zerasaw wrote:Yeah I was going to do a quadratic but I plotted them and it was pretty dang planar. I was able to derive the equation by taking the differential vectors of 2 points on the plane and calculating their cross product and substituting into the general equation of a plane.
6th place at regionals beating a Mentor and Solon team; they completely screwed up the height and weight of the bottle. I had the equation to 1cm accuracy the day before, plugged in the numbers and 40cm off the ground...but if 28cm gets you 6th place then you know something's wrong with the event.
I will say this. While the bungee drop equations are really, really planar, with planar models matching theoretically perfect models with a correlation of .99, there is still some error. The x1 column indicates mass, the x2 column indicates Drop Height, and the Y value indicates theoretically perfect length of bungee cord that one should measure for the distance for a partially elastic bungee cord. The "Predicted Y Values" indicates the Y values that would be given based on a multilinear regressional equation based off all of these theoretically ideal measurements. If a planar equation was really perfectly accurate, the "Predicted Y Values" would perfectly, if not almost, perfectly match the Y values columns. And that is mostly the case. Most of the "Predicted Y Values" are off the "Perfect Y Values" by roughly 2 cm. However, the one that is noticeable off is the .3 kg 2 meter one, which is off by 10 cm (note that this is also an issue in partially elastic bungee's where a .3 kg 2 meter drop usually forces a measurement into the elastic portion and causes a different "k" value). The R^2 value is .99, indicating still a relatively strong correlation. However, it is still a bit off from ideal. However, this also does not mean that the bungee equations are perfect. They are also approximations which are only valid within certain intervals, before the bungee cord starts to stray from the Hooke's Law equations.
(Note, the same comparison between ideal bungee equations and 3 dimensional analysis was done for the entire bungee equations, and, in fact, that yielded an ever lower correlation to the ideal bungee equation for entirely elastic bungee's with a correlation of .98 rather than .99. While this may seem ok, the deviation from ideal ranged between 2 cm to up to 20 cm. For example, the ideal bungee equation yielded the ideal measured bungee length for .1 kg 5 meters with a Young's Modulus of 10 being 3.22 meters, however the 3 dimensional regressional planar equation yielded a predicted y value of 3.06, suggesting that while 3 dimensional planar equations are accurate, they still are not accurate enough.
