The distances covered by a freely falling body in its first, second, third, ……, `b^(th)` seconds of its motion a)forms an arithmetic progression b) forms a geometric progression c) do not form any well defined series d) form a series corresponding to the difference of square root of the successive natural numbers.

Options

A. forms an arithematic progression
B. forms a geometric progression
C. do not form any well defined series
D. form a series corresponding to the difference of square root of the successive natural numbers.

Explanation

To solve the problem, we need to analyze the distances covered by a freely falling body in each second of its motion. The distance covered in the first, second, third, and so on, seconds can be calculated using the formula for distance in uniformly accelerated motion.

### Step-by-Step Solution:

1. **Understanding the Motion**:
A freely falling body is under the influence of gravity. The distance covered in the nth second can be calculated using the formula:
\[ x_n = \frac{1}{2} g (n^2 - (n-1)^2) \]
where \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

2. **Calculating Distances for Each Second**:
- For the **1st second** (\( n = 1 \)):
\[ x_1 = \frac{1}{2} g (1^2 - 0^2) = \frac{1}{2} g \cdot 1 = \frac{g}{2} \approx 4.9 \, \text{m} \]
- For the **2nd second** (\( n = 2 \)):
\[ x_2 = \frac{1}{2} g (2^2 - 1^2) = \frac{1}{2} g (4 - 1) = \frac{3g}{2} \approx 14.7 \, \text{m} \]
- For the **3rd second** (\( n = 3 \)):
\[ x_3 = \frac{1}{2} g (3^2 - 2^2) = \frac{1}{2} g (9 - 4) = \frac{5g}{2} \approx 24.5 \, \text{m} \]

3. **Generalizing the Distance Formula**:
The distance covered in the nth second can be expressed as:
\[ x_n = \frac{1}{2} g (2n - 1) \]
This shows that the distances are increasing by a constant amount each second.

4. **Finding the Differences**:
The distances \( x_1, x_2, x_3, \ldots \) can be expressed as:
- \( x_1 = 4.9 \)
- \( x_2 = 14.7 \)
- \( x_3 = 24.5 \)

The differences between successive distances are:
- \( x_2 - x_1 = 14.7 - 4.9 = 9.8 \)
- \( x_3 - x_2 = 24.5 - 14.7 = 9.8 \)

5. **Conclusion**:
Since the differences between the distances covered in successive seconds are constant, the distances form an arithmetic progression.

### Final Answer:
The correct option is:
**a) forms an arithmetic progression.**