Ball 1 Stephen Ball Lab Partner: Gustavo Larramendi Bungee Report 6 November 2014 Exploring the Effect of Bungee Length Manipulation on Acceleration Applied to Jumper Introduction In order to experience a thrilling bungee jump, a jumper should experience as much time in free fall as possible. This parameter may be analyzed via the jumper s maximum deceleration during the jump; a higher amount of deceleration means the jumper was slowed down over a shorter distance, allowing for more relative time in free fall. However, the parameters of the Bungee Challenge dictate that the jumper may experience no more than three times acceleration due to gravity (3 X 9.81 m/s 2, or 29.43 m/s 2 ). In this experiment, we examined the relationship between bungee cord length and the maximum acceleration experienced by the jumper. This connection relates to Newton s second law of F total = mass X acceleration; acceleration is greatest when maximum force is applied, so maximum acceleration in a bungee system can be modeled via equation 1: F TOTAL = m X a, so a = F total /m F TOTAL = F bungee,max - mg Equation 1 : a max = (F b,max /m)-g Methods We approached the posed question with the following set up, outlined in Fig. 1. First, the analog force sensor was hung at a height of 2.14 m on a stand clamped to a lab table; the
Ball 2 sensor was calibrated to tare for the mass of the hanging portion of bungee cord (2.00 m). Using the Capstone program, we changed the sensor s Gain Property to 10X for an improved signal to noise ratio, then changed its Sample Rate to 2.00 khz. A tape measure was then hung from the same stand with its zero mark exactly lined up with the bottom of the force sensor s hook (henceforth referred to as the zero point). We decided to test the bungee length jumper acceleration relationship over the course of five bungee length trial groups, with 5 trials in each group. L refers to the bungee length, including the loops tied on either end; for this experiment, our 5 trial Figure 1: Set-up parameters. groups are defined by L values of.11 m,.36 m,.66 m,.79 m, and.99 m. All loop lengths were kept under.02 m to minimize related interference. For each trial, one loop of the bungee cord was hung from the force sensor, while the hanging mass (henceforth referred to as the jumper ) was hung from the bottom bungee loop. We started recording the force sensor readings in Capstone, and then dropped the jumper from the zero point; all jumpers were dropped with the bottom of the bungee loop lined up with this zero point. Force recording was stopped after the first rebound; we determined the maximum upward force (F b,max ) by highlighting the entire data set and turning on the Maximum display within Capstone.
F b,max (N) Ball 3 Results Trial Group L F b,max a max (m, ±.01m) (N, ±.001N) (m/s 2, ±.01m/s 2 ) 1 0.11 1.574 21.67 2 0.36 1.540 20.99 3 0.66 1.508 20.35 4 0.79 1.472 19.63 5 0.99 1.448 19.15 We first present our results for the a max L relationship, which we found to have a negative linear correlation. In other words, as bungee length increased, the maximum acceleration experienced by the jumper decreased by 2.88 ±.21 m/s 2 for every additional meter of L. This relationship is depicted graphically in Fig. 3, a max (m/s 2 ) 22 21.5 21 20.5 20 19.5 along with equation 2 that describes the relationship. 19 Figure 2: Results Data Table. Table exhibits relevant bungee length, force exerted by the bungee cord, and the maximum acceleration experienced by the jumper. a max vs. L a max = -2.8795L + 22.033 0 0.2 0.4 0.6 0.8 1 1.2 L (m) Figure 3: a max vs. L. Graphical representation of the relationship between maximum acceleration experienced by the jumper and bungee cord length. Equation 2: a max = -2.8795L + 22.033 1.59 1.54 1.49 1.44 F b,max vs. L F b = -0.144L + 1.5922 0 0.2 0.4 0.6 0.8 1 1.2 L(m) A similar correlation exists for the F b,max L relationship. As bungee length increased, F b experienced by the jumper decreased by 0.144 ±.01 N for every additional meter of L. Figure 4:F b vs. L. Graphical representation of the relationship between F b,max and bungee cord length.
Ball 4 Fig. 4 depicts this association graphically, and equation 3 describes the relationship. Equation 3: F b,max = -0.144L + 1.5922 The uncertainty for the relationships given in both equations 2 and 3 is 7% based on an Excel linear regression. Discussion We should first emphasize that this data and our equations may only be applied to a system with a.050 kg jumper. However, our work shows that this small scale experiment with any mass can produce equations that are able to predict maximum acceleration experienced by a jumper of said mass. There is one major condition that must be met, however; F b,max must be greater than m jumper X g in order to guarantee rebound of the system. Once that condition has been met, calculated maximum acceleration values can be used to determine if the parameters of the bungee challenge have been met. In order for this to be true, a max must be less than three times acceleration due to gravity (29.43 m/s 2 ) so as to prevent any damage to the egg. Our a max of 21.67 m/ s 2 in the.050 kg system was only 74% of this maximum allotted acceleration; therefore, further research with regards to acceleration should incorporate static cords and packaged bungees in order to figure out how to create the most thrilling experience. Our experimentally determined equations both had uncertainties of 7%, leaving room for improvement to this experiment as well. My hands were not a completely reliable release system for the jumper; I was constantly straining to make sure the bungee loop lined up with the zero point and that I released the jumper from a stand-still. This variance most likely contributed the most to uncertainty, so an automated drop system could provide slightly more
Ball 5 accurate data. Additionally, we recognize that we may not be able to extrapolate acceleration rates for much higher L values. While a linear model works quite well for explaining the acceleration variance for L from.1 to 1 m (R 2 =.985), the relationship may change for much higher bungee lengths. Conclusion Our experiment demonstrates that one of the main parameters for the Bungee Challenge can easily be checked using small scale force sensor trials. Maximum force readings may be used to calculate the maximum acceleration of the egg, which can be compared against the 3 X g threshold to ensure safe acceleration rates. Seeing as larger bungee cords lead to smaller acceleration rates, the results of this experiment indicate that further research should be done regarding the effects of incorporating static cords and bungee bundling on overall acceleration. Keeping all of these factors in mind, we recommend that Bungee Challenge participants use Newton s second law and its derived experimental equations to determine a thrilling yet safe bungee cord length.