Read the given statements and select the correct option.
Statement 1 : Reducing the initial velocity to half the initial value, the stopping distance of moving object decreases by a factor of 2 (for the same deceleration).
Statement 2 : Stopping distance is proportional to the square of the initial velocity .

Options

A. Both statements 1 and 2 are true and statement 2 is the correct explanation of statement 1.
B. Both statements 1 and 2 are true but statement 2 is not the correct explanation of statement 1.
C. Statement 1 is true but statement 2 is false.
D. Statement 1 is false but statement 2 is true.

Explanation

To solve the problem, we need to analyze the two statements provided and determine their validity based on the principles of motion.

### Step-by-Step Solution:

1. **Understanding Stopping Distance**:
The stopping distance (S) of an object can be derived from the equation of motion:
\[ v^2 = u^2 + 2aS \]
where \(v\) is the final velocity (0 when the object stops), \(u\) is the initial velocity, \(a\) is the acceleration (deceleration in this case), and \(S\) is the stopping distance.

2. **Rearranging the Equation**:
Since the final velocity \(v\) is 0 when the object stops, we can rearrange the equation to find the stopping distance:
\[ 0 = u^2 + 2aS \implies S = \frac{u^2}{-2a} \]
Here, since \(a\) is negative (deceleration), we can express it as:
\[ S = \frac{u^2}{2|a|} \]
This shows that stopping distance is directly proportional to the square of the initial velocity (\(u^2\)).

3. **Analyzing Statement 1**:
Statement 1 claims that if the initial velocity is reduced to half, the stopping distance decreases by a factor of 2.
- If the initial velocity is halved, the new initial velocity \(u' = \frac{u}{2}\).
- The new stopping distance \(S'\) can be calculated as:
\[ S' = \frac{(u/2)^2}{2|a|} = \frac{u^2/4}{2|a|} = \frac{u^2}{8|a|} \]
- The original stopping distance was \(S = \frac{u^2}{2|a|}\).
- Therefore, the new stopping distance \(S'\) is:
\[ S' = \frac{S}{4} \]
This means the stopping distance decreases by a factor of 4, not 2. Hence, **Statement 1 is false**.

4. **Analyzing Statement 2**:
Statement 2 states that stopping distance is proportional to the square of the initial velocity.
- From our earlier derivation, we found that:
\[ S \propto u^2 \]
This confirms that **Statement 2 is true**.

5. **Conclusion**:
- Statement 1 is false.
- Statement 2 is true.

### Final Answer:
- **Statement 1**: False
- **Statement 2**: True