A body is dropped from a certain height.

Options

A. the time taken to travel first half of the height is greater than that for second half.
B. time time taken to travel first half of the height is less than that for second half
C. the time taken to travel first half of the height is equal to that for second half
D. any one of the above three situations may be correct depending upon the value of height.

Explanation

To solve the problem of a body dropped from a certain height and to determine the time taken to travel the first half of the height compared to the second half, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A body is dropped from a height \( h \). We need to analyze the time taken to cover the first half of the height (\( h/2 \)) and the second half of the height (\( h/2 \)).

2. **Using Kinematic Equations**:
- The kinematic equation for an object under free fall (with initial velocity \( u = 0 \)) is given by:
\[ s = ut + \frac{1}{2} a t^2 \]
- Here, \( s \) is the distance traveled, \( a \) is the acceleration due to gravity (\( g \)), and \( t \) is the time taken.

3. **Calculating Time for the First Half**:
- For the first half of the height (\( h/2 \)):
\[ \frac{h}{2} = 0 \cdot t_1 + \frac{1}{2} g t_1^2 \]
Simplifying this gives:
\[ h = g t_1^2 \implies t_1 = \sqrt{\frac{h}{g}} \]

4. **Calculating Time for the Second Half**:
- For the second half of the height, the body has already fallen \( h/2 \) and has a velocity \( v \) when it reaches this point. We need to find the time \( t_2 \) to fall the second half.
- The velocity at the end of the first half can be calculated using:
\[ v = u + at = 0 + g t_1 = g \sqrt{\frac{h}{g}} = \sqrt{gh} \]
- Now, using the kinematic equation for the second half:
\[ \frac{h}{2} = v t_2 + \frac{1}{2} g t_2^2 \]
Substituting \( v = \sqrt{gh} \):
\[ \frac{h}{2} = \sqrt{gh} t_2 + \frac{1}{2} g t_2^2 \]

5. **Comparing Times**:
- From the analysis, we can see that the body accelerates as it falls. Thus, it takes more time to cover the first half of the height than the second half because the velocity increases as the body falls.
- Therefore, we conclude that:
\[ t_1 > t_2 \]

6. **Final Conclusion**:
- The time taken to travel the first half of the height is greater than that of the second half. Thus, the correct option is:
\[ t_1 > t_2 \]