Pixar created one of the most popular movies for people of all ages in 2009. This film, titled Up, is a computer-animated adventure film which revolves around an elderly man named Carl Fredricksen and a young boy scout named Russell. The film follows the adventures of these two after 78-year-old Carl attempts to fly to South America simply by tying thousands of helium filled balloons to his home. But how realistic is this story? We can use the physics of buoyancy to try to gain a better understanding of just how many balloons Carl would require to lift his home off of the ground.
The photo above can help to provide an image of the situation. Ignoring air resistance, there are two general forces at work in this scenario. The force keeping Carl’s house grounded is the force due to gravity (FG). This force is simply the mass of the house and all of its contents multiplied by the acceleration due to gravity. The other force at work is the buoyant force (FB) of the helium balloons. This force can be calculated by multiplying the difference in density of air and helium by the volume of the balloon and the acceleration due to gravity. The calculations below include the calculation of the force due to gravity and the buoyant force of one helium balloon. By using the equation FG=(FB)*(number of balloons), we can solve for the number of balloons needed to generate enough force to lift Carl’s home off the ground.
Calculation for force due to gravity:
Assuming Carl’s house has a mass of 54,500. kg
FG=ma à FG=(54,500 kg) x (9.81 m/s2)= 534,645 N à 5.35x105 N
Calculation for buoyant force of one helium balloon:
Assuming the helium balloon has a diameter of 0.3 meters, and that the density of air is 1.225 kg/m3, and that the density of helium is 0.1786 kg/m3.
VBalloon= (4/3)pi(r)3 = (4/3)x(pi)x(0.15 meters)3 = 0.014 m3
FB=(density of air – density of helium) x (V of balloon) x (acceleration from gravity)
FB=(1.225kg/m3 – 0.1786kg/m3) x (0.014137 m3) x (9.8 m/s2) = 0.145 N
Calculation for number of balloons needed:
FG=FB x number of balloons
534,645 N = (0.145 N ) x n à number of balloons = 3,687,207 = 3.69 x 106 balloons would be needed!
As you can see from the calculations above, Carl would need to purchase almost four million balloons in order for the buoyant force of his balloon bouquet to overcome the force of gravity on the house. It seems that Carl may have been better off buying a plane ticket.