A car is moving towards east with a speed of 25 kmh^-1. To the driver of the car, a bus appears to move towards north with a speed of $25 \sqrt{3} {kmh}^{- 1}$ . What is the actual velocity of the bus?

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$50 {kmh}^{- 1}, 30^{\circ} E of N$

$50 {kmh}^{- 1}, 30^{\circ} N of E$

$25 {kmh}^{- 1}, 30^{\circ} E of N$

$25 {kmh}^{- 1}, 30^{\circ} N of E$

answer is A.

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Detailed Solution

$\begin{array}{l} {\vec{v}}_{c} = 25 \hat{i}, {\vec{v}}_{b / c} = 25 \sqrt{3} \hat{j} \\ {\vec{V}}_{b / c} = {\vec{V}}_{b} - {\vec{V}}_{c} \Rightarrow {\vec{V}}_{b} = {\vec{V}}_{b / c} + {\vec{V}}_{c} \\ \Rightarrow {\vec{v}}_{b} = 25 \hat{i} + 25 \sqrt{3} \hat{j} \\ |{\vec{v}}_{b}| = \sqrt{25^{2} + (25 \sqrt{3})^{2}} = 50 {kmh}^{- 1} \\ \tan θ = \frac{25}{25 \sqrt{3}} = \frac{1}{\sqrt{3}} \Rightarrow θ = 30^{\circ} \end{array}$