to relative velocity of "B" with respect to "A".

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".

#### Questions & Answers

List the application of projectile
Luther Reply
How can we take advantage of our knowledge about motion?
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pls explain what is dimension of 1in length and -1 in time ,what's is there difference between them
Mercy Reply
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show that 1w= 10^7ergs^-1
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Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
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Amar
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isaac Reply
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Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
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Mustapha Reply
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Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
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x=5.8-3.22 x=2.58
sajjad

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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