Question

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

• A. $50{\text{km h}}^{-1},{30}^{\circ }$ East of North
• B. $50{\text{km h}}^{-1},{30}^{\circ }$ North of East
• C. $25{\text{km h}}^{-1},{30}^{\circ }$ East of North
• D. $25{\text{km h}}^{-1},{30}^{\circ }$ North of East

Answer 1

The correct option is A $50{\text{km h}}^{-1},{30}^{\circ }$ East of North

Velocity of car,${\overrightarrow{v}}_c=25\hat{i}$
Velocity of bus w.r.t car, ${\overrightarrow{v}}_{b/c}=25\sqrt{3}\hat{j}$
Then, ${\overrightarrow{v}}_{b/c}={\overrightarrow{v}}_b-{\overrightarrow{v}}_c\Rightarrow {\overrightarrow{v}}_b={\overrightarrow{v}}_{b/c}+{\overrightarrow{v}}_c$

$\Rightarrow {\overrightarrow{v}}_b=25\hat{i}+25\sqrt{3}\hat{j}$
$\left|{\overrightarrow{v}}_b\right|=\sqrt{{25}^2+(25\sqrt{3}{)}^2}=50\text{km/h}$

Now,
$\tan \theta =\frac{25}{25\sqrt{3}}=\frac{1}{\sqrt{3}}\Rightarrow \theta ={30}^{\circ }$
Therefore direction is ${30}^{\circ }$ East of North