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 \( v_1 \) and \( v_2 \) that maximizes their relative velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Relative Velocity**:
The relative velocity \( \vec{v}_{\text{rel}} \) of one particle with respect to another is given by the vector difference of their velocities:
\[ \vec{v}_{\text{rel}} = \vec{v}_1 - \vec{v}_2 \]

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

3. **Maximizing the Magnitude**:
To find the maximum value of \( |\vec{v}_{\text{rel}}| \), we need to analyze the expression:
\[ |\vec{v}_{\text{rel}}| = \sqrt{|\vec{v}_1|^2 + |\vec{v}_2|^2 - 2 |\vec{v}_1| |\vec{v}_2| \cos \theta} \]
The term \( -2 |\vec{v}_1| |\vec{v}_2| \cos \theta \) contributes negatively, so to maximize the magnitude, we need to minimize \( \cos \theta \).

4. **Finding the Angle**:
The minimum value of \( \cos \theta \) occurs when \( \theta = 180^\circ \) (or \( \pi \) radians), which corresponds to the two velocities being in opposite directions. In this case:
\[ |\vec{v}_{\text{rel}}| = |\vec{v}_1| + |\vec{v}_2| \]

5. **Conclusion**:
Therefore, the angle between the velocities \( v_1 \) and \( v_2 \) that maximizes their relative velocity is:
\[ \theta = 180^\circ \text{ or } \pi \text{ radians} \]

### Final Answer:
The angle between their velocities is \( 180^\circ \) or \( \pi \) radians.

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