A particle moves with constant speed 𝑣 along a regular hexagon ABCDEF in the same order, then the magnitude of the average velocity for its motion from A to
 

Let side length of the hexagon is 𝑎
 

A) Magnitude of displacement from A to F is 𝑎
Time taken to reach from A to F is 𝑡 = 5𝑎/𝑣
$\Rightarrow {\mathrm{v}}_{\text{avg }}=\frac{\mathrm{a}}{\mathrm{t}}=\frac{\mathrm{a}}{5\mathrm{a}/\mathrm{v}}=\mathrm{v}/5$
B) Magnitude of displacement from A to D is 2𝑎
Time taken to reach from A to D is 3𝑎/𝑣
$\Rightarrow {\mathrm{v}}_{\text{avg }}=\frac{2\mathrm{a}}{3\mathrm{a}/\mathrm{v}}=\frac{2\mathrm{v}}{3}$
C) Magnitude of displacement from A to C is √3𝑎
Time taken to reach from A to C is 2𝑎/𝑣
$\Rightarrow {\mathrm{v}}_{\mathrm{avg}}=\frac{\sqrt{3}\mathrm{a}}{2\mathrm{a}/\mathrm{v}}=\frac{\sqrt{3}\mathrm{v}}{2}$
D) Magnitude of displacement from A to B is 𝑎
Time taken to reach from A to B is 𝑎/𝑣
$\Rightarrow {\mathrm{v}}_{\mathrm{avg}}=\frac{\mathrm{a}}{\mathrm{a}/\mathrm{v}}=\mathrm{v}$