Question

An arc of radius $6\text{m}$ subtends an angle ${120}^{\circ }$ at the centre. If an ant travels on the arc from one end to another end in $4\text{sec}$, what will be the magnitude of average velocity of the ant ?

• A. $2\text{m/s}$
• B. $\frac{\sqrt{3}}{2}\text{m/s}$
• C. $3\text{m/s}$
• D. $\frac{3\sqrt{3}}{2}\text{m/s}$

Answer 1

The correct option is D $\frac{3\sqrt{3}}{2}\text{m/s}$

${\text{Velocity}}_{\text{avg}}=\frac{\text{Displacement}}{\text{Time taken}}$
From the diagram,

The displacement of ant is $AB$.
Now from $\Delta AOB$, draw the perpendicular bisector $OD$ on $AB$
$\Rightarrow AB=OA\sin {60}^{\circ }+OB\sin {60}^{\circ }$
$\because OA=OB=r$
$\Rightarrow AB=2OA\sin {60}^{\circ }$
$\Rightarrow AB=2r\sin {60}^{\circ }=2\times 6\times \frac{\sqrt{3}}{2}=6\sqrt{3}\text{m}$
Time taken to complete journey is $\Delta t=4\text{s}$
Hence,
${\text{Velocity}}_{\text{avg}}=\frac{6\sqrt{3}\text{m}}{4\text{s}}$
$\therefore {\text{Velocity}}_{\text{avg}}=\frac{3\sqrt{3}}{2}\text{m/s}$