A stone falls freely under gravity. It covered distances `h_1, h_2` and `h_3` in the first `5` seconds. The next `5` seconds and the next `5` seconds respectively. The relation between `h_1, h_2` and `h_3` is :

Options

A. `h_(1)=2h_(2)=3h_(3)`
B. `h_(1)=(h_(2))/(3)=(h_(3))/(5)`
C. `h_(2)=3h_(1)` and `h_(3)=3h_(2)`
D. `h_(1)=h_(2)=h_(3)`

Explanation

To solve the problem of a stone falling freely under gravity and covering distances \( h_1 \), \( h_2 \), and \( h_3 \) in the first 5 seconds, the next 5 seconds, and the next 5 seconds respectively, we can use the equations of motion.

### Step-by-Step Solution:

1. **Understanding the motion**:
The stone is falling freely under gravity, which means it starts from rest (\( u = 0 \)) and accelerates downwards with an acceleration equal to \( g \) (acceleration due to gravity).

2. **Calculate \( h_1 \)**:
The distance covered in the first 5 seconds can be calculated using the equation of motion:
\[ h_1 = ut + \frac{1}{2}gt^2 \]
Since the initial velocity \( u = 0 \):
\[ h_1 = \frac{1}{2}gt^2 \]
Substituting \( g = 10 \, \text{m/s}^2 \) and \( t = 5 \, \text{s} \):
\[ h_1 = \frac{1}{2} \times 10 \times (5)^2 = \frac{1}{2} \times 10 \times 25 = 125 \, \text{m} \]

3. **Calculate \( h_2 \)**:
The distance covered in the next 5 seconds (from \( t = 5 \) to \( t = 10 \) seconds) can be calculated as:
\[ h_2 = \text{Total distance in 10 seconds} - \text{Distance in first 5 seconds} \]
First, calculate the total distance in 10 seconds:
\[ \text{Total distance} = \frac{1}{2}gt^2 = \frac{1}{2} \times 10 \times (10)^2 = \frac{1}{2} \times 10 \times 100 = 500 \, \text{m} \]
Thus,
\[ h_2 = 500 - 125 = 375 \, \text{m} \]

4. **Calculate \( h_3 \)**:
The distance covered in the next 5 seconds (from \( t = 10 \) to \( t = 15 \) seconds) can be calculated similarly:
\[ h_3 = \text{Total distance in 15 seconds} - \text{Distance in first 10 seconds} \]
First, calculate the total distance in 15 seconds:
\[ \text{Total distance} = \frac{1}{2}gt^2 = \frac{1}{2} \times 10 \times (15)^2 = \frac{1}{2} \times 10 \times 225 = 1125 \, \text{m} \]
Thus,
\[ h_3 = 1125 - 500 = 625 \, \text{m} \]

5. **Establishing the relationship**:
Now we have:
\[ h_1 = 125 \, \text{m}, \quad h_2 = 375 \, \text{m}, \quad h_3 = 625 \, \text{m} \]
To find the relationship between \( h_1, h_2, \) and \( h_3 \):
\[ h_2 = 3h_1 \quad (\text{since } 375 = 3 \times 125) \]
\[ h_3 = 5h_1 \quad (\text{since } 625 = 5 \times 125) \]

### Final Relation:
Thus, the relationship between the distances is:
\[ h_2 = 3h_1 \quad \text{and} \quad h_3 = 5h_1 \]