A hot-air balloon of mass M is descending vertically with downward acceleration of magnitude a. How much mass (ballast) must be thrown out to give the balloon an upward acceleration of magnitude a? Assume that the upward force from the air (the lift) does not change because of the decrease in mass.

Explanation

To solve the problem of how much ballast must be thrown out of a hot-air balloon to give it an upward acceleration of magnitude \( a \), we can follow these steps:

### Step 1: Understand the Forces Acting on the Balloon
When the balloon is descending with a downward acceleration \( a \), the forces acting on it are:
- The weight of the balloon, \( W = Mg \) (where \( M \) is the mass of the balloon and \( g \) is the acceleration due to gravity).
- The upward lift force from the air, \( F_A \).

### Step 2: Apply Newton's Second Law for Downward Motion
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