A particle has a velocity u towards east at `t = 0`. Its acceleration is towards west and is constant. Let `x_(A)` and `x_(B)` be the magnitude of displacement in the first 10 seconds and the next 10 seconds:

Options

A. `x _(A) lt x _(B)`
B. `x _(A) =x_(B)`
C. `x_(A) gt x_(B)`
D. the information is insufficient to decide the ralation of `x _(A) and x _(B)`

Answer

To solve the problem, we need to find the displacements \( x_A \) and \( x_B \) for a particle moving with an initial velocity \( u \) towards the east and a constant acceleration \( a \) towards the west.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle starts with an initial velocity \( u \) towards the east.
- The acceleration \( a \) is constant and directed towards the west (opposite to the direction of the initial velocity).

2. **Displacement in the First 10 Seconds**:
- We use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
- For the first 10 seconds (\( t = 10 \) seconds):
\[ x_A = u \cdot 10 + \frac{1}{2} (-a) \cdot (10)^2 \]
- Simplifying this gives:
\[ x_A = 10u - 50a \]
- This is our **Equation 1**.

3. **Displacement in the Next 10 Seconds**:
- For the next 10 seconds, the total time is now 20 seconds.
- Using the same equation of motion:
\[ x_B = u \cdot 20 + \frac{1}{2} (-a) \cdot (20)^2 \]
- Simplifying this gives:
\[ x_B = 20u - 200a \]
- This is our **Equation 2**.

4. **Finding the Displacement for the Next 10 Seconds**:
- The displacement \( x_B \) for the next 10 seconds can also be expressed as:
\[ x_B = x_{total} - x_A \]
- The total displacement after 20 seconds is:
\[ x_{total} = x_B = 20u - 200a \]
- Therefore:
\[ x_B = (20u - 200a) - (10u - 50a) \]
- Simplifying this gives:
\[ x_B = 10u - 150a \]

5. **Comparing \( x_A \) and \( x_B \)**:
- We have:
\[ x_A = 10u - 50a \quad \text{(from Equation 1)} \]
\[ x_B = 10u - 150a \quad \text{(from Equation 3)} \]
- To compare \( x_A \) and \( x_B \):
- Subtract \( x_B \) from \( x_A \):
\[ x_A - x_B = (10u - 50a) - (10u - 150a) = 100a \]
- Since \( a \) is positive (acceleration towards the west), \( x_A > x_B \).

### Conclusion:
Thus, the relationship between the displacements is:
\[ x_A > x_B \]

### Final Answer:
The correct option is that \( x_A \) is greater than \( x_B \).