The terminal velocity of a balloon depends on several factors, including its shape, size, weight, and the density of the surrounding air. As a balloon falls through the air, it experiences two main forces: gravity, which pulls it downward, and air resistance, which opposes its motion. As the balloon's speed increases, the air resistance increases until it eventually reaches a point where the two forces are balanced, and the balloon no longer accelerates. This point is known as the terminal velocity.

Mathematically, the terminal velocity of a balloon can be calculated using the following formula:

Vd = √(2mg / (ρACd))

where:

Vd: Terminal velocity of the balloon

m: Mass of the balloon

g: Acceleration due to gravity

ρ: Density of the surrounding air

A: Cross-sectional area of the balloon

Cd: Drag coefficient of the balloon

In general, the terminal velocity of a balloon is quite low, especially for small, lightweight balloons. For example, a typical party balloon may have a terminal velocity of only a few centimeters per second. Larger balloons, such as weather balloons, can have terminal velocities of several meters per second.

However, it's important to note that the terminal velocity is not a fixed value. It can vary depending on changes in the ambient conditions. For instance, if the air resistance decreases, such as at higher altitudes, the terminal velocity of the balloon will increase.