We are all familiar with the scene from Up, a few balloons, a little helium and the small, idyllic, house is freed from its conventional constraints and floats off into the sky…
Could this scene be recreated without the somewhat questionable Pixar physics? And, how many balloons would you actually need to set your own house free? The first thing we should determine is the relationship between the mass of the house and the lifting power of the balloons.
The total mass of the things being lifted must be less than the total lifting power. We can expand this relationship further to include the mass of the individual balloons and strings attaching the balloons to the house.
Where n (positive integer) is the total number of balloons.
We can rearrange this to make the mass of the house the subject.
To simplify this inequality to give us the relationship between the number of balloons and the mass of the house we should calculate the lifting power, the mass of a balloon and the mass of the string.
An averaged sized balloon of diameter 30cm has a volume of 4.5×10⁻³π m³. The air density at room temperature and pressure is 1.20kgm⁻³, and the density of helium under the same conditions is 0.176kgm⁻³ this gives an overall lifting power of:
Lifting Power of 1 balloon: 0.0145kg
After a quick google, we can find the mass of 1 balloon.
Mass of 1 balloon: 0.0017kg
The mass of the individual strings connecting the balloons to the house depends on the total number of balloons as average string length varies with the size of the balloon cluster.
We can substitute values in and express this relationship in terms of n as follows.
We can use this all to make a single equation, describing the relationship between the mass of the house and the total number of balloons.
We can plot this relationship graphically to find where the maximum mass of the house lies.
The graph shows that there is only a small feasible region, the maximum mass of the house is 8049kg which would require 2,520,162 balloons. A small brick house weighs around 35 tonnes so it is probably not possible to lift your house with balloons. However, it would be plausible to lift a small wooden house (much like a shed) similar to the one in Up.
Something else we should look at is the cost of the helium require to lift the building, enough helium to fill 2 and a half million balloons will not be cheap, the US governments regulate the price of helium, charging $3.03/cubic meter giving us a total cost to fill those balloons at $34,000 which is rather expensive for an unreliable transportation method.
In conclusion, the scene in the film is possible provided that the house is empty that it weighs less than 8 tonnes and that you have enough money to buy 11,000 cubic meters of helium.