D. Experiment Summary

Transcription

1 Deriving Equations for a Dynamic Bungee Egg Drop Experiment D. Experiment Summary The purpose of this lab was to explore the dynamic aspect of a bungee as opposed to the static aspects. Our goal was to find a relationship between the equilibrium length of our bungee and the height from which our mass was dropped. In our experiment we actively sought out the minimum height from which varying masses were dropped so that they would not hit the ground. We used the same three equilibrium lengths and three masses from our first bungee lab experiment so that our numbers could compare well. We dropped each mass attached to each equilibrium length while changing the height until it came close to hitting the floor. This data is useful for determining a relationship between our bungee cord and the way it behaves as height and equilibrium length increase. Dropping the masses at each height without it hitting the ground allowed for a dynamic observation without using stop motion video. Because we have had issues with the camera in the past, we constructed this experiment where technology constraints would not interfere with our data. Our three trials produced linear trends when graphing the minimum height from the ground vs equilibrium length. We then took the slopes and y-intercepts from the three equations L = h , L = h , and L = h which correspond to the masses kg,.1180 kg, and.1322 kg respectively and graphed them against these masses. In doing this we can produce an equation that has slopes and intercepts that accurately relate to any given mass plugged into the equation. This produced the two equations Slope = (Mass) and Y-intercept = (Mass) With these models we can plug in a mass and find both slope and y-intercept to create a new model in y=mx+b form where y is EQ length and x is the height from which the mass is dropped. A linear regression analysis was ran for both models and the model for slope vs mass shows an uncertainty of (± m) and the model for y-intercept vs mass shows an uncertainty of (±.0395 m). The main sources of error were from our estimations of height; there could have been a more precise height that was right at the equilibrium of the ground and not hitting the ground. Another source of error was not having precise enough of measurements while raising the height and measuring our bungee length. Despite possible error, these equations are the most useful findings from our experiment because they allow us to directly calculate the EQ bungee length needed when dropping a mass (our egg) from a given height (the science center balcony). We will be ready to test our our findings when given the egg mass and drop height when prompted to during the next lab.

2 B. Experimental Diagram **typo above, EQ length should say full stretched length ** Figure 1: Diagram of Dynamic Bungee Experiment The diagram here shows how we ran each trial per equilibrium length. We held each mass at an estimated drop height which changed per mass per EQ length and then dropped it so that it just barely missed the ground. We measured drop height and EQ length with a tape measure and measured our masses with a scale. We wrapped our excess bungee around the hanging pole so that it would not interfere with the dropping mass.

3 C. Quantitative Data and Analysis Equilibrium Length m Equilibrium Length m Ground (m) Mass 1 (kg) (±0.0001) Mass 2 (kg) (±0.0001) Mass 3 (kg) (±0.0001) Ground (m) Mass 1 (kg) (±0.0001) Mass 2 (kg) (±0.0001) Mass 3 (kg) (±0.0001) Equilibrium Length m Ground (m) Mass 1 (kg) (±0.0001) Mass 2 (kg) (±0.0001) Mass 3 (kg) (±0.0001) Figure 2: Table of Experimental Values The data above shows the values gathered from our three trials. At each set equilibrium bungee length, we dropped 3 masses from various heights until we found the minimum height from which it could be dropped without reaching the ground. We used the same three mass values for each trial. EQ Length (m) EQ Length vs Minimum Height From Ground L = h L = h L = h Height (m) Linear (Mass kg) Linear (Mass kg) Linear (Mass kg) Figure 3: Graph of EQ Length vs. Minimum Height The trend lines here each correspond to the three specific masses used in our experiment. The linear relationship between length and height shows that the longer the bungee length, the higher you must drop a mass to prevent it from hitting the ground. If a height were given for the three masses shown, we would be able to find a corresponding bungee length that would allow us to drop the mass without it hitting the ground.

4 (±0.0001) Slope Y Intercept Figure 4: Table of Calculated Values These values correspond to the three equations derived from the data plotted in Figure 3 above. Separately graphing these slopes and y-intercepts in relation to mass allows us to find a relationship between our data and any given mass. Slope Value Slope vs Mass Slope = (mass) Figure 5: Graph of Calculated Slopes vs. Masses The linear relationship here shows that we can calculate a slope for a new equation that our data from figure 3 is tied to. With any given mass we would be able to find a slope that could be plugged into a general y=mx+b equation where y is our EQ length and x is our height from which the mass is dropped. Y-Intercept Value (m) Y-Intercept vs Mass Y-intercept = (mass) Figure 6: Graph of Calculated Y-Intercepts vs. Masses The linear relationship here shows that we can calculate a y-intercept for a new equation that our data from figure 3 is tied to. With any given mass we would be able to find a y-intercept that could be plugged into a general y=mx+b equation where y is our EQ length and x is our height from which the mass is dropped.

5 Experimental Values of Interest: The main experimental values of interest we found were the models for producing slope, Slope = (mass) , and y-intercept, Y-intercept = (mass) A linear regression analysis was run for both models and the model for slope vs mass shows an uncertainty of (± m) and the model for y-intercept vs mass shows an uncertainty of (±.0395 m). When plugging in any value for mass, these equations will produce a slope and y- intercept that can be paired together in a new y=mx+b structured equation where y is EQ length of bungee and x is height from which mass is dropped. Models: Slope = (mass) Uncertainty- ± (m) Y-intercept = (mass) Uncertainty- ±.0395 (m) Quantitative Error Analysis: The uncertainties from our models were found from our linear regression analysis. The values were ± (m) for the model for slope and ±.0395 (m) for the model for y-intercept. The main sources of error were from our estimations of height. We used sight and sound to estimate the best height from which the mass could be dropped without touching the ground. Because we only used subjective visual and audio cues, there could have been a more precise height that was right at the equilibrium of the ground and not hitting the ground. A way to solve this problem would have been recording the drop and looking at the distance between the mass and the ground at maximum bungee stretch. Another source of error was not having precise enough of measurements while raising the height and measuring our bungee length. At one point we had to hold the bungee by hand so that it was high enough to not hit the ground. This was imprecise and not constant compared to attaching it to a fixed rod. Having a fixed attachment to drop from would have solved this problem, but unfortunately we were limited in height. We did not have time to compare our model at the end of lab, but before the egg drop we will test our equations with a random mass in the lab. This will allow for a good comparison and check of how well our models represents the behavior of our bungee. Despite our small range of error, hopefully our model will check out before dropping the egg from the top of the science center balcony. On my honor, I have neither given nor received any unacknowledged aid on this assignment. Pledged,