I haven’t seen the Pixar Movie “Up” yet, so don’t spoil it for me. I have, however, seen the trailer. In my usual fashion, I have to find something to complain about. There is this scene where the old man releases balloons out of the house.

What is wrong with this scene? Also, would that be enough balloons to make the house float? Here is a shot of the balloons coming out of the house.

Ok, I was already wrong. The first time I saw this trailer I thought the balloons were stored in his house. After re-watching in slow motion, it seems the balloons were maybe in the back yard held down by some large tarps. This is better than what I originally thought. Oh well, let me answer the question even though it is wrong. What if he had the balloons in his house and then released them? Would that make the house float more? Here is a diagram:

So, would one of these float more than the other? What makes things float? I have talked about this in more detail in the Mythbusters and the lead balloon, so I will just say that there is a buoyancy force when objects displace air or a fluid. This buoyancy force can be calculated with Archimedes’ principle which states: The buoyancy force is equal to the weight of the fluid displaced. The easiest way to make sense of this is to think of some water floating in water. Of course water floats in water. For floating water, it’s weight has to be equal to it’s buoyant force. Now replace the floating water with a brick or something. The water outside the brick will have the exact same interactions that they did with the floating water. So the brick will have a buoyancy force equal to the weight of the water displaced. For a normal brick, this will not be enough to make it float, but there will still be a buoyant force on it. Mathematically, the buoyant force can be written as:

Ok, back to the UP house. What is being displaced? What is the mass of the object. It really is not as clear in this case. What is clear is the thing that is providing the buoyancy is the air. So, the buoyancy force is equal to the weight of the air displaced. What is displacing air? In this case, it is mostly the house, all the stuff in the house, the balloons and the helium in the balloons. In the two cases above, the volume of the air displaced does not change. This is because the balloons are in the air in the house. (Remember, I already said that I see that this NOT how it was shown in the movie). So, if you (somehow) had enough balloons to make your house fly and you put them IN your house, your house would float before you let them outside.

### How many balloons would you need to make the house float?

I realized while writing this that I am once again too slow. Others have already calculated this. First is the The Science and Entertainment Exchange. The other excellent coverage of this is from Wired.com. I think with both of these I will just describe how you could do this and leave it as a homework problem. You would need to estimate:

- The size of the house.
- The mass of the house (I would assume the whole house is like 10% wood and use the volume of the house and the density of wood – my first guess)
- The volume of the air displaced. Again use the density of wood above.
- The size of each balloon. A typical house hold balloon probably has a diameter of about 30-40 cm. You would also need to know the mass of the rubber in each balloon. This shouldn’t be too hard as you could get this from a deflated balloon (first guess 5 grams).
- The above could give you an estimated calculation for the buoyant force from each ballon plus its weight, or the net force from each balloon. You could just estimate this also.
- You would also need to estimate the amount of string needed, it would have a non-negligible weight.

Since I got “scooped” on my original investigation, I will give two bonus topics

### Why doesn’t the balloon house keep rising?

The reason the balloon reaches a certain height is that the buoyant force is not constant with altitude. As the balloon rises, the density of the air decreases. This has the effect of a lower buoyant force. At some point, the buoyant force and the weight are equal and the balloon no longer changes in altitude.

### When the boy throws the GPS out the window, is it modeled correctly?

I noticed this when I was watching the preview again. I realized that it was set up perfectly for video analysis of motion. So, here I go. Here is a screen shot of the scene in question:

To analyze this video, I used my favorite Tracker Video (it’s free and runs in Windows, Mac OS X and Linux). To set the scale I said the height of the house was 10. 10 what? I don’t know, but 10. It really doesn’t matter. I could estimate the scale by estimating the size of the house or by assuming the vertical acceleration is 9.8 m/s^{2}. In this case, you will see that is not necessary. Here is a plot of the horizontal position as a function of time.

This is a shot from Video Tracker’s built in analysis tools. They are really good, I should have been using these the whole time (I used to export the data to Vernier’s Logger Pro). I fit a function to this data, randomly choosing a parabolic fit. Is this ok? No. The horizontal position should be a straight line indicating constant velocity in the x-direction. Why would it behave this way? I don’t know. Air resistance would not be enough to make it behave this way unless it was really light. If you want to model this for a homework problem and estimate the mass the GPS would have to have to have a motion like this, let me know.

Now, here is the vertical position.

Again, I fit a quadratic function to the data. If the object is in free fall, the only significant force would be the gravitational force. This would give the object a constant vertical acceleration such that it would have a position as a function of time as:

Since the data seems to fit ok, I can assume an acceleration of g and use that to find the size of the house. If you need a refresher course on finding the acceleration from a position plot, check this out. Anyway, the function I fit to the vertical data says that the acceleration would be 2*(-1.662 U/s^{2}) where U is the distance unit in the video. I will assume the time step between frames is correct. This gives:

So, going back to the scaling, this would make the height of the house 29.5 meters or 96.79 feet. I don’t think so.

I can see it now. A high school class is finally learning kinematics and getting excited. The teacher says “hey that movie Up was awesome, lets do some video analysis of that GPS out the window.” You can imagine what happens next. We will have a generation of kids growing up not understanding kinematics.

If you can model the hairs on the head of a man in an animation, don’t you think you could use Newtonian mechanics to plot the position of the GPS? I don’t know, maybe it would have fallen too fast or something. Oh well.