Statement-1: Weight of a empty balloon measured in air is `w_(1)`. If air at atmospheric pressure is filled inside balloon and again weight of balloon is measured. Weight of balloon is `w_(2)` is second case. Then `W_(2)` is equal to `w_(1)`
Statement-2: Upthrust is equal to weight of the fluid displaced by the body.

Options

A. Statement-1, is true, Statement-2 is true, Statement-2 is correct explanation for statement-1
B. Statement-1 is true, statement-2 is true, statement-2 is NOT a correct explantion for statement-1
C. statement-1 is true, statement-2 is false
D. statement-1 is false statement-2 is true

Answer

To analyze the given statements, let's break down the problem step by step.

### Step 1: Understanding Statement 1
- **Statement 1**: The weight of an empty balloon measured in air is \( w_1 \). When air at atmospheric pressure is filled inside the balloon, the weight of the balloon is measured again as \( w_2 \). The statement claims that \( w_2 \) is equal to \( w_1 \).

### Step 2: Analyzing the Weight of the Balloon
- When the balloon is empty, its weight is simply \( w_1 \).
- When the balloon is filled with air, the weight of the air inside the balloon contributes to the total weight. However, this air displaces an equal volume of air from the atmosphere.

### Step 3: Understanding Buoyant Force
- According to Archimedes' principle, the buoyant force (upthrust) acting on the balloon is equal to the weight of the fluid (in this case, air) displaced by the balloon.
- If the weight of the air inside the balloon is \( W_0 \), then the buoyant force acting on the balloon is also \( W_0 \).

### Step 4: Balancing Forces
- When the balloon is filled with air, the weight of the balloon plus the weight of the air inside it is balanced by the buoyant force.
- Thus, we can write the equation:
\[ w_2 = w_1 + W_0 - W_0 \]
Simplifying this gives:
\[ w_2 = w_1 \]
- Therefore, the first statement is true.

### Step 5: Understanding Statement 2
- **Statement 2**: Upthrust is equal to the weight of the fluid displaced by the body.
- This is a direct application of Archimedes' principle and is universally accepted in fluid mechanics.

### Conclusion
- Both statements are independently true. Statement 1 is true because the weight of the balloon remains the same when filled with air due to the balance of forces. Statement 2 is true as it correctly describes the principle of buoyancy.

### Final Answer
Both Statement 1 and Statement 2 are true, and Statement 2 provides the correct explanation for Statement 1.

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