Question

To a driver going east in a car in the velocity of $40\mathrm{km}/\mathrm{h}$. A bus appears to move towards the north with a velocity of $40\sqrt{3}\mathrm{km}/\mathrm{h}$. what is the actual velocity and direction of motion of the bus?

Answer 1

Step 1: Given information

The velocity of the car = $40\mathrm{km}/\mathrm{h}$

The velocity of the bus with respect to the car = $40\sqrt{3}\mathrm{km}/\mathrm{h}$

Step 2: To find

We have to find the actual velocity and direction of motion of the bus.

Step 3: Finding the velocity of the bus

$\Rightarrow \sqrt{{{(\mathrm{Velocity}\mathrm{of}\mathrm{the}\mathrm{car})}^2+(\mathrm{Velocity}\mathrm{of}\mathrm{the}\mathrm{bus}\mathrm{with}\mathrm{respect}\mathrm{to}\mathrm{the}\mathrm{car})}^2}$

By substituting the values,

$$\begin{gathered} \Rightarrow \sqrt{{\left(40\right)}^2+{\left(40\sqrt{3}\right)}^2} \\ \Rightarrow \sqrt{1600+4800} \\ \Rightarrow \sqrt{6400} \\ \Rightarrow 80\mathrm{km}/\mathrm{h} \end{gathered}$$

Step 4: Finding the direction of the motion of the bus

$$\begin{gathered} \Rightarrow \mathrm{tan\beta }=\frac{40}{40\sqrt{3}} \\ \Rightarrow =\frac{1}{\sqrt{3}} \\ \Rightarrow =30° \end{gathered}$$

Therefore, the actual velocity of the bus is $80\mathrm{km}/\mathrm{h}$and the direction of motion is $30°$.