Question

A hot air balloon including the envelope, gondola, burner and fuel and one passenger has a total mass of 500 kg. The air outside the balloon at 200C and has a density of 1.24 kg/m3. The air inside the envelope is heated to a temperature of 1200C, at which it has a density of
0.95 kg/m3. What volume must the envelope expand in order to lift the balloon into the air?

Answer 1

To find the volume the envelope must expand to lift the balloon into the air, we can use the principle of buoyancy. The buoyant force is equal to the weight of the air displaced by the balloon.

First, we need to calculate the weight of the balloon:
Weight of balloon = mass of balloon * acceleration due to gravity
Weight of balloon = 500 kg * 9.8 m/s^2
Weight of balloon = 4900 N

Next, we need to calculate the volume of air displaced by the balloon:
Density of air outside = 1.24 kg/m^3
Density of air inside = 0.95 kg/m^3

Buoyant force = weight of displaced air
Buoyant force = Density of air outside * g * Volume - Density of air inside * g * Volume
Buoyant force = (1.24 kg/m^3 - 0.95 kg/m^3) * 9.8 m/s^2 * Volume
Buoyant force = 0.29 kg/m^3 * 9.8 m/s^2 * Volume
Buoyant force = 2.842 kg*m/s^2 * Volume

Now, we know that the buoyant force is equal to the weight of the balloon:
2.842 kg*m/s^2 * Volume = 4900 N
Volume = 4900 N / 2.842 kg*m/s^2
Volume = 1723 m^3

Therefore, the envelope must expand to a volume of 1723 m^3 in order to lift the balloon into the air.