A balloon of total mass 1000 Kg float motionless over the earth's surface. If 100 kg of sand ballast are thrown over board, the balloon starts to rise with an acceleration of

Options

A. `10 m//s^(2)`
B. `9.8 m//s^(2)`
C. `1.09 m//s^(2)`
D. `4.9 m//s^(2)`

Answer

To solve the problem, we will use Newton's second law of motion and the concept of buoyancy.

### Step-by-Step Solution:

1. **Identify the Initial Conditions**:
- The total mass of the balloon when it is floating motionless is \( m_{initial} = 1000 \, \text{kg} \).
- The weight of the balloon is given by \( W_{initial} = m_{initial} \cdot g = 1000 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 9800 \, \text{N} \).

2. **Determine the Buoyant Force**:
- Since the balloon is floating motionless, the buoyant force \( F_b \) acting on it is equal to its weight:
\[ F_b = W_{initial} = 9800 \, \text{N} \]

3. **Calculate the New Mass After Throwing Sand**:
- When 100 kg of sand is thrown overboard, the new mass of the balloon becomes:
\[ m_{final} = 1000 \, \text{kg} - 100 \, \text{kg} = 900 \, \text{kg} \]

4. **Calculate the Weight of the Balloon After Throwing Sand**:
- The new weight of the balloon is:
\[ W_{final} = m_{final} \cdot g = 900 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 8820 \, \text{N} \]

5. **Determine the Net Force Acting on the Balloon**:
- The net force \( F_{net} \) acting on the balloon when the sand is thrown is given by the difference between the buoyant force and the new weight:
\[ F_{net} = F_b - W_{final} = 9800 \, \text{N} - 8820 \, \text{N} = 980 \, \text{N} \]

6. **Apply Newton's Second Law to Find Acceleration**:
- According to Newton's second law, \( F_{net} = m_{final} \cdot a \), where \( a \) is the acceleration of the balloon:
\[ 980 \, \text{N} = 900 \, \text{kg} \cdot a \]
- Solving for \( a \):
\[ a = \frac{980 \, \text{N}}{900 \, \text{kg}} \approx 1.09 \, \text{m/s}^2 \]

### Final Answer:
The balloon starts to rise with an acceleration of approximately \( 1.09 \, \text{m/s}^2 \).