A body starts from rest and moves with constant acceleration for t s. It travels a distance `x_1` in first half of time and `x_2` in next half of time, then

Options

A. `x_2 = 3x_1`
B. `x_2 = x_1`
C. `x_2 = 4x_1`
D. `x_2 = 2x_1`

Explanation

To solve the problem, we need to find the relationship between the distances \( x_1 \) and \( x_2 \) traveled by a body moving with constant acceleration over two halves of a total time \( T \).

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The body starts from rest, which means the initial velocity \( u = 0 \).
- The total time of motion is \( T \).
- The first half of the time is \( \frac{T}{2} \) and the second half is also \( \frac{T}{2} \).

2. **Distance in the First Half of Time**:
- Using the equation of motion:
\[ x_1 = ut + \frac{1}{2} a t^2 \]
- Since \( u = 0 \), we have:
\[ x_1 = \frac{1}{2} a \left(\frac{T}{2}\right)^2 = \frac{1}{2} a \frac{T^2}{4} = \frac{a T^2}{8} \]

3. **Total Distance in the Entire Time**:
- For the total time \( T \):
\[ x_{total} = ut + \frac{1}{2} a T^2 \]
- Again, since \( u = 0 \):
\[ x_{total} = \frac{1}{2} a T^2 \]

4. **Distance in the Second Half of Time**:
- The distance traveled in the second half of the time \( x_2 \) can be found by subtracting \( x_1 \) from the total distance:
\[ x_2 = x_{total} - x_1 \]
- Substituting the values we found:
\[ x_2 = \frac{1}{2} a T^2 - \frac{a T^2}{8} \]
- Finding a common denominator:
\[ x_2 = \frac{4a T^2}{8} - \frac{a T^2}{8} = \frac{3a T^2}{8} \]

5. **Finding the Relation Between \( x_1 \) and \( x_2 \)**:
- Now we have:
\[ x_1 = \frac{a T^2}{8} \]
\[ x_2 = \frac{3a T^2}{8} \]
- To find the ratio \( \frac{x_1}{x_2} \):
\[ \frac{x_1}{x_2} = \frac{\frac{a T^2}{8}}{\frac{3a T^2}{8}} = \frac{1}{3} \]
- Therefore, we can conclude that:
\[ x_2 = 3 x_1 \]

### Final Result:
The relationship between the distances \( x_1 \) and \( x_2 \) is:
\[ x_2 = 3 x_1 \]