Relative Velocity - Practice Questions & MCQ

• Rate of change in position of one object with respect to another object with time is defined as relative velocity of one object with another.

• Formula-

           Relative  velocity of object A with respect to object B .

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

- Relative velocity of $A$ with respect to $B$ is velocity of $A$ as observed by $B$.
- Case of Relative velocity
1. When $A$ and $B$ are moving along a straight line in the same direction.
$\overrightarrow{V_A}=$ Velocity of object $A$.
$\overrightarrow{V_B}=$ Velocity of object $B$.
Then, relative velocity of A w.r.t B is

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

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

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

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

Relative velocity of $A$ with respect to $B$ is.

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

3. Relative Velocity when bodies moving at an angle $\theta$ to each other
- Relative velocity of a body, A with respected body B

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

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

$V_A=$ velocity of $A$
$V_B=$ velocity of $B$
Where, $\theta=$ angle between $A$ and $B$
- If $\overrightarrow{V_{A B}}$ makes an angle $\beta$ with the direction of $\overrightarrow{V_A}$, 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

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