Dot Physics

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.

i-d75b4b46d5ec1617dc68e0ae22387771-up-balloon-1.jpg

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:

i-bcea1568c4dbdda588b1c76fdc86ec74-balloon-house-diagram.jpg

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:

i-bdc91755f0fe4e2400f78a097de38324-f-bouyancy-equation.jpg

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:

i-8f3b82f15d12ef720b3c0c0c20448ee0-gps-throw.jpg

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/s2. In this case, you will see that is not necessary. Here is a plot of the horizontal position as a function of time.

i-73a61176d8af6b291706676d86558950-x-video-data.jpg

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.

i-4defd025aa357abffe6e2d77a77acdc3-y-data-video.jpg

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:

i-8c61dc65068c19488808e29f09b8ec73-kinematic-example.jpg

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/s2) where U is the distance unit in the video. I will assume the time step between frames is correct. This gives:

i-6c1ac164af1b7a1c98c3f88895525051-1-u-in-meters.jpg

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.

Comments

  1. #1 Gina
    June 3, 2009

    Slightly more details from the movie:
    The old man goes back inside the house just before it takes off.
    There are many empty helium tanks on the front lawn, so it’s not clear where he filled them.
    The shot returns to the house which is breaking away from it’s various moorings (a whole other problem).
    Later, you see that the balloons are tethered to the fireplace grate through the chimney.
    Finally, regarding cost and experience, the old man has spent his life in the noble career of selling helium balloons to tourists, so maybe he had wholesaler’s pricing.

  2. #2 Guy Srinivasan
    June 3, 2009

    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?

    You could, of course, but the goal is not in fact realistic physics in either the GPS case or the hair case. The goal is tricking the audience’s brains into a) loving the movie during* and after the movie, and b) having memories after the movie of the movie looking realistic.

    *maybe

  3. #3 Steven Peters
    June 3, 2009

    I haven’t seen the movie yet either, but I wouldn’t be too upset with them for muddying the physics a bit, since it is a cartoon after all. Road runner can run across gaps that make the coyote fall, and run into tunnels painted on a wall while the coyote hits the wall. I don’t think we need to hold cartoons that responsible.

  4. #4 sylvia martinez
    June 3, 2009

    Sometimes real physics is boring looking. If you’ve ever seen an actor fall down vs. a real person falling down you get the point. Willy E. Coyote falling into the canyon is funny because of the completely wrong physics. The delay creates anticipation and delivers the punch line.

    When they make games or movies they model things because it makes it easier to get close to realism, but then the designers back off and undo the reality to add impact, suspense or other drama. They wouldn’t bother modeling a single thing falling, it’s cheaper to do it by hand. But for hair, it’s easier and cheaper to use mathematically accurate models. But I’m sure in some scenes they modified the final output by hand if the hair looked “funny”.

  5. #5 Rhett
    June 3, 2009

    @Steven,

    Speaking of road runner reminded me of this awesome site – http://www.lghs.net/teachers/science/burns/rrphysics/Road_Runner_Physics/Movie.html

  6. #6 Chad
    June 3, 2009

    Movie was great. During a few particular scenes I questioned why the characters were doing something with the house, but I won’t bring it up yet for the sake of not posting spoilers yet.

  7. #7 Anonymous Coward
    June 4, 2009

    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.

    With regard to the vertical motion: only with a bad teacher. Perhaps the movie (which I haven’t seen) is taking place on a different planet, where the gravitational constant at the planet’s surface is smaller than on earth? The student could use estimates of the house size to figure out g on that planet. Perhaps the lower value of g on that planet might also explain how the house could be floated with such a paltry volume of balloons.

    It would be a great way of distinguishing things that are fundamental and universal (Newton’s laws) from things that are specific to the surface of the earth (little g, the buoyancy of air).

    With regard to the horizontal position: this is a bit trickier to reconcile. Maybe the kid throws a mean curve ball?. Or the air is quite dense on this planet, explaining both the rapid damping of the GPS motion (you’d need to refit the vertical motion including damping to see if this is a consistent explation) and the high buoyancy of the balloons.

  8. #8 Jon H
    December 29, 2009

    “The teacher says “hey that movie Up was awesome, lets do some video analysis of that GPS out the window.” “

    Why, is Up the first movie ever made where something is thrown? There are probably millions of hours of easily accessible video footage of projectiles more interesting than a thrown cartoon GPS.

    It’s not a simulation, it’s art. If it’s not ‘correct’, it’s because they modified the path for aesthetic reasons.

    Cartoons also exaggerate the elastic deformation of objects when they collide. Because it’s funny. Big deal.

  9. #9 Lauren
    April 15, 2010

    Actually, Pixar did have engineers figure the math for just how many balloons it would take to hold up the house in the film. The number was so astronomical, it would have looked ridiculous and been much more difficult to animate. So, they consciously scrapped the notion of accuracy regarding the hot air balloon and went with a number that looks better on film.

  10. #10 hobo bob
    August 11, 2010

    more importantly, ITS A FUCKING MOVIE!

  11. #11 C.Lee
    October 10, 2010

    I think maybe you should consider a position with pixar for doing these types of “calculations” for their films!
    In case you didn’t know, they are located in Emeryville, CA…quite close to Berkeley and San Francisco… ;-) Pretty nice areas if you ask me…. but then again I am biased since I live there! LOL

  12. #12 Becky Costello
    April 13, 2011

    Dear Rhett,
    I found you while googling the topic of “the real up movie house”, because I had read an article about someone who really attempted to preform this. I was fascinated at first by all the thought & calculating you must have done for this piece. Then I remembered the “Analytical Man” I am married to…LoL!! I am a very Artsy Fartsy person & studied 16th Century Glass in College so I live on the other side of the mind. I know so many times he & I will be having the same experience & yet see it so very differently. All I can say is I pray that when you finally saw the movie, you saw it with the heart of a 6 six year old. Saw the adventure, the lesson of the wisdom of an older generation, the lesson of learning to love again after such terrible loss, the lesson of good verses evil & mostly the humor throughout the whole experience. Sometimes that is life, a house that could in no way be picked up & carried off yet it is….. As far as whether “mathematically is it possible~~~ well in the famous words of someone else “Frankly Scarlett, I don’t give a…Oh never-mind that is your line…..Hugz!!

  13. #13 Becky Costello
    April 13, 2011

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