Question:

Six small ants are each at the vertex of regular hexagon of side 60 cm. The 1st sets out towards the 2nd , the 2nd towards the 3rd … and the 6th towards the 1st, with uniform speed of 5 cm s^{-1}. During their motion each of them always heads towards its respective target ant. How much time has elapsed and what distance do the ants cover before they meet?

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Answer:

24 s, 120 cm.

The motion of the ants will be such that they always form a regular hexagon, which rotates and shrinks over time until it vanishes at the centre. While every ant will be moving in a complicated curved path, we can simplify matter by focusing on just the motion directed towards the centre of the hexagon.

At the start, each ant is 60 cm away from the centre of the hexagon. And their component of velocity in the direction towards the centre is 5 cos 60° = =2.5 cm s^{-1}. So it will take them 60 cm ÷ 2.5 cm s^{-1} = 24 s before they meet.

Since they move at a constant speed of 5 cm s^{-1}, each ant would have moved a total distance of 5 cm s-1 × 24 s = 120 cm.

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