A boat has velocity \( (3\hat{i} + 4\hat{j}) \) m/s with respect to ground. River water flows with velocity \( (-3\hat{i} - 4\hat{j}) \) m/s. What is the boat's velocity relative to water?

Relative Velocity Problem

Question: A boat has velocity \( (3\hat{i} + 4\hat{j}) \) m/s with respect to ground. River water flows with velocity \( (-3\hat{i} - 4\hat{j}) \) m/s. What is the boat's velocity relative to water?

Options:

• (1) 8
• (2) \( 5\sqrt{2} \)
• (3) \( 6\hat{i} + 8\hat{j} \)
• (4) \( -6\hat{i} - 8\hat{j} \)

Solution

Formula:
\( \vec{v}_{\text{boat/water}} = \vec{v}_{\text{boat/ground}} - \vec{v}_{\text{water/ground}} \)

Calculation:
\( (3\hat{i} + 4\hat{j}) - (-3\hat{i} - 4\hat{j}) \)
\( = (3+3)\hat{i} + (4+4)\hat{j} \)
\( = 6\hat{i} + 8\hat{j} \)

Answer: Option (3) \( \boxed{6\hat{i} + 8\hat{j}} \)