A stone is allowed to fall freely from rest. The ratio of the time taken to fall through the first metre and the second metre distance is

Options

A. `sqrt(2)-1`
B. `sqrt(2)+1`
C. `sqrt(2)`
D. None of these

Explanation

To solve the problem of finding the ratio of the time taken to fall through the first meter and the second meter for a stone falling freely from rest, we can follow these steps:

### Step 1: Understand the motion
The stone is falling freely under the influence of gravity, which means it starts from rest (initial velocity \( u = 0 \)) and accelerates at \( g \) (acceleration due to gravity).

### Step 2: Use the equation of motion
The distance fallen in time \( t \) is given by the equation:
\[ s = ut + \frac{1}{2} g t^2 \]
Since the initial velocity \( u = 0 \), this simplifies to:
\[ s = \frac{1}{2} g t^2 \]

### Step 3: Calculate the time taken to fall the first meter
For the first meter (\( s_1 = 1 \) m):
\[ 1 = \frac{1}{2} g t_1^2 \]
Rearranging gives:
\[ t_1^2 = \frac{2}{g} \]
Taking the square root:
\[ t_1 = \sqrt{\frac{2}{g}} \]

### Step 4: Calculate the time taken to fall the first two meters
For the first two meters (\( s_2 = 2 \) m):
\[ 2 = \frac{1}{2} g t_2^2 \]
Rearranging gives:
\[ t_2^2 = \frac{4}{g} \]
Taking the square root:
\[ t_2 = \sqrt{\frac{4}{g}} = \frac{2}{\sqrt{g}} \]

### Step 5: Calculate the time taken to fall the second meter
The time taken to fall through the second meter is the total time to fall 2 meters minus the time to fall the first meter:
\[ t_{2m} = t_2 - t_1 \]
Substituting the values we found:
\[ t_{2m} = \frac{2}{\sqrt{g}} - \sqrt{\frac{2}{g}} \]
To simplify, we can express \( t_1 \) in terms of a common denominator:
\[ t_{2m} = \frac{2 - \sqrt{2}}{\sqrt{g}} \]

### Step 6: Find the ratio of the times
Now, we need the ratio of the time taken to fall through the first meter to the time taken to fall through the second meter:
\[ \text{Ratio} = \frac{t_1}{t_{2m}} = \frac{\sqrt{\frac{2}{g}}}{\frac{2 - \sqrt{2}}{\sqrt{g}}} \]
This simplifies to:
\[ \text{Ratio} = \frac{\sqrt{2}}{2 - \sqrt{2}} \]

### Step 7: Rationalize the denominator
To rationalize the denominator:
\[ \text{Ratio} = \frac{\sqrt{2}(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} = \frac{2\sqrt{2} + 2}{2^2 - (\sqrt{2})^2} = \frac{2\sqrt{2} + 2}{2} \]
This simplifies to:
\[ \text{Ratio} = \sqrt{2} + 1 \]

### Final Answer
Thus, the ratio of the time taken to fall through the first meter to the time taken to fall through the second meter is:
\[ \text{Ratio} = \sqrt{2} + 1 \]