A balloon of mass m is rising up with an acceleration a.

Options

A. The upthrust on the balloon is `m(g+a)`.
B. The upthrust on the balloon is ma.
C. `ma//(2a+g)` mass must be detached in order to double its acceleration.
D. m/2 mass must be detached in order to double its acceleration.

Answer

To solve the problem of a balloon of mass \( m \) rising with an acceleration \( a \), we will analyze the forces acting on the balloon and apply Newton's second law of motion.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Balloon**:
- The weight of the balloon, \( W \), acting downwards is given by:
\[ W = mg \]
- The buoyant force (upthrust) acting upwards, which we denote as \( U \).

2. **Apply Newton's Second Law**:
- According to Newton's second law, the net force acting on the balloon is equal to the mass of the balloon multiplied by its acceleration:
\[ F_{\text{net}} = ma \]
- The net force acting on the balloon can also be expressed as the difference between the buoyant force and the weight:
\[ F_{\text{net}} = U - mg \]

3. **Set Up the Equation**:
- Equating the two expressions for net force, we have:
\[ U - mg = ma \]
- Rearranging this gives:
\[ U = mg + ma \]

4. **Determine the Buoyant Force**:
- From the equation derived, we can express the buoyant force as:
\[ U = m(g + a) \]
- This indicates that the buoyant force must be equal to the weight of the balloon plus the force required to accelerate it upwards.

5. **Consider Doubling the Acceleration**:
- If we want to double the acceleration to \( 2a \), we need to find the new mass of the balloon after detaching some mass \( m' \).
- The new mass of the balloon will be \( m - m' \), and the new equation for the buoyant force will be:
\[ U = (m - m')g + (m - m') \cdot 2a \]

6. **Set Up the New Equation**:
- Setting the new buoyant force equal to the new weight gives:
\[ U = (m - m')g + (m - m') \cdot 2a \]
- Since the buoyant force remains the same, we can equate it to the original buoyant force:
\[ m(g + a) = (m - m')g + (m - m') \cdot 2a \]

7. **Solve for \( m' \)**:
- Expanding and rearranging the equation will help us isolate \( m' \):
\[ mg + ma = mg - m'g + 2ma - 2m'a \]
- Simplifying this leads to:
\[ ma + m'g = 2ma - 2m'a \]
- Collecting terms involving \( m' \) gives:
\[ m' (g + 2a) = ma \]
- Finally, solving for \( m' \) gives:
\[ m' = \frac{ma}{g + 2a} \]

### Conclusion:
Thus, the mass that needs to be detached to double the acceleration of the balloon is given by:
\[ m' = \frac{ma}{g + 2a} \]