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)`

Explanation

To solve the problem, we need to analyze the forces acting on the balloon that is falling with a constant acceleration. Let's break it down step by step.

### Step 1: Identify the forces acting on the balloon
The balloon has two main forces acting on it:
1. The weight of the balloon (W), which acts downward.
2. The air resistance (R), which acts upward against the motion of the balloon.

### Step 2: Write the equation of motion
Since the balloon is falling with a constant acceleration (a), we can apply Newton's second law of motion. The net force acting on the balloon can be expressed as:
\[ \text{Net Force} = \text{Weight} - \text{Air Resistance} \]
This can be written mathematically as:
\[ W - R = ma \]
where:
- \( W \) is the weight of the balloon,
- \( R \) is the air resistance,
- \( m \) is the mass of the balloon,
- \( a \) is the acceleration of the balloon.

### Step 3: Relate mass to weight
The weight (W) of the balloon can be expressed in terms of mass (m) and gravitational acceleration (g):
\[ W = mg \]
From this, we can express the mass as:
\[ m = \frac{W}{g} \]

### Step 4: Substitute mass in the equation of motion
Now, substituting \( m \) in the equation of motion:
\[ W - R = \left(\frac{W}{g}\right) a \]

### Step 5: Rearranging the equation to find air resistance
Rearranging the equation to solve for air resistance (R):
\[ R = W - \left(\frac{W}{g}\right) a \]

### Step 6: Factor out W
Factoring out W from the right-hand side:
\[ R = W \left(1 - \frac{a}{g}\right) \]

### Final Answer
Thus, the magnitude of the air resistance is:
\[ R = W \left(1 - \frac{a}{g}\right) \]

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