A balloon with its contents weighing 160 N is moving down with an acceleration of g/2 `ms^(-2)` . The mass to be removed from it so that the balloon moves up with an acceleration of-g/3 `ms^(-2)` is

Options

A. 5 kg
B. 10 kg
C. 6 kg
D. 3 kg

Answer

To solve the problem step by step, we will analyze the forces acting on the balloon and apply Newton's second law of motion.

### Step 1: Determine the mass of the balloon
The weight of the balloon and its contents is given as \( W = 160 \, \text{N} \).

Using the formula for weight:
\[ W = mg \]
where \( g \) is the acceleration due to gravity. Assuming \( g = 10 \, \text{m/s}^2 \):
\[ 160 = m \cdot 10 \]
\[ m = \frac{160}{10} = 16 \, \text{kg} \]

### Step 2: Analyze the downward motion
The balloon is moving down with an acceleration of \( \frac{g}{2} = \frac{10}{2} = 5 \, \text{m/s}^2 \).

Applying Newton's second law:
\[ F_{\text{net}} = mg - F_{\text{air}} = ma \]
where \( F_{\text{air}} \) is the air resistance acting upward.

Substituting the known values:
\[ 160 - F_{\text{air}} = 16 \cdot 5 \]
\[ 160 - F_{\text{air}} = 80 \]
Solving for \( F_{\text{air}} \):
\[ F_{\text{air}} = 160 - 80 = 80 \, \text{N} \]

### Step 3: Analyze the upward motion
Now, we want the balloon to move upward with an acceleration of \( -\frac{g}{3} = -\frac{10}{3} \, \text{m/s}^2 \).

Let \( m' \) be the new mass of the balloon after removing some contents. The weight of the balloon after removing mass is \( m' g \).

Using Newton's second law again:
\[ F_{\text{air}} - m'g = m' \left(-\frac{g}{3}\right) \]
Substituting \( F_{\text{air}} = 80 \, \text{N} \):
\[ 80 - m'g = -\frac{m'g}{3} \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ 80 = m'g - \frac{m'g}{3} \]
Factoring out \( m'g \):
\[ 80 = m'g \left(1 - \frac{1}{3}\right) \]
\[ 80 = m'g \left(\frac{2}{3}\right) \]
Solving for \( m' \):
\[ m' = \frac{80 \cdot 3}{2g} = \frac{240}{20} = 12 \, \text{kg} \]

### Step 5: Calculate the mass to be removed
The initial mass of the balloon was \( 16 \, \text{kg} \), and after removing some mass, it is now \( 12 \, \text{kg} \).

The mass to be removed \( \Delta m \) is:
\[ \Delta m = 16 - 12 = 4 \, \text{kg} \]

### Final Answer
The mass to be removed from the balloon to achieve the desired upward acceleration is \( 4 \, \text{kg} \).

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