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 displacements in the first 10 seconds and the next 10 seconds

Options

A. `x_Altx_B`
B. `x_A=x_B`
C. `x_Agtx_B`
D. the information is insufficient to decide the relation of `x_A with x_B`.

Answer

To solve the problem, we need to analyze the motion of the particle under constant acceleration. The particle starts with an initial velocity \( u \) towards the east and has a constant acceleration towards the west.

### Step-by-step Solution:

1. **Understanding the Initial Conditions**:
- At \( t = 0 \), the particle has an initial velocity \( u \) (towards the east).
- The acceleration \( a \) is constant and directed towards the west (which we can consider as negative).

2. **Displacement in the First 10 Seconds**:
- We can use the equation of motion for displacement under constant acceleration:
\[ x_A = u t + \frac{1}{2} a t^2 \]
- For the first 10 seconds, \( t = 10 \) seconds:
\[ x_A = u \cdot 10 + \frac{1}{2} a \cdot (10)^2 \]
- This simplifies to:
\[ x_A = 10u + 50a \]

3. **Finding the Velocity After 10 Seconds**:
- The velocity after 10 seconds can be calculated using:
\[ v = u + at \]
- Substituting \( t = 10 \):
\[ v = u + 10a \]

4. **Displacement in the Next 10 Seconds**:
- Now, we need to find the displacement from \( t = 10 \) seconds to \( t = 20 \) seconds. The initial velocity for this interval is \( v \) (which we found above):
\[ x_B = v \cdot t + \frac{1}{2} a t^2 \]
- Here, \( t = 10 \) seconds (for the next interval):
\[ x_B = (u + 10a) \cdot 10 + \frac{1}{2} a \cdot (10)^2 \]
- This simplifies to:
\[ x_B = 10(u + 10a) + 50a = 10u + 100a + 50a = 10u + 150a \]

5. **Comparing Displacements**:
- Now we have:
\[ x_A = 10u + 50a \]
\[ x_B = 10u + 150a \]
- To compare \( x_A \) and \( x_B \):
\[ x_B - x_A = (10u + 150a) - (10u + 50a) = 100a \]
- Since \( a \) is negative (acceleration towards the west), \( x_B < x_A \).

### Conclusion:
- Therefore, \( x_A > x_B \).