A body is allowed to fall from a height of `100 m`. If the time taken for the first `50 m` is `t_(1)` and for the remaining `50 m` is `t_(2)`.
The ratio of time to reach the ground and to reach first half of the distance is .

Options

A. `sqrt3: 1`
B. `sqrt 2:1`
C. `5:2`
D. `1: sqrt3`

Explanation

To solve the problem, we will use the equations of motion under gravity. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a body falling from a height of 100 meters. The time taken to fall the first 50 meters is \( t_1 \) and the time taken to fall the remaining 50 meters is \( t_2 \). We need to find the ratio of the total time to reach the ground \( t \) to the time taken to reach the first half of the distance \( t_1 \).

### Step 2: Use the Equation of Motion for the First 50 Meters
For the first 50 meters, we can use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Here,
- \( s = 50 \, \text{m} \) (distance fallen),
- \( u = 0 \, \text{m/s} \) (initial velocity, since it starts from rest),
- \( a = g = 10 \, \text{m/s}^2 \) (acceleration due to gravity),
- \( t = t_1 \) (time taken to fall the first 50 m).

Substituting these values into the equation:
\[ 50 = 0 \cdot t_1 + \frac{1}{2} \cdot 10 \cdot t_1^2 \]
This simplifies to:
\[ 50 = 5 t_1^2 \]
Dividing both sides by 5:
\[ t_1^2 = 10 \]
Taking the square root:
\[ t_1 = \sqrt{10} \, \text{s} \]

### Step 3: Use the Equation of Motion for the Total Distance (100 Meters)
Now, we will find the total time \( t \) to fall the entire 100 meters using the same equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Here,
- \( s = 100 \, \text{m} \),
- \( u = 0 \, \text{m/s} \),
- \( a = g = 10 \, \text{m/s}^2 \),
- \( t = t \) (total time).

Substituting these values:
\[ 100 = 0 \cdot t + \frac{1}{2} \cdot 10 \cdot t^2 \]
This simplifies to:
\[ 100 = 5 t^2 \]
Dividing both sides by 5:
\[ t^2 = 20 \]
Taking the square root:
\[ t = \sqrt{20} \, \text{s} \]

### Step 4: Find the Ratio \( \frac{t}{t_1} \)
Now we can find the ratio of the total time to the time taken for the first half:
\[ \frac{t}{t_1} = \frac{\sqrt{20}}{\sqrt{10}} = \sqrt{\frac{20}{10}} = \sqrt{2} \]

### Conclusion
Thus, the ratio of the time to reach the ground to the time to reach the first half of the distance is:
\[ \frac{t}{t_1} = \sqrt{2} : 1 \]