A body starts from rest and travels with uniform acceleration the time taken by the body to cover the whole distance is t. Then the time taken the body to cover the second half of the distance is

Options

A. `tsqrt(2)`
B. `t(1-(1)/(sqrt(2)))`
C. `t(1 + (1)/(sqrt(2)))`
D. `t((1)/(sqrt(2))-1)`

Explanation

To solve the problem, we need to determine the time taken by a body to cover the second half of the distance when it starts from rest and travels with uniform acceleration. Let's break this down step by step.

### Step 1: Define the total distance and time
Let the total distance traveled by the body be \( s \) and the total time taken to cover this distance be \( t \).

### Step 2: Use the equation of motion
Since the body starts from rest (initial velocity \( u = 0 \)) and travels with uniform acceleration \( a \), we can use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting \( u = 0 \):
\[ s = \frac{1}{2} a t^2 \]

### Step 3: Calculate the distance for the first half
The distance covered in the first half of the journey is \( \frac{s}{2} \). We can use the same equation of motion for this distance, but we need to find the time taken for the first half, which we will denote as \( t_1 \):
\[ \frac{s}{2} = u t_1 + \frac{1}{2} a t_1^2 \]
Again substituting \( u = 0 \):
\[ \frac{s}{2} = \frac{1}{2} a t_1^2 \]

### Step 4: Relate the two equations
From the equation for the total distance \( s \):
\[ s = \frac{1}{2} a t^2 \]
Substituting this into the equation for the first half:
\[ \frac{1}{2} \left(\frac{1}{2} a t^2\right) = \frac{1}{2} a t_1^2 \]
This simplifies to:
\[ \frac{1}{4} a t^2 = \frac{1}{2} a t_1^2 \]
Cancelling \( \frac{1}{2} a \) from both sides (assuming \( a \neq 0 \)):
\[ \frac{1}{2} t^2 = t_1^2 \]

### Step 5: Solve for \( t_1 \)
Taking the square root of both sides gives:
\[ t_1 = \frac{t}{\sqrt{2}} \]

### Step 6: Calculate the time for the second half
The total time \( t \) is the sum of the time taken for the first half and the second half:
\[ t = t_1 + t_2 \]
Thus, we can express \( t_2 \) (the time taken for the second half) as:
\[ t_2 = t - t_1 \]
Substituting \( t_1 \):
\[ t_2 = t - \frac{t}{\sqrt{2}} = t \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Conclusion
Therefore, the time taken to cover the second half of the distance is:
\[ t_2 = t \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Final Answer
The correct option is:
**Option 2: \( t \left(1 - \frac{1}{\sqrt{2}}\right) \)**