A body is allowed to fall from a height of `10 m`. If the time taken for the first `50 m` is `t_(1)` and for the remaining `50 s`,is `t_(2)`.
Which is correct?

Options

A. `t_(1)=t_(2)`
B. `t_(1) le t_(2)`
C. `t_(1) lt t_(2)`
D. `t_(1).t_(2)`

Explanation

To solve the problem, we need to analyze the motion of a body falling freely under the influence of gravity. The body falls from a height of 10 m, and we need to find the time taken for the first 50 m (denoted as \( t_1 \)) and the time taken for the remaining distance (denoted as \( t_2 \)).

### Step-by-Step Solution:

1. **Understanding the motion**:
The body is falling freely, so we can use the equations of motion under uniform acceleration due to gravity. The acceleration \( g \) is approximately \( 9.81 \, \text{m/s}^2 \).

2. **Finding \( t_1 \)**:
The distance fallen in the first 50 m can be calculated using the equation:
\[ s = \frac{1}{2} g t_1^2 \]
where \( s = 50 \, \text{m} \). Rearranging the equation gives:
\[ 50 = \frac{1}{2} g t_1^2 \implies t_1^2 = \frac{100}{g} \implies t_1 = \sqrt{\frac{100}{g}} = \frac{10}{\sqrt{g}} \]

3. **Finding the total time to fall 10 m**:
The total distance fallen is 10 m, so we use the same equation:
\[ 10 = \frac{1}{2} g T^2 \implies T^2 = \frac{20}{g} \implies T = \sqrt{\frac{20}{g}} = \frac{2\sqrt{5}}{\sqrt{g}} \]

4. **Finding \( t_2 \)**:
The time taken to fall the remaining distance (10 m - 50 m = -40 m) can be calculated as:
\[ t_2 = T - t_1 \]
We need to find the time taken to fall the remaining distance of 50 m, which can be calculated as:
\[ t_2 = T - t_1 = \frac{2\sqrt{5}}{\sqrt{g}} - \frac{10}{\sqrt{g}} = \frac{2\sqrt{5} - 10}{\sqrt{g}} \]

5. **Comparing \( t_1 \) and \( t_2 \)**:
To compare \( t_1 \) and \( t_2 \), we can analyze the expressions:
- \( t_1 = \frac{10}{\sqrt{g}} \)
- \( t_2 = \frac{2\sqrt{5} - 10}{\sqrt{g}} \)

Since \( \sqrt{5} \) is approximately 2.236, we find:
\[ 2\sqrt{5} \approx 4.472 \implies 2\sqrt{5} - 10 \approx -5.528 \]
Thus, \( t_2 \) is negative, indicating that \( t_1 > t_2 \).

### Conclusion:
From our calculations, we conclude that \( t_1 > t_2 \).

### Final Answer:
The correct statement is: \( t_1 > t_2 \).

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