The displacement-time graph for two particles A and B are straight lines inclined at angles of `30^(@)` and `60^(@)` with the time axis. The ratio of velocities of `V_(A) :V_(B)` is

Options

A. `1:2`
B. `1:sqrt(3)`
C. `sqrt(3):1`
D. `1:3`

Answer

To solve the problem, we need to determine the velocities of two particles A and B based on the angles of their displacement-time graphs.

### Step-by-Step Solution:

1. **Understanding the Graphs**:
- The displacement-time graph for particle A is inclined at an angle of \(30^\circ\) with the time axis.
- The displacement-time graph for particle B is inclined at an angle of \(60^\circ\) with the time axis.

2. **Identifying the Velocity**:
- The velocity of a particle can be calculated from the slope of the displacement-time graph.
- Mathematically, the velocity \(V\) is given by:
\[ V = \frac{\Delta D}{\Delta t} \]
- In the context of the graph, this is equivalent to the slope of the line, which can be expressed in terms of the tangent of the angle of inclination.

3. **Calculating Velocities**:
- For particle A, the velocity \(V_A\) can be expressed as:
\[ V_A = \tan(30^\circ) \]
- For particle B, the velocity \(V_B\) can be expressed as:
\[ V_B = \tan(60^\circ) \]

4. **Finding the Ratio of Velocities**:
- We need to find the ratio of the velocities \(V_A : V_B\):
\[ \frac{V_A}{V_B} = \frac{\tan(30^\circ)}{\tan(60^\circ)} \]

5. **Using Known Values of Tangents**:
- We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \quad \text{and} \quad \tan(60^\circ) = \sqrt{3} \]
- Substituting these values into the ratio:
\[ \frac{V_A}{V_B} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{3} \]

6. **Final Ratio**:
- Therefore, the ratio of the velocities \(V_A : V_B\) is:
\[ V_A : V_B = 1 : 3 \]

### Summary:
The ratio of the velocities of particles A and B is \(1 : 3\).