#### Problem Of the week: 20th September 2024

#### Rajas N

A hot air balloon uses a gas flame to heat the air it contains to a temperature required to enable it to hover at a small distance above ground level. The mass of the balloon, ropes, basket, and riders is 240 kg and the volume of the balloon is 1100 m³. The temperature of the surrounding air is 15°C and its density is 1.23 kg/m³

To what temperature does the air in the balloon need to be heated?

The buoyant force is equal to the weight of the air displaced

The relationship between density and temperature is:

Density A / Density B = Temperature B / Temperature A

Density A / Density B = Temperature B / Temperature A

We know the buoyant force is equal to the weight of the air displaced so

M of displaced air = density of surrounding air x V of balloon, so

buoyant force = density of surrounding air x V of balloon x acceleration due to gravity

The total mass of the balloon system is the balloon’s mass + the mass of the surrounding air so

Mass of balloon system = 240 kg + density of inside of balloon x V of balloon

Since it is hovering, the buoyant force is equal to the mass of the balloon system x g

So the whole equation is

(240 + density of inside of balloon x V of balloon) x g = density of surrounding air x V of balloon x g

We can cancel g from both sides so

240 + density of inside of balloon x V of balloon = density of surrounding air x V of balloon

So after rearranging,

density of inside of balloon = density of surrounding air – (240 / V of balloon)

So, density of inside of balloon = 1.23 – (240 / 1100) = 1.012 kg/m³

The relationship between density and temperature is this:

Density of inside of balloon / density of surrounding air =

temperature of surrounding area / temperature of inside of balloon

Hence in kelvin, 15°C = 288°K,

temperature of inside of balloon = (288 x 1.23) / 1.012 = 350°K

350°K = 77°C

Thus, the air in the balloon needs to be heated to 77°C

M of displaced air = density of surrounding air x V of balloon, so

buoyant force = density of surrounding air x V of balloon x acceleration due to gravity

The total mass of the balloon system is the balloon’s mass + the mass of the surrounding air so

Mass of balloon system = 240 kg + density of inside of balloon x V of balloon

Since it is hovering, the buoyant force is equal to the mass of the balloon system x g

So the whole equation is

(240 + density of inside of balloon x V of balloon) x g = density of surrounding air x V of balloon x g

We can cancel g from both sides so

240 + density of inside of balloon x V of balloon = density of surrounding air x V of balloon

So after rearranging,

density of inside of balloon = density of surrounding air – (240 / V of balloon)

So, density of inside of balloon = 1.23 – (240 / 1100) = 1.012 kg/m³

The relationship between density and temperature is this:

Density of inside of balloon / density of surrounding air =

temperature of surrounding area / temperature of inside of balloon

Hence in kelvin, 15°C = 288°K,

temperature of inside of balloon = (288 x 1.23) / 1.012 = 350°K

350°K = 77°C

Thus, the air in the balloon needs to be heated to 77°C