Hot Air Baloon Problem

• notorious big
In summary: D2*(T2+273)/T1W2-W1 = mg = (D2-D1)gV = D2gV*(1-T2/(T2+273))In summary, the problem involves using the ideal gas law to find the temperature of the air inside a hot-air balloon in order to achieve the desired lift. This is done by comparing the weight of the air inside and outside the balloon, which is affected by the temperature and density of the air. By using the given values and equations, the temperature of the air inside the balloon can be calculated.

notorious big

I'm confused as to what equation I need to use for the following problem:

A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1820 m3 and the required lift is 2983 N (weight of equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the desired lift. Assume that the outside air temperature is 0oC and that air is an ideal gas under these conditions. Express your answer in oC. The density of air at STP is 1.29 kg/m3.

Any help would be greatly appreciated.

notorious big said:
I'm confused as to what equation I need to use for the following problem:

A hot-air balloon achieves its buoyant lift by heating the air inside the balloon, which makes it less dense than the air outside. Suppose the volume of a balloon is 1820 m3 and the required lift is 2983 N (weight of equipment and passenger). Calculate the temperature of the air inside the balloon which will produce the desired lift. Assume that the outside air temperature is 0oC and that air is an ideal gas under these conditions. Express your answer in oC. The density of air at STP is 1.29 kg/m3.

Any help would be greatly appreciated.

In most problems involving the ideal gas law, the number of molecules of gas is constant, and P, V and T vary in some way. In this problem, P and V are essentially constant and changing the temperature affects the number of molecules in the balloon, which affects the density of the air.

The buoyant force on the balloon is the weight of air at STP that is displaced by the balloon. The weight of the air inside the balloon is less because of the higher temperature. The difference must equal the weight of the passenger and equipment.

Thanks for the reply Dan. I've been trying for hours to figure out what your explanation means, but I can't figure out how to find the weight of the air inside or outside of the baloon.

notorious big said:
Thanks for the reply Dan. I've been trying for hours to figure out what your explanation means, but I can't figure out how to find the weight of the air inside or outside of the baloon.

PV = nRT

P and V are constant assuming the balloon is fully expanded and it stays at about the same altitude. Let T1 be the temperature of the air in the balloon and T2 be the temperature of the outside air, with n1/V = number density of molecules inside and n2/V = number density of molecules outside.

P = (n1/V)RT1 = (n2/V)RT2

(n1/V)/(n2/V) = T2/T1

The ratio of the number densities is the inverse of the ratio of the temperatures. The ratio of the number densities is the same as the ratio of the mass densities because it is the same air, just heated up. The mass density of the outside air is given, and the volume of the balloon is given. From that you can calculate the weight of the air displaced by the balloon. The weight of the air inside must be less than the outside by the amount of the buoyant force. You can find the weight of the inside air as a function of temperature using the density ratio and the outside temperature (STP means standard temperature and pressure 0C and 1 atmosphere). Since you know how much it has to weigh, you can calculate the temperature it must have

W2 = outside air weight = D2(g)V; D2 = given density
W1 = inside air weight = D1(g)V; D1 = hot air density

D1/D2 = T2/T1
D1 = D2*T2/T1

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What is the "Hot Air Balloon Problem"?

The "Hot Air Balloon Problem" is a mathematical puzzle that involves finding the optimal path for a hot air balloon to travel from point A to point B, taking into account wind direction and speed.

What makes the "Hot Air Balloon Problem" challenging?

The "Hot Air Balloon Problem" is challenging because it requires complex calculations to determine the best route for the balloon to take. It also involves considering different variables such as wind speed and direction, which can change throughout the journey.

What are some real-world applications for the "Hot Air Balloon Problem"?

The "Hot Air Balloon Problem" has practical applications in fields such as aviation and meteorology. It can be used to calculate the most efficient flight path for a plane or the trajectory of a weather balloon.

What are some strategies for solving the "Hot Air Balloon Problem"?

There are various strategies for solving the "Hot Air Balloon Problem", including using mathematical equations and computer algorithms. Some approaches involve breaking the journey into smaller segments and calculating the optimal path for each segment.

What are some limitations of the "Hot Air Balloon Problem"?

The "Hot Air Balloon Problem" is a simplified model and does not take into account all variables, such as air density and temperature. It also assumes a perfectly spherical Earth, which may not be accurate for longer journeys. Additionally, unexpected events like storms or turbulence can make the calculations less reliable.