MEIntroduction and Background Information What would happen if an asteroid the size of a school bus hit New York City? Would it affect citizens across the nation or be localized? What if the asteroid was the size of the Empire State Building, or an entire continent? Can an equation model the size of a crater or the damage caused by different sizes of celestial projectiles? While modeling asteroids may seem arbitrary, it could serve a very real purpose. If the damage caused by projectiles could be modelled by an equation, it might save lives in the unfortunate case of a dangerously large projectile hitting Earth. Meteorites, or projectiles that make it through the atmosphere and onto Earth’s surface, can pose a significant threat to large amounts of people depending on several factors including size. According to NASA, a projectile large enough to threaten life on Earth is very rare and only appears every few million years, but objects the size of a football field come along about every 2,000 years and cause significant damage (“Asteroid Fast Facts”). They also state that any projectile larger than two kilometers would cause worldwide damage (“Asteroid Fast Facts”). Therefore, it is important to understand how the size of a projectile affects the damage it causes when predicting damage caused by one of the many asteroids and other objects that have the potential to hit or come into contact with Earth. To investigate and model the impact of different sizes of space projectiles, I will be using the Impact: Earth! simulation from Purdue University. The simulation allows users to change several factors of the collision, including the diameter, density, impact, angle, distance, and velocity of the object. I will be testing how a variance in size with all other variables constant affects several outcomes of the impact, including crater size and energy produced. Method The simulation I am utilizing for this investigation allows users to change the size of a potential projectile, and correlates their diameter to familiar objects. The sizes range from a mere school bus to entire continents or planets. I will be testing four sizes: The Empire State Building (381 meters), the city of London (44.75 kilometers), a small planet (2390 km), and Asia (7535.82 km). These sizes should give me a wide enough range to acquire varying results at each diameter. With each different diameter, I will be running the simulation with the rest of my variables controlled: my density will remain at 1500 kilograms per meters cubed, which is consistent with porous rock, according to the simulation. The impact angle and velocity will stay at 45 degrees and 30 kilometers per second, respectively. I will keep the target area as sedimentary rock, and the distance from impact at 1000 kilometers. By keeping the variables other than size constant, I should be able to get a clear picture of how a change in size of the projectile affects Earth. After running the simulation with the differences in diameter of the projectiles, I will interpret the results in several categories. I will try to find the correlation between size of projectile and crater size, energy produced before atmospheric entry, and the percent of Earth’s mass melted or damaged. While I assume that the crater size will have a direct correlation with the size of the asteroid, it may not follow a linear pattern. I will need to model the correlation of energy produced and percent of the planet’s mass damaged in order to find an equation or pattern that they follow. I will model the effects by graphing them and finding the line of best fit, therefore finding an equation that could predict the crater size, energy produced, or damage to the Earth in the case of a celestial body coming into contact with Earth. Investigation and Results I modeled each crater diameter with otherwise constant variables in order to record my data. I then entered the data into Excel and created a chart for each variable. size of projectile (diameter in km)size of crater (diameter in km)0.3814.8644.75209239046507535.2811,400size of projectile (diameter in km)energy produced before atmospheric entry (Joules)0.3811.95 x 101944.753.17 x 102523904.82 x 10307535.281.51 x 1032size of projectile (diameter in km)percent of Earth’s mass damaged/melted0.381044.75023902.767535.2886.53It should be noted that the energy produced before atmospheric entry is in scientific notation. In addition, data involving the damage to Earth’s mass is represented in percentages – the first two projectile sizes are too small to cause any significant damage to Earth’s mass, rendering them non-applicable; I represented them with zeros for the sake of graphing and modeling the information later. I used the data from the tables to create the following three scatter plots of data.It should be noted again that the second graph is noted in scientific notation, which is formatted slightly differently in Excel. The number added to the decimal times E is the exponent to which ten is raised. In addition, it is important to know that the graphs are not directly comparable as they are all on different scales due to the use of different units. Based on the patterns of the scatterplot, I will consider the first graph (crater size) to follow a linear pattern with the other two graphs following an exponential pattern. In order to find the line of best fit for the first graph, I used a standard linear best fit equation with the points of (0.381, 4.86) and (7535.28, 11400). y2- y1x2- x1=slope11400-4.867535.28-.381= 11395.147534.619=1.512=slope I then plugged in the slope and the point (0.381, 4.86) to find the y-intercept.4.86=1.512(.381)+b4.86=.576+b4.284=bTherefore, the equation would be:y=1.512x+4.284However, this equation produces a logical fallacy in terms of the variable being investigated: A crater with a diameter of zero is not possible, and it would not create a crater measuring 4.284 km in diameter. Therefore, the variable may be modeled more accurately with an exponential equation where the y value for an x of zero would be zero. To see which equation would fit the scatter plot of the data most accurately, I used Microsoft Excel to create trendlines to see which best fit the data. In the following graphs, I applied an exponential (in green) and polynomial (in blue) trendline. However, Excel was unable to create an exponential trendline for the graph containing data on the mass of Earth melted or damaged; therefore, I only represented the data with a polynomial equation. From viewing these three graphs, it is evident that the polynomial trendline fits the data points from each scatterplot most accurately; however, I wanted to ensure that the line was accurate by calculating the R-squared value of each line. Using Excel, I found the R-squared value of each polynomial equation to be .999, 1, and 1, respectively. The two exponential equations have lower R-squared values, .6135 and .6154. These values indicate that the polynomial line is the most accurate fit for the data points generated by the simulation, as an R-squared value closer to 1 reflects a more accurate model.Conclusion Although I was unable to find a linear equation for best fit, I used Microsoft Excel and was successful in creating a polynomial trendline that fit the data almost perfectly, with R-squared values extremely close to 1. While this equation might not be used in an emergency situation, I enjoyed investigating it because I find asteroids and their effects on planets interesting. By using the simulation and modeling the data I got from it, I was able to see in more detail how destructive asteroids can be, even if the chances of them hitting Earth are extremely rare. By modelling the data using scatter plots and trendlines, I was able to explore an area of interest through mathematical processes. Works Cited”Asteroid Fast Facts.” NASA, 31 Mar. 2014, www.nasa.gov/mission_pages/asteroids/overview/fastfacts.html.Impact: Earth!, Purdue University, www.purdue.edu/impactearth/.