A stone falls freely rest. The distance covered by it in the last second is equal to the distance covered by it in the first 2 s. The time taken by the stone to reach the ground is

Options

A. 5 s
B. 12 s
C. 15 s
D. 8 s

Explanation

To solve the problem, we will use the equations of motion under uniform acceleration due to gravity. The stone is falling freely from rest, which means its initial velocity (u) is 0.

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to find the total time (t) taken by the stone to fall such that the distance covered in the last second is equal to the distance covered in the first 2 seconds.

2. **Define Variables**:
- Let the total time of fall be \( t \).
- The initial velocity \( u = 0 \) (since the stone falls from rest).
- The acceleration \( g \) (acceleration due to gravity) is approximately \( 9.81 \, \text{m/s}^2 \).

3. **Distance Covered in the Last Second**:
The distance covered in the last second can be calculated using the formula:
\[ d_{\text{last}} = S(t) - S(t-1) \]
where \( S(t) \) is the distance covered in \( t \) seconds, given by:
\[ S(t) = \frac{1}{2} g t^2 \]
Therefore,
\[ d_{\text{last}} = \frac{1}{2} g t^2 - \frac{1}{2} g (t-1)^2 \]

4. **Distance Covered in the First 2 Seconds**:
The distance covered in the first 2 seconds is:
\[ d_{\text{first 2s}} = S(2) = \frac{1}{2} g (2)^2 = 2g \]

5. **Set Up the Equation**:
According to the problem, the distance covered in the last second equals the distance covered in the first 2 seconds:
\[ d_{\text{last}} = d_{\text{first 2s}} \]
Substituting the expressions we derived:
\[ \frac{1}{2} g t^2 - \frac{1}{2} g (t-1)^2 = 2g \]

6. **Simplify the Equation**:
Factor out \( \frac{1}{2} g \):
\[ \frac{1}{2} g \left( t^2 - (t^2 - 2t + 1) \right) = 2g \]
This simplifies to:
\[ \frac{1}{2} g (2t - 1) = 2g \]
Dividing both sides by \( g \) (assuming \( g \neq 0 \)):
\[ \frac{1}{2} (2t - 1) = 2 \]

7. **Solve for t**:
Multiply both sides by 2:
\[ 2t - 1 = 4 \]
Add 1 to both sides:
\[ 2t = 5 \]
Divide by 2:
\[ t = \frac{5}{2} = 2.5 \, \text{s} \]

### Final Answer:
The time taken by the stone to reach the ground is \( t = 2.5 \) seconds.

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