91Ó°ÊÓ

To calculate the average density of the heated gases inside a hot-air balloon, we need to begin by examining the concept of total lift. The air inside the balloon must be buoyant enough to lift not only the balloon itself, but also any additional weight it carries, like passengers and equipment.
The formula for average density is derived from the relationship between mass, volume, and density. We use:

• Total weight lifted (sum of all weights the balloon carries)
• Weight of the displaced air (buoyant force minus weight of all lifted content)

First, the total weight being lifted was calculated as 5800 N. By determining the weight of air inside with the formula \[W_{\text{air}} = F_b - \text{total lift} = 26562.06 \, \text{N} - 5800 \, \text{N} = 20762.06 \, \text{N}\]we next find the mass of the heated air: \[m_{\text{air}} = \frac{W_{\text{air}}}{g}\]. This formula allows us to transition from weight to mass, making it possible to calculate density, using the balloon's volume. Thus, the average density of the air is found by the formula: \[\rho_{\text{gas}} = \frac{m_{\text{air}}}{V}\].
This result provides insight into how light the heated gas becomes compared to the surrounding air.

Buoyant force is a fundamental concept in physics, particularly when dealing with floating or submerged objects. It refers to the force exerted by a fluid (like air or water) that supports the weight of an object.
For a hot-air balloon, this force determines how much the balloon can lift. The buoyant force can be calculated using the formula: \[ F_b = \rho_{\text{air}} \times g \times V \].
Where:

• \(\rho_{\text{air}}\) is the density of the surrounding air
• \(g\) is the acceleration due to gravity
• \(V\) is the volume of the balloon

This equation highlights that the lift (buoyant force) depends not only on the balloon's size (volume) but also on the density of the air it moves through.
When the calculated buoyant force exceeds the weight the balloon needs to lift, the balloon will rise. This principle keeps hot-air balloons afloat and allows them to carry passengers and cargo.

The density of heated gases in a hot-air balloon plays a pivotal role in the balloon's ability to fly. By heating the air inside, the density decreases compared to the cooler air outside.
This difference in density is the key to generating lift. Heated air expands, takes up more space, and as a result, the same mass of air occupies a larger volume. This means it becomes less dense.
The challenge is achieving the right balance:

• If the density of the heated air becomes too low, the balloon won't rise beyond a certain altitude because it lacks support.
• If not heated enough, the air's density remains too high, preventing sufficient lift.

By calculating and adjusting the internal temperature of the air, balloon pilots achieve the optimal average density, ensuring a steady ascent and controlled flight. This is crucial for maintaining balance and safety during a journey. Understanding the dynamics of heated gas density equips pilots with the knowledge to make informed decisions about altitude and speed control.