The Physics of the Movie UP

The movie Up took a thought that most kids had and made it a reality: could you use balloons to make yourself fly in the sky? In the movie, Carl Fredrickson is being forced to move out of his house due to a company buying the land it is on. So instead of moving himself, he moves the whole house, flying off to South America just as he and his wife once dreamed. According to Pixar animators, there were 20,622 balloons used in the scenes of the house flying. But how many helium filled balloons would it take to actually make the house fly?

First, we have to determine the mass of the house and through this the force that gravity exerts on it. The average weight for a two-story house is between 36287 kg and 72574 kg. Since Carl's house looks to be on the smaller side, I have chosen to mass the house at 37000 kg. Using F=ma, multiplying 37000 kg by the acceleration of gravity which is 9.8 m/s^2 the gravitational force on the house ends up to be 362600 N. I am going to ignore foundation since the house does not fly with its foundation, as well as the force it took to break free of its foundation.

To calculate the upwards force of a helium balloon, I found the average volume of a carnival helium balloon is 0.028 m^3, as the balloons in the movie appear to be this type. Using the buoyant force equation, F(buoyant) = pVg, I multiplied 1.2 kg/m^3 (density of the atmosphere) with 0.028m^3 and 9.8 m/s^2 and got 0.33 N. A helium-filled balloon of this size can hold up to 0.0014 kg, meaning it exerts an upward force of 0.014 N. Subtracting 0.33 N from the total force, 0.014 N, the downward force due to gravity from the balloon can be found to be 0.316 N.

Now knowing the upwards force a balloon exerts, the total number of balloons can be calculated. This can be found by dividing the downwards force of the house by the upwards force of a balloon. The total amount of balloons is calculated to be 25.9 million balloons, 1250 times more than the 20,622 balloons in the original scene.