A stone is dropped from a certain heitht which can reach the ground in `5 s`. It is stopped aftre `3 s` of its fall and then it is again released. The total time taken by the stone to reach the ground will be .

Options

A. ` 6s`
B. ` 6.5 s`
C. ` 7 s`
D. ` 7.5 s`

Explanation

To solve the problem step by step, we will analyze the motion of the stone in two parts: the first part where it falls for 3 seconds and is then stopped, and the second part where it is released again to reach the ground.

### Step 1: Calculate the distance fallen in the first 3 seconds
The stone is dropped from rest, so its initial velocity \( u = 0 \). The distance fallen in time \( t \) under gravity can be calculated using the formula:
\[ S = ut + \frac{1}{2} g t^2 \]
Where:
- \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)),
- \( t = 3 \, \text{s} \).

Substituting the values:
\[ S_3 = 0 + \frac{1}{2} \times 10 \times (3)^2 = \frac{1}{2} \times 10 \times 9 = 45 \, \text{m} \]

### Step 2: Calculate the total distance fallen in 5 seconds
Using the same formula for the total time of 5 seconds:
\[ S_5 = 0 + \frac{1}{2} \times 10 \times (5)^2 = \frac{1}{2} \times 10 \times 25 = 125 \, \text{m} \]

### Step 3: Calculate the remaining distance after 3 seconds
The remaining distance after 3 seconds of fall is:
\[ \text{Remaining Distance} = S_5 - S_3 = 125 \, \text{m} - 45 \, \text{m} = 80 \, \text{m} \]

### Step 4: Calculate the time taken to fall the remaining distance
After being stopped for 3 seconds, the stone is released again from rest to fall the remaining distance of 80 m. We will use the same formula:
\[ S = ut + \frac{1}{2} g t^2 \]
Where \( S = 80 \, \text{m} \), \( u = 0 \), and \( g = 10 \, \text{m/s}^2 \).

Setting up the equation:
\[ 80 = 0 + \frac{1}{2} \times 10 \times t^2 \]
This simplifies to:
\[ 80 = 5 t^2 \implies t^2 = \frac{80}{5} = 16 \implies t = \sqrt{16} = 4 \, \text{s} \]

### Step 5: Calculate the total time taken to reach the ground
The total time taken by the stone to reach the ground is the sum of the time it fell before being stopped and the time it took to fall the remaining distance:
\[ \text{Total Time} = 3 \, \text{s} + 4 \, \text{s} = 7 \, \text{s} \]

### Final Answer
The total time taken by the stone to reach the ground is **7 seconds**.

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