Question

A hot-air balloon has a volume of \(2979 \mathrm{~m}^{3}\). The density of the air outside the balloon is \(1.205 \mathrm{~kg} / \mathrm{m}^{3}\). The density of the hot air inside the balloon is \(0.9441 \mathrm{~kg} / \mathrm{m}^{3}\). How much weight can the balloon lift (including its own weight)?

Short answer

Based on the solution above, calculate the weight (including its own weight) the hot-air balloon can lift. Answer: The hot-air balloon can lift a weight of 7622.782 N, including its own weight.

Step-by-step solution

Calculate the weight of the air inside the balloon

To find the weight of the air inside the balloon, we need to find its mass. We can find the mass by multiplying the balloon's volume with the density of the hot air inside. \(Mass_{inside} = Volume \times Density_{inside}\) where \(Volume = 2979 \mathrm{~m}^{3}\) \(Density_{inside} = 0.9441 \mathrm{~kg/m}^{3}\) \(Mass_{inside} = 2979 \mathrm{~m}^{3} \times 0.9441 \mathrm{~kg/m}^{3} = 2813.0359 \mathrm{~kg}\) Now, we can find the weight of the air inside the balloon using the mass and acceleration due to gravity (g): \(Weight_{inside} = Mass_{inside} \times g\) where \(g = 9.81 \mathrm{~m/s}^{2}\) \(Weight_{inside} = 2813.0359 \mathrm{~kg} \times 9.81 \mathrm{~m/s}^{2} = 27591.702 \mathrm{~N}\)

Calculate the weight of the air displaced by the balloon

The balloon displaces an equal volume of air outside which is equal to its own volume. Thus, we can find the mass and weight of the air displaced by the balloon in a similar manner as we did for the air inside the balloon: \(Mass_{displaced} = Volume \times Density_{outside}\) where \(Density_{outside} = 1.205 \mathrm{~kg/m}^{3}\) \(Mass_{displaced} = 2979 \mathrm{~m}^{3} \times 1.205 \mathrm{~kg/m}^{3} = 3589.795 \mathrm{~kg}\) Now, we can find the weight of the air displaced by the balloon: \(Weight_{displaced} = Mass_{displaced} \times g\) \(Weight_{displaced} = 3589.795 \mathrm{~kg} \times 9.81 \mathrm{~m/s}^{2} = 35214.484 \mathrm{~N}\)

Calculate the buoyant force acting on the balloon

The buoyant force can be calculated as the difference between the weight of the air displaced by the balloon and the weight of the air inside the balloon: \(Buoyant\_Force = Weight_{displaced} - Weight_{inside}\) \(Buoyant\_Force = 35214.484 \mathrm{~N} - 27591.702 \mathrm{~N} = 7622.782 \mathrm{~N}\)

Calculate the weight the balloon can lift

Before finding the weight the balloon can lift, we need to know the weight of the balloon itself. According to the details given in the question, it should be considered as a part of the weight that the balloon can lift. Let's say the weight of the hot-air balloon (including its own weight) is \(W\). The buoyant force acting on the balloon must be equal to the total weight it can lift (including its own weight). Therefore, \(W = Buoyant\_Force\) \(W = 7622.782 \mathrm{~N}\) Hence, the balloon can lift a weight of 7622.782 N, including its own weight. It's important to note that if we want to find the weight the balloon can lift excluding its own weight, we would have to subtract the weight of the balloon from this value. However, the exercise does not provide the weight of the balloon itself, so we cannot calculate that.