Relative Velocity
You have no doubt observed how a car that is moving slowly forward appears to be moving backward when you pass it. In general when two observers measure the velocity of a moving body they get different results if one observer is moving relative to the other. The velocity seen by a particular observer is called the velocity relative to that observer, or simply relative velocity.
When two objects A and B are moving with different velocity then the velocity of one object A with respect to another object B is called relative velocity of object A with respect to B.
Hence, relative velocity is defined as the time rate of change of relative position of one object with respect to another.
The concept of relative velocity is analysed as follows:
a. Expression for relative velocity
Consider two objects A and B, moving with uniform velocities $ \displaystyle {{v}_{1}}$ and $ \displaystyle {{v}_{2}}$ along parallel straight tracks in the same direction. Let $ \displaystyle {{x}_{{01}}}$ and $ \displaystyle {{x}_{{02}}}$ be their displacements from the origin at t = 0. If at any time t, $ \displaystyle {{x}_{1}}$ and $ \displaystyle {{x}_{2}}$ are the displacements (distances) of the two objects with respect to the origin of the position axis. then for the object A,
$ \displaystyle {{x}_{1}}={{x}_{{01}}}+{{v}_{1}}t\text{ }…..\text{(1)}$
and for object B,
$ \displaystyle {{x}_{2}}={{x}_{{02}}}+{{v}_{2}}t\text{ }…..\text{(2)}$
Subtracting the above equations we have
$ \displaystyle {{x}_{2}}-{{x}_{1}}=\left( {{{x}_{{02}}}-{{x}_{{01}}}} \right)+\left( {{{v}_{2}}-{{v}_{1}}} \right)t\text{ }…..\text{(3)}$
where, $ \displaystyle {{x}_{{02}}}-{{x}_{{01}}}={{x}_{0}}$, initial displacement of object B w.r.t. to object A at t = 0.
$ \displaystyle {{x}_{2}}-{{x}_{1}}=x$, relative displacement of object B w.r.t. to object A at time t.
Thus, relation 3 can be written as
$ \displaystyle x={{x}_{0}}+\left( {{{v}_{2}}-{{v}_{1}}} \right)t$
or $ \displaystyle x-{{x}_{0}}=\left( {{{v}_{2}}-{{v}_{1}}} \right)t$
$ \displaystyle \frac{{x-{{x}_{0}}}}{t}={{v}_{2}}-{{v}_{1}}\text{ }…..\text{(4)}$
Here, L.H.S. of relation 4 gives the time rate of change of position of object B w.r.t object A i.e., the relative velocity of object B w.r.t. object A.
$ \displaystyle \therefore $ relative velocity of object B w.r.t. object A,
$ \displaystyle {{v}_{{BA}}}={{v}_{2}}-{{v}_{1}}\text{ }…..\text{(5)}$
= velocity of object B – velocity of object A.
b. Position-time graph in relative velocity
From 5 we know that relative velocity $ \displaystyle {{v}_{{BA}}}$ will be zero, negative or positive depending upon the relative magnitude of $ \displaystyle {{v}_{1}}\text{ and }{{v}_{2}}$.
(i) If two objects A and B are moving with same velocity i.e. $ \displaystyle {{v}_{1}}={{v}_{2}}$ or $ \displaystyle \left( {{{v}_{2}}-{{v}_{1}}} \right)=0$ then from 4, $latex \displaystyle x-{{x}_{0}}=0\text{ or }x={{x}_{0}}$ i.e. the two objects will remain always constant distance apart, which is equal to the relative distance between them initially at t = 0. Therefore, their position graph are parallel straight lines as shown in the figure below:
But the graph for relative displacement $ \displaystyle \left( {x-{{x}_{0}}} \right)=x\left( t \right)$ with time $ \displaystyle \left( t \right)$ will be straight line PQ parallel to time axis.
(ii) If $ \displaystyle {{v}_{2}}>{{v}_{1}}\text{ i}\text{.e}\text{. }\left( {{{v}_{2}}-{{v}_{1}}} \right)$ is positive, then from equation 4 we note that $latex \displaystyle \left( {x-{{x}_{0}}} \right)$ is positive. It means the relative separation between two objects will increase by an amount $latex \displaystyle \left( {{{v}_{2}}-{{v}_{1}}} \right)$ after each unit of time. Therefore, their positive-time graph will open out gradually.
(iii) If $ \displaystyle {{v}_{1}}>{{v}_{2}}\text{ i}\text{.e}\text{. }\left( {{{v}_{2}}-{{v}_{1}}} \right)$ is negative, then from equation 4 we note that $ \displaystyle \left( {x-{{x}_{0}}} \right)$ is negative. It means the separation between the two objects will go on decreasing by the amount $ \displaystyle \left( {{{v}_{1}}-{{v}_{2}}} \right)$ after each unit of time. After some time the two objects will meet and then the object B, which was to the right of A will get more and more to the left of A.
The position time graph of this motion will be as soon in figure. The time coordinates corresponding to point of intersection gives their time of meeting and the corresponding position coordinates give the position of meeting.
Determination of relative velocity
The basic rule used for the determination of relative velocity of one object with respect to another is stated as follows:
When two objects A and B are in relative motion the relative velocity of object A with respect to object B can be obtained by imposing equal and opposite velocity of B on both A and B, so that B is brought to rest. The resultant of two velocity is of A gives the relative velocity of object A with respect to object B.
To understand we consider the following examples. Let $ \displaystyle \overrightarrow{{{{v}_{A}}}}$ and $ \displaystyle \overrightarrow{{{{v}_{B}}}}$ be the uniform velocities of the objects A and B, where $ \displaystyle {{v}_{A}}>{{v}_{B}}$.
(i) When the two objects are moving along parallel state line in the same direction i.e. angle between them is $ \displaystyle {{0}^{\circ }}$
The above figure shows the object A and B are moving towards the right. To find the relative velocity of object A w.r.t B, superimpose velocity $ \displaystyle -\overrightarrow{{{{v}_{B}}}}$ on both the objects A and B. Due to which , the velocity of object B becomes zero i.e. the object B is brought to rest and the velocity of object A becomes $ \displaystyle \overrightarrow{{{{v}_{A}}}}+\left( {-\overrightarrow{{{{v}_{B}}}}} \right)$ i.e. $ \displaystyle \overrightarrow{{{{v}_{A}}}}-\overrightarrow{{{{v}_{B}}}}$. Hence, relative velocity of object A w.r.t. B is given by
$ \displaystyle \overrightarrow{{{{v}_{{AB}}}}}=\overrightarrow{{{{v}_{A}}}}-\overrightarrow{{{{v}_{B}}}}\text{ }$
Since, $ \displaystyle {{v}_{{AB}}}$, $ \displaystyle {{{v}_{A}}}$ and $ \displaystyle {{{v}_{B}}}$ all are in the same direction, we can write
$ \displaystyle {{v}_{{AB}}}={{v}_{A}}-{{v}_{B}}$
Thus, if two objects are moving in the same direction, of relative velocity of one object with respect to another is equal to difference in magnitude of two velocity.
(ii) When the two objects are moving along parallel straight lines in opposite directions i.e., angle between them is $ \displaystyle {{180}^{\circ }}$. The figure below shows that the object A is moving towards the right and the object B is moving towards the left. To find the relative velocity of object A w.r.t. B, superimpose velocity $ \displaystyle -\overrightarrow{{{{v}_{B}}}}\text{ }$ on both the objects A and B.
Due to which, the velocity of object B becomes zero i.e., the object B is brought to rest and the velocity of object A becomes $ \displaystyle \overrightarrow{{{{v}_{A}}}}+\left( {-\overrightarrow{{{{v}_{B}}}}} \right)$ i.e. $ \displaystyle \overrightarrow{{{{v}_{A}}}}-\overrightarrow{{{{v}_{B}}}}$. Therefore, the relative velocity of object A w.r.t object B is given by
$ \displaystyle \overrightarrow{{{{v}_{{AB}}}}}=\overrightarrow{{{{v}_{A}}}}-\overrightarrow{{{{v}_{B}}}}$
since, the direction of $ \displaystyle \overrightarrow{{{{v}_{B}}}}\text{ }$ is opposite to that of $ \displaystyle \overrightarrow{{{{v}_{A}}}}$ then magnitude of $ \displaystyle \overrightarrow{{{{v}_{{AB}}}}}$, will be
$ \displaystyle {{v}_{{AB}}}={{v}_{A}}+{{v}_{B}}$
Thus, if two objects are moving in opposite directions, the magnitude of relative velocity of one object with respect to other is equal to the sum of magnitude of their velocity.
(iii) When two objects are moving at an angle
Let $ \displaystyle \theta $ be the angle between the directions of motion of the objects A and B, moving with velocities $ \displaystyle {{v}_{A}}$ and $ \displaystyle {{v}_{B}}$, where $ \displaystyle \overrightarrow{{{{v}_{A}}}}=\overrightarrow{{OQ}}\text{ and }\overrightarrow{{{{v}_{B}}}}=\overrightarrow{{OP}}$. To find the relative velocity of object A w.r.t. B, superimpose velocity $ \displaystyle -\overrightarrow{{{{v}_{B}}}}\left( {=\overrightarrow{{O{P}’}}} \right)$ on both the objects A and B.
Due to which the object B is brought to rest and object A possesses two velocities $ \displaystyle {{v}_{A}}$along OQ and $ \displaystyle {{v}_{B}}$along OP’, inclined at an angle $ \displaystyle \left( {{{{180}}^{\circ }}-\theta } \right)$. The relative velocity of object A w.r.t B is the resultant of velocities $ \displaystyle {{v}_{A}}\text{ and }{{v}_{B}}$ acting at an angle $ \displaystyle \left( {{{{180}}^{\circ }}-\theta } \right)$, which will be represented by the diagonal OR of the parallelogram OQRP’, according to parallelogram law of vectors.
In magnitude, the relative velocity of A w.r.t. B is given by
$ \displaystyle \begin{array}{l}{{v}_{{AB}}}=\sqrt{{v_{A}^{2}+v_{B}^{2}+2{{v}_{A}}{{v}_{B}}\cos \left( {{{{180}}^{\circ }}-\theta } \right)}}\\=\sqrt{{v_{A}^{2}+v_{B}^{2}+2{{v}_{A}}{{v}_{B}}\cos \theta }}\end{array}$
If $ \displaystyle \overrightarrow{{{{v}_{{AB}}}}}$ makes an angle $ \displaystyle \beta $ with the direction of $ \displaystyle \overrightarrow{{{{v}_{A}}}}$ then
$ \displaystyle \begin{array}{l}\tan \beta =\dfrac{{{{v}_{B}}\sin \left( {{{{180}}^{\circ }}-\theta } \right)}}{{{{v}_{A}}+{{v}_{B}}\cos \left( {{{{180}}^{\circ }}-\theta } \right)}}\\=\dfrac{{{{v}_{B}}\sin \theta }}{{{{v}_{A}}+{{v}_{B}}\cos \theta }}\end{array}$
