A balloon is descending with a constant acceleration a, less than the acceleration due to gravity g. The weight of the balloon, with its basket and contents, is w. What weight w, should be released so that the balloon will begin to accelerate upward with constant acceleration a? Neglect air resistance.

Answer

To solve the problem step by step, we will analyze the forces acting on the balloon and derive the required weight to be released for the balloon to start ascending with a constant acceleration \( a \).

### Step 1: Understand the forces acting on the balloon
The balloon is descending with a constant acceleration \( a \) which is less than the acceleration due to gravity \( g \). The forces acting on the balloon are:
- The weight of the balloon \( W \) acting downward.
- The buoyant force \( F_B \) acting upward.

Since the balloon is descending with an acceleration \( a \), we can write the equation of motion as:
\[ F_B - W = -m a \]
where \( m \) is the mass of the balloon system.

### Step 2: Express the buoyant force
The buoyant force \( F_B \) can be expressed as:
\[ F_B = m g \]
where \( m \) is the mass of the balloon system.

### Step 3: Set up the equation for upward acceleration
When we release a weight \( W_0 \) from the balloon, the new weight of the balloon becomes \( W - W_0 \). For the balloon to accelerate upwards with acceleration \( a \), we can write:
\[ F_B - (W - W_0) = m a \]

### Step 4: Substitute the buoyant force
Substituting \( F_B \) in the equation gives:
\[ m g - (W - W_0) = m a \]

### Step 5: Rearrange the equation
Rearranging the equation, we get:
\[ m g - W + W_0 = m a \]
This can be rearranged to find \( W_0 \):
\[ W_0 = W - m a + m g \]

### Step 6: Express mass in terms of weight
Since weight \( W = m g \), we can express \( m \) as:
\[ m = \frac{W}{g} \]
Substituting this back into the equation for \( W_0 \):
\[ W_0 = W - \frac{W}{g} a + W \]
This simplifies to:
\[ W_0 = 2W - \frac{W a}{g} \]

### Step 7: Final expression for \( W_0 \)
Factoring out \( W \):
\[ W_0 = W \left(2 - \frac{a}{g}\right) \]

### Conclusion
The weight \( W_0 \) that should be released from the balloon so that it begins to accelerate upward with a constant acceleration \( a \) is:
\[ W_0 = W \left(2 - \frac{a}{g}\right) \]