Lab 3 & 4 Cratering Parts 1 and 2 - Notebook Template

Cratering (Part 1) Autumn 2023 Name: Marco Sanchez Name: Yuchan Ahn Name: Gabe Kinner Date: Oct 25, 2023 Course: 13100/2 Lab Section / TA Name: Joseph Noonan Initial observations As your group begins, what do you notice? What things will be important to keep in mind as you design and conduct the experiment? What (if anything) do you learn from the group discussion that informs how you will take data? - Important to keep in mind how heavy the balls are. - We notice sizable differences in the crater size with different masses and sizes of the balls dropped. - We decided to measure the masses of the balls and the diameters of the craters to cm and kg. Model 1: K=Mgh=U g= 9.8 m/s^2 How tall is the height? 1/2 meters Model 2: K proportional to volume needing to be moved Department of Physics, University of Chicago

Collecting data and plotting Record your data here, and make sure to plot as you go. Use this space also to record observations and thoughts, including details about your procedure (including pictures, if it would help) and how you are minimizing (and quantifying) your uncertainties. Mass of the balls: Ball 1 2 3 4 5 Mass (± 0.0001 kg) 0.0084 kg 0.0035 kg 0.0011 kg 0.0005 kg 0.0001 kg Expected Energy (J) 0.5*9.8*0.0084 =0.041 0.017 0.0054 0.0025 0.00049 Diameter of Craters Made by Different Impactors Diameter (m) ± 0. 001𝑚 Trial 0.0084 kg ball 0.0035 kg ball 0.0011 kg ball 0.0005 kg ball 0.0001 kg ball 1 .054 .044 .030 .024 .017 2 .055 .044 .030 .025 .017 3 .054 .043 .033 .027 .018 4 .055 .043 .030 .025 .018 5 .054 .043 .030 .024 .019 Avg . 0544 ± 0.0002 0.0434 ± 0.0002 0.0306± 0.0006 0.0250 ± 0.0005 0.0178 ± 0.0004 Department of Physics, University of Chicago

Department of Physics, University of Chicago

Procedure ● Mass of the ball bearings were taken using an electronic scale of resolution 0.1g. ● Height of the ball’s being dropped was measured from the sand to the max height of ejection, resulting in 0.5 meters. ● The sand was flattened using a waffle grated rake and tapping the box until the surface was level. ● Magnets were used to fix the ball bearing at a height of 0.5m and the ball bearing was released by lifting the magnet. ● Diameter of the crater was measured by using a compass so that its two legs touched the opposite ends of the crater. The compass was then put against a ruler (of resolution 0.1cm) to find the diameter. ● The ball bearing was then retrieved using a sieve. ● The procedure was repeated five times for each ball bearing. ● A light was used to accentuate the contrast between the crater and the surrounding sand. Department of Physics, University of Chicago

● Looking at the Xi 2 /v (reduced Xi 2 ), it gives us how well the model fits the data. V (degrees of freedom) is given by the number of data points subtracted from the number of fit parameters. A value of reduced chi-squared that is approx. 1 means that the model is a “good fit” for the data. A larger value than 1 means that either the model does not fit the data, or that the uncertainty calculated from the data is too low. If the value of the reduced chi-squared is significantly less than 1, then the data’s uncertainties are overestimated and/or more data is needed to test the model. ● Both of the models have one free parameter, if we increased the number of free parameters, it can lead to overfitting. This makes the model overly complex and fit noise in the data rather than the true underlying patterns. Above is the plot of the model plotted with the best fitting parameters A and B found with the python code. Department of Physics, University of Chicago

● Above is the plot of residuals for the two functions for each energy value. From this plot, we can infer that the y=Bx 1/4 is the better model because its residuals are smaller than that of y=Ax 1/3 for every point, meaning that the function is closer to every data point at every point thus being a better fit to the data. When looking at the reduced chi-squared we can also see that the y=Bx 1/4 model has a value much closer to 1 than the y=Ax 1/3 model, 6.88 to 83.51 respectively. ● There is no recognizable trend in the residuals, they seem to fluctuate unpredictably. ● Next, the python tested to fit a more general function such that both the exponent itself (x) and the constant (C) is a free parameter, increasing the free parameters in the model to two. And below is the result we found. Department of Physics, University of Chicago

● Fitting a more general function to the data, we found a new model which was similar to the previous model of y=Bx 1/4 but had a smaller reduced chi-squared value. Thus, by adding a new free parameter, we have found a better fitting model which does not overfit (which we know is true since it closely assembles the power ¼ model). BIG BALL DROP Mass of ball: 66g +- .05g Height: second floor landing (4.68 +- .03 m) and/or third floor landing (8.78 +- .03c m) d= B best K 1/4 . Predicted crater diameter: Potential energy= 3.06 J (second floor) d= 15.90 cm +- 0.04 cm PE= 5.75 J (third floor) d= 18.62 cm +- 0.05 cm Actual Second floor meteor diameter: 16 +- .5 cm Our |t’|= 0.20 Third floor meteor diameter: 18 +- .5 cm Our |t’|=1.2 Energy of Sedan and Chixulub crater found using our model Model used: d=12.0227KE 1/4 Diameter of Sedan crater: 390 m ⇒ 39000 𝑐𝑚 Energy of sedan crater: KE= (3900/12.0227) 4 =1.11*10 10 J The number we got is not in the same order of magnitude as the crater. The energy converted to heat would decrease the diameter of the crater so we would have to be smaller than that which we predicted, which means we would have gotten a smaller KE, which we did. Diameter of Chixulub crater: 150km ⇒ 1. 5 * 10 7 𝑐𝑚 Energy of Chixulub crater: KE=(1.5*10 7 /12.0227) 4 = 2.42*10^24 This value is within the range of the suggested kinetic energy of the impactor which is 3*10 23 and 6*10 25 J. It is in agreement, but it could have not been due to the angle the meteor was falling at. Department of Physics, University of Chicago