A balloon of mass `M` is descending at a constant acceleration `alpha`. When a mass m is released from the balloon it starts rising with the same acceleration `alpha` Assuming that its volume does not change what is the value of m. ?

Options

A. `(alpha)/(alpha+g)M`
B. `(2alpha)/(alpha+g)M`
C. `(alpha +g)/(alpha)M`
D. `(alpha + g)/(2alpha)M`

Explanation

To solve the problem step by step, we will analyze the forces acting on the balloon and the mass when it is released.

### Step 1: Analyze the Balloon's Descent
When the balloon of mass \( M \) is descending with a constant acceleration \( \alpha \), we can apply Newton's second law. The forces acting on the balloon are:
- The gravitational force acting downward: \( F_g = Mg \)
- The buoyant force acting upward: \( F_b \)

Since the balloon is descending with an acceleration \( \alpha \), we can write the equation of motion as:
\[ Mg - F_b = M \alpha \]
This is our **Equation 1**.

### Step 2: Analyze the Mass When Released
When the mass \( m \) is released from the balloon, it starts rising with the same acceleration \( \alpha \). The forces acting on the mass \( m \) are:
- The gravitational force acting downward: \( F_g = mg \)
- The buoyant force acting upward: \( F_b \)

For the mass \( m \) rising with an acceleration \( \alpha \), we can write the equation of motion as:
\[ F_b - mg = m \alpha \]
This is our **Equation 2**.

### Step 3: Relate the Two Equations
From **Equation 1**, we have:
\[ F_b = Mg - M\alpha \]

Substituting \( F_b \) from **Equation 1** into **Equation 2** gives us:
\[ (Mg - M\alpha) - mg = m \alpha \]

### Step 4: Simplify the Equation
Rearranging the equation, we get:
\[ Mg - M\alpha - mg = m \alpha \]
\[ Mg - mg = M\alpha + m\alpha \]
Factoring out \( m \) from the right side:
\[ Mg = m(g + \alpha) + M\alpha \]

### Step 5: Solve for \( m \)
Now we can isolate \( m \):
\[ m(g + \alpha) = Mg - M\alpha \]
\[ m = \frac{Mg - M\alpha}{g + \alpha} \]

### Step 6: Final Expression for \( m \)
Factoring out \( M \) gives us:
\[ m = \frac{M(g - \alpha)}{g + \alpha} \]

### Conclusion
Thus, the value of \( m \) is:
\[ m = \frac{M(g - \alpha)}{g + \alpha} \]