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 step by step, we will analyze the motion of the body in two halves of the time interval \( t \).

### Step 1: Understanding the motion
The body starts from rest and moves with constant acceleration \( a \) for a total time \( t \). The distance traveled in the first half of the time \( (t/2) \) is \( x_1 \) and in the second half of the time \( (t/2) \) is \( x_2 \).

### Step 2: Calculate the velocity at the end of the first half
Using the first equation of motion:
\[ V_1 = U + a \cdot t_1 \]
where \( U = 0 \) (initial velocity), \( t_1 = \frac{t}{2} \).
Thus,
\[ V_1 = 0 + a \cdot \frac{t}{2} = \frac{a t}{2} \]

### Step 3: Calculate the distance \( x_1 \) for the first half
Using the third equation of motion:
\[ x_1 = U \cdot t_1 + \frac{1}{2} a t_1^2 \]
Substituting \( U = 0 \) and \( t_1 = \frac{t}{2} \):
\[ x_1 = 0 + \frac{1}{2} a \left(\frac{t}{2}\right)^2 = \frac{1}{2} a \cdot \frac{t^2}{4} = \frac{a t^2}{8} \]

### Step 4: Calculate the velocity at the end of the total time
Using the first equation of motion again for the total time \( t \):
\[ V = U + a \cdot t = 0 + a \cdot t = a t \]

### Step 5: Calculate the distance \( x_2 \) for the second half
Now, we need to calculate the distance traveled from point B to point C (the second half):
Using the third equation of motion:
\[ x_2 = V_1 \cdot t_2 + \frac{1}{2} a t_2^2 \]
where \( t_2 = \frac{t}{2} \) and \( V_1 = \frac{a t}{2} \):
\[ x_2 = \left(\frac{a t}{2}\right) \cdot \frac{t}{2} + \frac{1}{2} a \left(\frac{t}{2}\right)^2 \]
Calculating this:
\[ x_2 = \frac{a t^2}{4} + \frac{1}{2} a \cdot \frac{t^2}{4} = \frac{a t^2}{4} + \frac{a t^2}{8} = \frac{2a t^2}{8} + \frac{a t^2}{8} = \frac{3a t^2}{8} \]

### Step 6: Relate \( x_1 \) and \( x_2 \)
Now we have:
- \( x_1 = \frac{a t^2}{8} \)
- \( x_2 = \frac{3a t^2}{8} \)

We can express \( x_2 \) in terms of \( x_1 \):
\[ x_2 = 3x_1 \]

### Conclusion
Thus, the relationship between the distances traveled in the two halves of the time is:
\[ x_2 = 3x_1 \]