Relative Velocity - Definition, Formula, FAQs

To look outside the window of a moving train is to see that another stationary train appears to move backwards. How does a stationary train appear to move? Behind it lies a very crucial concept of relative motion, which will help us explain why objects seem to move differently in different frames. Suppose you are driving a car and you overtake the other car from behind. What happens is that the driver from the car behind you sees the car coming in the backward direction and eventually goes back. But the person standing on the ground doesn’t see it as the car is moving backwards, although the driver behind sees it that way.

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This Story also Contains

1. What Is Relative Velocity?
2. Solved Example Based on Relative Velocity
3. Summary

In this article, we will cover the concept of relative velocity This concept falls under the broader category of kinematics which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of six questions have been asked on this concept. And for NEET two questions were asked from this concept.

What Is Relative Velocity?

The rate of change in the position of one object concerning another object with time is defined as the relative velocity of one object with another.

Mathematically, relative velocity can be represented as:

Relative velocity of object A concerning object B.

$ \vec{V}_{A B}=\vec{V}_A-\vec{V}_B$

The relative velocity of A concerning B is the velocity of A as observed by B.

Case of Relative Velocity

let's start with some important case

• When A and B are moving along a straight line in the same direction.

$ \begin{aligned}
& \overrightarrow{V_A}=\text { Velocity of object } A . \\
& \overrightarrow{V_B}=\text { Velocity of object } B \text {. }
\end{aligned}$

Then, the relative velocity of A w.r.t B is:

$
\vec{V}_{A B}=\vec{V}_A-\vec{V}_B
$
$\vec{V}_{A B}, \vec{V}_A, \vec{V}_B$ all are in the same direction. (If $\vec{V}_A>\vec{V}_{B \text { ) }}$

And Relative velocity of B w.r.t A is

$\begin{aligned}
\vec{V}_{B A} & =\vec{V}_B-\vec{V}_A \\
\& \vec{V}_{A B} & =-\vec{V}_{B A}
\end{aligned}$

• When A & B are moving along with straight line in the opposite direction.

The relative velocity of A concerning B is.

$\begin{aligned}
& \vec{V}_{A B}=\vec{V}_A-\vec{V}_B \\
& V_{A B}=V_A+V_B
\end{aligned}$

• Relative Velocity when bodies moving at an angle to each other

Relative velocity of a body, A with respected body B

$
\begin{aligned}
V_{A B} & =\sqrt{V_A^2+V_B^2+2 V_A V_B \cos (180-\theta)} \\
& =\sqrt{V_A^2+V_B^2-2 V_A V_B \cos (\theta)}
\end{aligned}
$

Where,
$V_A=$ velocity of $A$
$V_B=$ velocity of $B$
$\theta=$ angle between $A$ and $B$

• $\text { If } \overrightarrow{V_{A B}} \text { makes an angle } \beta \text { with the direction of } \overrightarrow{V_A} \text {, then }$

$ \begin{aligned}
& \tan \beta=\frac{V_B \sin (180-\theta)}{V_A+V_B \cos (180-\theta)} \\
& =\frac{V_B \cdot \sin \theta}{V_A-V_B \cos \theta}
\end{aligned}$

• If two bodies are moving at right angles to each other

Relative Velocity of A with respect to B is

$V_{A B}=\sqrt{V_A^2+V_B^2}$

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Solved Example Based on Relative Velocity

Example 1: A train is moving at 50 km/h and a man is running at 20 km/h in a direction opposite to the direction of the train. The relative velocity of the train (in km/h ) concerning man is

1) 70

2)30

3)60

4)40

Solution:

When A & B are moving along with straight line in the opposite direction.

$
\begin{aligned}
& \vec{V}_A=\text { Velocity of object } \mathrm{A} . \\
& \vec{V}_B=\text { Velocity of object } \mathrm{B} \text {. }
\end{aligned}
$

The relative velocity of $A$ concerning $B$ is.
$
\vec{V}_{A B}=\vec{V}_A-\left(-\vec{V}_B\right)
$
$-\vec{V}_B$ is because of the opposite direction.
$
\vec{V}_{A B}=\vec{V}_A+\left(\vec{V}_B\right)
$

As per question
$
\overrightarrow{V_R}=V_T+V_M
$

= ( 50 + 20 ) km/h

= 70 km/h

Hence, the answer is option (1).

$V_{A B}=\sqrt{V_A^2+V_B^2+2 V_A V_B \cos (180-\theta)}$

Example 2: Person A is moving along east and B is moving along north. The relative velocity of A concerning B is :$(\left.V_A=10 \mathrm{~m} / \mathrm{s}, V_B=10 \sqrt{3} \mathrm{~m} / \mathrm{s}\right)$

1) 20 m/s along north

2) 20 m/s along south

3) 20 m/s along 120⁰ with east

4) None of the above

Solution:

$\begin{aligned}
& \overrightarrow{V_A}=\text { Velocity of object } A \text {. } \\
& \overrightarrow{V_B}=\text { Velocity of object } B .
\end{aligned}$

$\text { The relative velocity of } \mathrm{B} \text { wrt to } \mathrm{A} \text { is } V_{BA}$

$\begin{aligned}
& \overrightarrow{V_{B / A}}=\overrightarrow{V_B} \cdot \overrightarrow{V_A}=10 \sqrt{3} \hat{j}-10 \hat{\imath} \\
& \left|\overrightarrow{V_{B / A}}\right|=\sqrt{100+300}=20 \mathrm{~m} / \mathrm{s} \\
& \tan \theta=\frac{V_B}{V_A}=\frac{10 \sqrt{3}}{10}=\sqrt{3} \\
& \text { So } \theta=60^{\circ} \text { from west } \\
& \text { i.e } \theta=120^{\circ} \text { from east }
\end{aligned}$

Hence, the answer is option (3).

Example 3: Rain is falling vertically downward with a speed of 4 km/h. A girl moves on a straight road with a velocity of 3km/h. The apparent speed (in m/s) of rain with respect to the girl is:

1) 5

2) 4

3) 3

4) 7

Solution :

$
\overrightarrow{V_{R / G}}=\overrightarrow{V_R}-\overrightarrow{V_G}
$
$\mathrm{V}_{\mathrm{R}}=$ Velocity of rain wrt ground, $\mathrm{V}_{\mathrm{G}}=$ Velocity of girl wrt ground
$
\left|\overrightarrow{V_{R / G}}\right|=\sqrt{V_R^2+V_G^2}=\sqrt{3^2+4^2}=5 \mathrm{~km} / \mathrm{h}
$

Hence, the answer is option (1).

Example 4: A particle A is moving along north with a speed of 3 m/s and another particle B is moving with a velocity of 4 m/s at 60^o with north, then the velocity of B as seen by A is

1) $5 \mathrm{~m} / \mathrm{s}$
2) $3.5 \mathrm{~m} / \mathrm{s}$
3) $\sqrt{13} \mathrm{~m} / \mathrm{s}$
4) $\sqrt{15} \mathrm{~m} / \mathrm{s}$

Solution:

The relative velocity of a body, A with respected body B when the two bodies moving at an angle $\theta$ is:

$\begin{aligned}
& V_{A B}=\sqrt{V_A^2+V_B^2+2 V_A V_B \cos (180-\theta)} \\
& =\sqrt{V_A^2+V_B^2-2 V_A V_B \cos (\theta)} \\
& \left|\vec{V}_{A B}\right|=\sqrt{V_A^2+V_B^2-2 V_A V_B \cdot \cos \theta} \\
& =\sqrt{3^2+4^2-2 \times 3 \times 4 \times \frac{1}{2}}=\sqrt{13} \mathrm{~m} / \mathrm{s} \\
&
\end{aligned}$

Hence, the answer is option (3).

Example 5: A particle A is moving along the x-axis and a particle B is moving along the y-axis. The speed of A is 6m/s and that of B is 8 m/s, then the velocity (in m/s) of A concerning B is:

1) 10

2) 12

3) 8

4) 14

Solution:

Relative Velocity of $\mathrm{A}$ with respect to $\mathrm{B}$ is
$
\begin{aligned}
& V_{A B}=\sqrt{V_A^2+V_B^2} \\
& V_{A B}=\sqrt{V_A^2+V_B^2}=\sqrt{8^2+6^2}=10 \mathrm{~m} / \mathrm{s}
\end{aligned}
$

Hence, the answer is option (1).

Summary

Relative motion describes how one object's movement is observed from another object's reference point. It's crucial in fields like aviation, maritime, and driving, where understanding relative velocity ensures safe and efficient operations. For example, drivers must consider the relative speeds of other vehicles to change lanes safely. This concept also applies to boat-river and rain-man problems in kinematics.