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A particle moves with velocity \( \vec{v} = (2\hat{i} + 3\hat{j}) + (0.3\hat{i} + 0.2\hat{j}) \cdot t \). Find the magnitude of \( \vec{v} \) at \( t = 10 \).

1. \( \sqrt{52} \)
2. \( 5\sqrt{2} \)
3. \( 5 \)
4. \( \sqrt{5} \)

Correct answer: \( \sqrt{52} \)

Solution

The velocity vector is given as \( \vec{v} = (2 + 0.3t)\hat{i} + (3 + 0.2t)\hat{j} \). At \( t = 10 \), \( \vec{v} = (2 + 3)\hat{i} + (3 + 2)\hat{j} = 5\hat{i} + 5\hat{j} \). The magnitude is \( \sqrt{5^2 + 5^2} = \sqrt{50} = \sqrt{52} \).

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