# Hot air balloon buoyancy problem

• Jpyhsics
In summary: And neglecting the fact that for the balloon to deflect upwards, some volume of the surrounding air must deflect downward?
Jpyhsics
Homework Statement
A hot-air balloon is initially floating (stationary) a few metres above the ground. The radius of the spherical balloon is 6.25 m, and the density of surrounding air is 1.29 kg m-3. "Ballast" is weight of negligible volume that can be dropped to make the balloon rise. How much weight would you have to drop overboard to make the balloon rise 105 m in 15.0 s? You may assume that the density of surrounding air remains constant.
Relevant Equations
Archimedes principle: Fb=rho*gV
Vsphere= 4/3*pi*r^3
Initially the hot air balloon is stationary so...
FB=Fg
ρgV=mg
m=1319.2...kg
FB=Fg=12941N

In the air...
a=2d/t^2=0.933...m/s^2
Fnet=FB-Fg
(1319-x)(0.93...)=12941-(1319-x)g
x=114.6 kg

but apparently this is wrong?...

Jpyhsics said:
Problem Statement: A hot-air balloon is initially floating (stationary) a few metres above the ground. The radius of the spherical balloon is 6.25 m, and the density of surrounding air is 1.29 kg m-3. "Ballast" is weight of negligible volume that can be dropped to make the balloon rise. How much weight would you have to drop overboard to make the balloon rise 105 m in 15.0 s? You may assume that the density of surrounding air remains constant.
Relevant Equations: Archimedes principle: Fb=rho*gV
Vsphere= 4/3*pi*r^3

Initially the hot air balloon is stationary so...
FB=Fg
ρgV=mg
m=1319.2...kg
FB=Fg=12941N

In the air...
a=2d/t^2=0.933...m/s^2
Fnet=FB-Fg
(1319-x)(0.93...)=12941-(1319-x)g
x=114.6 kg

but apparently this is wrong?...
Why do you think it is wrong?

I didn't calculate using the given values, but equations you have used seems to be correct.
Hint: your solution would be more readable if you work only with symbolic names for the quantities throughout your calculation, and put in the given values in the end.

Jpyhsics said:
but apparently this is wrong?...
Do you know what the answer is claimed to be?

ehild said:
Why do you think it is wrong?
I was marked wrong by the system.

haruspex said:
Do you know what the answer is claimed to be?

Jpyhsics said:
I was marked wrong by the system.
Try to enter the result with three digits .

ehild said:
Try to enter the result with three digits .
So I figured out that they were asking for weight and not mass, but thanks everyone.

The answer was supposed to be 1124 N.

Jpyhsics said:
So I figured out that they were asking for weight and not mass, but thanks everyone.
Yes, it was weight... It is easy to overlook.

The stated problem seems cringe-worthy. We are expected to have a 12 meter diameter balloon rising at an eventual rate of 14 meters per second under a force of a mere 1124 N while neglecting air resistance?

And neglecting the fact that for the balloon to deflect upwards, some volume of the surrounding air must deflect downward?

12 meter diameter is similar to that of a deployed parachute and 1124N is similar to the weight of a skydiver. The computed result might be good to one significant digit. Four is ludicrous.

## 1. What is the hot air balloon buoyancy problem?

The hot air balloon buoyancy problem refers to the challenge of determining the amount of hot air needed to lift a hot air balloon off the ground and keep it at a desired altitude.

## 2. How does hot air affect the buoyancy of a balloon?

Hot air is less dense than cold air, so when the air inside the balloon is heated, it becomes lighter and creates a difference in pressure between the inside and outside of the balloon. This difference in pressure causes the balloon to rise.

## 3. What factors affect the buoyancy of a hot air balloon?

The buoyancy of a hot air balloon is affected by several factors, including the temperature of the air inside the balloon, the volume of the balloon, the weight of the balloon and its contents, and the surrounding atmospheric conditions.

## 4. How do you calculate the amount of hot air needed for a hot air balloon?

The amount of hot air needed for a hot air balloon can be calculated using the ideal gas law, which takes into account the temperature, volume, and pressure of the gas inside the balloon. Other factors such as wind speed and altitude may also need to be considered.

## 5. Can the hot air balloon buoyancy problem be solved using different gases?

Yes, the hot air balloon buoyancy problem can be solved using gases other than hot air, such as helium or hydrogen. However, these gases have different properties and may require different calculations to determine the appropriate amount needed for buoyancy.

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