Question

On an open ground, a motorist follows a track that turns to his left by an angle of ${60}^o$ after every $500m$. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.

Answer 1

The path followed by the motorist is a regular hexagon with side 500 m, as shown in the given figure
Let the motorist start from point A.
The motorist takes the third turn at D.
Magnitude of displacement $=AD=BC+EF=500+500=1000m$
Total path length $=AB+BC+CD=500+500+500=1500m$
The motorist takes the sixth turn at point A, which is the starting point.
Magnitude of displacement $=0$
Total path length $=6AB$
$=6\times 500=3000m$
The motorist takes the eight turn at point C
The magnitude of displacement $=AC$
From triangle law, the magnitude of displacement is $866.03m$ at an angle of $30$ with AC.
It means AC makes an angle 30 with the initial direction. Total path length $=8\times 500=4000m$.