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: sqrt3 `
C. ` sqrt3:1`
D. ` 1: 3`

Explanation

To solve the problem of finding the ratio of the velocities of two particles A and B based on their displacement-time graphs, we can follow these steps:

### Step 1: Understand the relationship between slope and velocity
The slope of a displacement-time graph represents the velocity of the particle. Therefore, if we have two angles, θ_A for particle A and θ_B for particle B, we can express their velocities in terms of the slopes of their respective graphs.

### Step 2: Write the formula for the ratio of velocities
The ratio of the velocities of particles A and B can be expressed as:
\[ \frac{V_A}{V_B} = \frac{\text{slope of A}}{\text{slope of B}} \]

### Step 3: Relate the slope to the angle
The slope of the displacement-time graph can be calculated using the tangent of the angle:
\[ \text{slope} = \tan(\theta) \]
Thus, we can rewrite the ratio of velocities as:
\[ \frac{V_A}{V_B} = \frac{\tan(\theta_A)}{\tan(\theta_B)} \]

### Step 4: Substitute the given angles
We are given:
- θ_A = 30°
- θ_B = 60°

Substituting these values into the equation gives:
\[ \frac{V_A}{V_B} = \frac{\tan(30°)}{\tan(60°)} \]

### Step 5: Calculate the tangent values
Using the known values of tangent:
- \(\tan(30°) = \frac{1}{\sqrt{3}}\)
- \(\tan(60°) = \sqrt{3}\)

Substituting these values into the equation:
\[ \frac{V_A}{V_B} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{3} \]

### Step 6: Express the ratio in simplest form
This can be simplified to:
\[ \frac{V_A}{V_B} = \frac{1}{3} \Rightarrow V_A : V_B = 1 : 3 \]

### Final Answer
Thus, the ratio of the velocities of particles A and B is:
\[ \boxed{1 : 3} \]

---