Relative Motion in Two Dimension – Page 2 – Rankers Physics
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Relative Motion in Two Dimension

Question 1:

A boat is sailing with a velocity \[\left( 3\hat{i}+4\hat{j} \right)\] with respect to ground and water in river is flowing with a velocity

\[\left( -3\hat{i}-4\hat{j} \right)\] . Relative velocity of the boat with respect to water is :

\[8\hat{j}\]
5āˆš2
\[6\hat{i}+8\hat{j}\]
\[-6\hat{i}-8\hat{j}\]
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Solution:

The relative velocity of the boat with respect to the water is given by subtracting the velocity of the water from the velocity of the boat.

\[
\vec{v}_{\text{bw}} = \vec{v}_{\text{boat}} - \vec{v}_{\text{water}}
\]

Given:
\[
\vec{v}_{\text{boat}} = 3\hat{i} + 4\hat{j}, \quad \vec{v}_{\text{water}} = -3\hat{i} - 4\hat{j}
\]

Now, subtract:

\[
\vec{v}_{\text{bw}} = (3\hat{i} + 4\hat{j}) - (-3\hat{i} - 4\hat{j})
\]
\[
\vec{v}_{\text{bw}} = 3\hat{i} + 4\hat{j} + 3\hat{i} + 4\hat{j}
\]
\[
\vec{v}_{\text{bw}} = 6\hat{i} + 8\hat{j}
\]

Thus, the relative velocity of the boat with respect to the water is:

\[ {6\hat{i} + 8\hat{j}} \]

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Question 2:

Four persons P, Q, R and S of same mass travel with same speed u along a square of side ‘d’ such that each one always faces the other. After what time will they meet each other ?

\[\frac{d}{u}\]
\[\frac{2d}{3u}\]
\[\frac{2d}{u}\]
dāˆš3u
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Solution:

To determine when the four persons \( P, Q, R, \) and \( S \) will meet, consider the following:

Relative Velocity:
- Each person moves with speed \( u \) towards the center of the square.

Configuration:
- As they face each other and move towards the center, their paths converge.

Effective Speed Towards Each Other:
- The effective speed of each person towards the center is \( u \cos 45^\circ = \frac{u}{\sqrt{2}} \) because they move diagonally.

Distance to Center:
- The distance from each person to the center of the square is \( \frac{d}{\sqrt{2}} \).

Time to Meet:
\[
t = \frac{\text{Distance}}{\text{Effective Speed}} = \frac{\frac{d}{\sqrt{2}}}{\frac{u}{\sqrt{2}}} = \frac{d}{u}
\]

Thus, the time after which they will all meet is \( \frac{d}{u} \).

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