A balloon of gross weight w newton is falling vertically downward with a constant acceleration `a(lt g)` . The magnitude of the air resistance is :

Options

A. w
B. `w(1+a/g)`
C. `w(1-a/g)`
D. `w (a)/g`

Answer

To solve the problem of finding the magnitude of air resistance acting on a balloon falling with a constant acceleration less than the acceleration due to gravity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Balloon:**
- The balloon has a weight \( W \) acting downwards due to gravity.
- There is an upward force due to air resistance, which we will denote as \( F_a \).

2. **Write the Equation of Motion:**
- Since the balloon is falling with a constant acceleration \( a \) (where \( a < g \)), we can apply Newton's second law of motion:
\[ F_{\text{net}} = m \cdot a \]
- The net force acting on the balloon can be expressed as:
\[ F_{\text{net}} = W - F_a \]
- Therefore, we can write:
\[ W - F_a = m \cdot a \]

3. **Express Mass in Terms of Weight:**
- The weight \( W \) is related to mass \( m \) by the equation:
\[ W = m \cdot g \quad \Rightarrow \quad m = \frac{W}{g} \]

4. **Substitute Mass into the Equation:**
- Substitute \( m \) into the equation of motion:
\[ W - F_a = \left(\frac{W}{g}\right) \cdot a \]

5. **Rearrange to Solve for Air Resistance:**
- Rearranging the equation to isolate \( F_a \):
\[ F_a = W - \left(\frac{W}{g}\right) \cdot a \]

6. **Factor Out \( W \):**
- Factor \( W \) out of the equation:
\[ F_a = W \left(1 - \frac{a}{g}\right) \]

7. **Final Expression for Air Resistance:**
- Thus, the magnitude of the air resistance \( F_a \) is:
\[ F_a = W \left(1 - \frac{a}{g}\right) \]

### Conclusion:
The magnitude of the air resistance acting on the balloon is \( F_a = W \left(1 - \frac{a}{g}\right) \).

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