Three particles A, B and C are situated at the vertices of an equilateral triangle ABC of side d at time `t=0.` Each of the particles moves with constant speed v. A always has its velocity along AB, B along BC and C along CA. At what time will the particles meet each other?

Explanation

To solve the problem of when the three particles A, B, and C will meet, we can follow these steps:

### Step 1: Understand the Motion of the Particles
Each particle moves with a constant speed \( v \) along the edges of the equilateral triangle. Particle A moves towards B, B moves towards C, and C moves towards A.

### Step 2: Analyze the Velocity Components
Since the triangle is equilateral, the angle between the direction of motion of each particle and the line connecting them is \( 60^\circ \).

- The velocity of particle A along AB is \( v \).
- The component of the velocity of particle B towards A (along BA) is \( v \cos(60^\circ) = \frac{v}{2} \).

### Step 3: Determine the Rate of Decrease of Separation
The separation between particles A and B decreases due to both A moving towards B and B moving towards A. The rate of decrease of separation \( d \) between A and B can be calculated as follows:

\[ \text{Rate of decrease of separation} = v + \frac{v}{2} = \frac{3v}{2} \]

### Step 4: Calculate the Time to Meet
The initial distance between particles A and B is \( d \). The time \( t \) taken for the particles to meet can be calculated using the formula:

\[ t = \frac{\text{Distance}}{\text{Rate}} = \frac{d}{\frac{3v}{2}} = \frac{2d}{3v} \]

### Conclusion
Thus, the time at which all three particles will meet is:

\[ t = \frac{2d}{3v} \]