Question

A boat is moving with velocity \[3\widehat{i}+4\widehat{j}\] with respect to the ground. The water in the river is moving with velocity $-3\widehat{i}-4\widehat{j}$ with respect to the ground. The relative velocity of the boat with respect to the water is
$\begin{align}
  & \text{A}\text{. 8}\widehat{j} \\
 & \text{B}\text{. }-6\widehat{i}-8\widehat{j} \\
 & \text{C}\text{. }6\widehat{i}+8\widehat{j} \\
 & \text{D}\text{. }6\widehat{i} \\
\end{align}$

Answer 1

Hint: Relative velocity or relative speed means how fast an object is moving with respect to another. The relative velocity between two objects is given by the difference between the velocities of the two objects.

Formula used:
If A and B are two objects moving with speed $\overrightarrow{{{v}_{A}}}\text{ and }\overrightarrow{{{v}_{B}}}$ with respect to the ground then.
The relative velocity of object A with respect to object B is
$\overrightarrow{{{v}_{AB}}}=\overrightarrow{{{v}_{A}}}-\overrightarrow{{{v}_{B}}}$ , and
The relative velocity of object B with respect to object A is
$\overrightarrow{{{v}_{BA}}}=\overrightarrow{{{v}_{B}}}-\overrightarrow{{{v}_{A}}}$

Complete answer:
According to the question the boat is moving with velocity \[3\widehat{i}+4\widehat{j}\]with respect to the ground. i.e.
${{\overrightarrow{v}}_{boat}}=3\widehat{i}+4\widehat{j}$ and
The water in the river is moving with velocity $-3\widehat{i}-4\widehat{j}$with respect to the ground. i.e.
${{\overrightarrow{v}}_{water}}=-3\widehat{i}-4\widehat{j}$
Then the relative velocity of the boat with respect to the water is given by,
${{\overrightarrow{V}}_{boat-water}}={{\overrightarrow{v}}_{boat}}-{{\overrightarrow{v}}_{water}}=3\widehat{i}+4\widehat{j}-\left( -3\widehat{i}-4\widehat{j} \right)=6\widehat{i}+8\widehat{j}$

So, the correct answer is “Option C”.

Additional Information:
The relative velocity of an object A with respect to object B, when both are in motion, is the rate of change of position of object A with respect to object B. And is given by the differences between the velocities of the two objects.
From above we know that
The relative velocity of object A with respect to object B is
$\overrightarrow{{{v}_{AB}}}=\overrightarrow{{{v}_{A}}}-\overrightarrow{{{v}_{B}}}$ , and
The relative velocity of object B with respect to object A is
$\overrightarrow{{{v}_{BA}}}=\overrightarrow{{{v}_{B}}}-\overrightarrow{{{v}_{A}}}$
Clearly $\overrightarrow{{{v}_{AB}}}=-\overrightarrow{{{v}_{BA}}}$ and $\overrightarrow{\left| {{v}_{AB}} \right|}=\overrightarrow{\left| {{v}_{BA}} \right|}$
Now the relative velocity of object A $={{\overrightarrow{v}}_{A}}+\left( -{{\overrightarrow{v}}_{B}} \right)$
i.e. The relative velocity of object A$=\text{ Velocity vector of A + Negative velocity vector of B}$.
Hence the relative velocity of object A with respect to object B is equal to the vector addition of velocity vector of A and the negative velocity vector of B.

Note:
During the calculation of relative velocity the direction of velocity should be taken into consideration. Also the addition and subtraction should be done vertically. Two objects moving opposite to each other will have more relative velocity than two objects moving in the same direction. Here direction plays a major role so the direction should be calculated in every case.