A body is dropped from a height H. The time taken to cover second half of the journey is

Options

A. `2sqrt((2H)/(g))`
B. `sqrt((H)/(g))`
C. `sqrt((H)/(g))(sqrt2-1)`
D. `sqrt((2H)/(g))xx(1)/((sqrt2-1))`

Answer

To solve the problem of finding the time taken to cover the second half of the journey when a body is dropped from a height \( H \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A body is dropped from a height \( H \).
- We need to find the time taken to cover the second half of the journey, which means from \( H/2 \) to \( H \).

2. **Total Time to Fall from Height \( H \)**:
- The total time \( T \) taken to fall from height \( H \) can be calculated using the formula:
\[ T = \sqrt{\frac{2H}{g}} \]
- Here, \( g \) is the acceleration due to gravity.

3. **Time to Fall from Height \( H/2 \)**:
- To find the time taken to fall from \( H/2 \) to \( H \), we can use the same formula for the height \( H/2 \):
\[ t = \sqrt{\frac{2(H/2)}{g}} = \sqrt{\frac{H}{g}} \]

4. **Calculating Time for the Second Half**:
- The time taken to cover the second half of the journey is the total time \( T \) minus the time \( t \) taken to fall the first half:
\[ \text{Time for second half} = T - t \]
- Substituting the values we found:
\[ \text{Time for second half} = \sqrt{\frac{2H}{g}} - \sqrt{\frac{H}{g}} \]

5. **Simplifying the Expression**:
- Factor out \( \sqrt{\frac{H}{g}} \):
\[ \text{Time for second half} = \sqrt{\frac{H}{g}} \left( \sqrt{2} - 1 \right) \]

6. **Final Answer**:
- Therefore, the time taken to cover the second half of the journey is:
\[ \text{Time for second half} = \sqrt{\frac{H}{g}} \left( \sqrt{2} - 1 \right) \]