Two particles are moving with velocities `v_(1) and v_2` . Their relative velocity is the maximum, when the angle between their velocities is

Options

A. zero
B. `pi//4`
C. `pi//2`
D. `pi`

Answer

To solve the problem of finding the angle between the velocities of two particles when their relative velocity is maximum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Relative Velocity**:
The relative velocity \( \vec{v}_{rel} \) of two particles moving with velocities \( \vec{v}_1 \) and \( \vec{v}_2 \) is given by:
\[ \vec{v}_{rel} = \vec{v}_1 - \vec{v}_2 \]

2. **Magnitude of Relative Velocity**:
The magnitude of the relative velocity can be calculated using the formula:
\[ |\vec{v}_{rel}| = |\vec{v}_1 - \vec{v}_2| = \sqrt{|\vec{v}_1|^2 + |\vec{v}_2|^2 - 2|\vec{v}_1||\vec{v}_2|\cos(\alpha)} \]
where \( \alpha \) is the angle between the two velocity vectors.

3. **Maximizing the Magnitude**:
To find the angle \( \alpha \) that maximizes the relative velocity, we need to maximize the expression:
\[ |\vec{v}_{rel}| = \sqrt{|\vec{v}_1|^2 + |\vec{v}_2|^2 - 2|\vec{v}_1||\vec{v}_2|\cos(\alpha)} \]

4. **Setting Up the Condition for Maximum**:
The expression inside the square root will be maximized when \( \cos(\alpha) \) is minimized. The minimum value of \( \cos(\alpha) \) is -1, which occurs when \( \alpha = 180^\circ \).

5. **Conclusion**:
Therefore, the angle \( \alpha \) that maximizes the relative velocity of the two particles is:
\[ \alpha = 180^\circ \]

### Final Answer:
The angle between the velocities \( v_1 \) and \( v_2 \) for maximum relative velocity is \( 180^\circ \).

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