A large number of particles are moving each with speed v having directions of motion randomly distributed. What is the average relative velocity between any two particles averaged over all the pairs?

Options

A. v
B. `(pi//4)v`
C. `(4//pi)v`
D. Zero

Explanation

To find the average relative velocity between any two particles moving with speed \( v \) and having randomly distributed directions, we can follow these steps:

### Step 1: Understand Relative Velocity
The relative velocity \( \mathbf{v_r} \) between two particles can be expressed as:
\[ \mathbf{v_r} = \mathbf{v_1} - \mathbf{v_2} \]
where \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are the velocities of the two particles.

### Step 2: Magnitude of Relative Velocity
If both particles have the same speed \( v \) and the angle between their directions is \( \theta \), the magnitude of the relative velocity can be calculated using the formula:
\[ |\mathbf{v_r}| = |\mathbf{v_1} - \mathbf{v_2}| = \sqrt{v^2 + v^2 - 2v^2 \cos \theta} \]
This simplifies to:
\[ |\mathbf{v_r}| = \sqrt{2v^2(1 - \cos \theta)} = v \sqrt{2(1 - \cos \theta)} \]

### Step 3: Use Trigonometric Identity
Using the trigonometric identity \( 1 - \cos \theta = 2 \sin^2(\theta/2) \), we can rewrite the expression for the magnitude of the relative velocity:
\[ |\mathbf{v_r}| = v \sqrt{2 \cdot 2 \sin^2(\theta/2)} = 2v \sin(\theta/2) \]

### Step 4: Average Over All Angles
To find the average relative velocity, we need to average this expression over all possible angles \( \theta \) from \( 0 \) to \( 2\pi \):
\[ \langle |\mathbf{v_r}| \rangle = \frac{1}{2\pi} \int_0^{2\pi} 2v \sin(\theta/2) d\theta \]

### Step 5: Change of Variable
Let \( u = \theta/2 \), then \( d\theta = 2 du \) and the limits change from \( 0 \) to \( \pi \):
\[ \langle |\mathbf{v_r}| \rangle = \frac{1}{2\pi} \int_0^{\pi} 4v \sin(u) du \]

### Step 6: Evaluate the Integral
The integral of \( \sin(u) \) is:
\[ \int \sin(u) du = -\cos(u) \]
Thus,
\[ \int_0^{\pi} \sin(u) du = [-\cos(u)]_0^{\pi} = -\cos(\pi) - (-\cos(0)) = 1 + 1 = 2 \]

### Step 7: Final Calculation
Substituting back into our average expression:
\[ \langle |\mathbf{v_r}| \rangle = \frac{1}{2\pi} \cdot 4v \cdot 2 = \frac{4v}{\pi} \]

### Conclusion
The average relative velocity between any two particles is:
\[ \langle |\mathbf{v_r}| \rangle = \frac{4v}{\pi} \]