21 December 2011

How many soda bottles in Brian's raft?

The question from Dec. 14:

Mr. Jacobs’ friend Brian Jackson saved two-liter soda bottles throughout his senior year of college.  During “Haverfest," he duct taped the bottles together to form a raft.  He then successfully floated himself out onto the duck pond.

Estimate how many bottles Brian used.  Explain your reasoning thoroughly and show all calculations for full credit.

While the majority does not always rule in physics, in this case "they" were right on.  My reasoning:

Call Brian 80 kg or so.  His weight is then 800 N.  That weight must be supported by the buoyant force, which is equal to the density of water times the displaced volume times g.  If each bottle is fully submerged, it displaces 2 L, or 0.002 cubic meters.  The buoyant force created by one bottle is then (1000 kg/m^3)(0.002 m^3)(10 N/kg) = 20 N.  To get to 800 N at 20 N per bottle, you'd need about 40 bottles.

What if Brian's not 80 kg?  Well, as I have to point out to people, 80 kg is a reasonable estimate for Brian, but college guys who drink soda are often heavier; and, in a recent development of Haverlore, I have discovered that Brian supported a second person on the raft as well.  Furthermore, even if Brian were 75 kg, 40 bottles would have to be nearly fully submerged, leaving essentially no safety margin, and getting Brian's feet* wet.  This is an order of magnitude estimate... why not double the estimate to 80 bottles or so?  Then the bottles are in the neighborhood of halfway underwater.  Brian can sit dry, he can bring a friend, he can eat at the COOP** all he wants; 80 bottles will support him.

* Or more likely, his tuckus
** The yummy snack bar... it used to be too expensive for me, but now I find out that students can make their parents pay for the COOP as part of these newfangled meal plans.  Ach, and nowadays students can access email from their rooms, too.

What about the weight of the soda bottles themselves?  Some students will tell me "the soda bottles are of negligible weight."  Okay, but are they?  What's the evidence?  

Some students found that an empty bottle has mass about 40-50 g, for a weight of about half a newton or so. That means that each bottle will only support 19.5 N of Brian rather than the 20 N previously conjectured.  

Does that mean, as some say, that the proper answer is "41 bottles?"  No, certainly not.  As discussed in the previous paragraph, the uncertainty in Brian's mass, and in just how much of the raft is submerged, far outweighs this 3% change due to the weight of the bottle.  The answer is still somewhere around 80 bottles, or better yet, some dozens of bottles. 

1 comment:

  1. Yes but I would have expected students to get the buoyancy faster by just looking at the mass of displaced water. A liter of water is a kg, so each bottle displaced about 2kg, so 40-50 bottles need to be fully submerged. Generally raft builders like to have another dry layer on top, so that any crushing of bottles doesn't reduce floation, so 100 bottles are a good number to estimate.