Four persons K,L,M,N are initially at the four corners of a square of side d. Each person now moves with a uniform speed v in such a way that K always moves directly towards L, L directly towards M, M directly towards N, and N directly towards K. The four persons will meet at a time............. .

Explanation

It is obvious from considerations of symmetry that at any moment of time the persons will be at the corners of a square whose side gradually decreases (Fig.4.53) and so they will finally meet at the centre of the square O. The speed of each person along the line joining his initial position and O will be `v cos 45^(@) = v//sqrt2`. And as each person has to walk a distance `d cos 45^(@) = d//sqrt2` to reach the centre, the four persons will meet at the centre of the square O after time
`t (d//sqrt2)/(v//sqrt2) = (d)/(v)`