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 ................

Step 1: Set up the initial configuration.

Four persons $K, L, M, N$ are positioned at the corners of a square of side $d$. Each person moves with a uniform speed $v$ such that their velocity vector is always directed toward the next person.

$$K \to L,\quad L \to M,\quad M \to N,\quad N \to K$$

Step 2: Analyze the relative velocity of approach.

Consider the line joining any two adjacent persons, for example, $K$ and $L$. The velocity of $K$ is directed entirely along the line $KL$. The velocity of $L$ is directed along $LM$, which is perpendicular to $KL$.

$$(\vec{v}_K)_{\text{along } KL} = v$$ $$(\vec{v}_L)_{\text{along } KL} = 0$$

The rate of change of the distance $s$ between $K$ and $L$ depends only on the component of their velocities along the line joining them.

$$\frac{ds}{dt} = -v$$

Step 3: Integrate to find the meeting time.

Since the rate of approach is constant, the distance $s$ at any time $t$ is given by integrating the rate of change from the initial distance $s(0) = d$.

$$s(t) = d - vt$$

The four persons will meet at the center of the square when the distance between them becomes zero ($s = 0$).

$$d - vt = 0$$ $$t = \frac{d}{v}$$

Step 4: Determine the distance covered by each person.

Using the uniform speed $v$ and the total time $t$, the distance covered by each person along their spiral path is calculated.

$$\text{Distance} = v \cdot t = v \cdot \frac{d}{v} = d$$

Answer: The four persons will meet at a time $\dfrac{d}{v}$.