|New Mexico Supercomputing Challenge|
Challenge Team Interim Report
Our project uses two primary equations as the basis for analyzing the aftermath of one-dimensional and two-dimensional collisions. These equations are:
These two equations apply to a collision given two conditions. First, the collision is perfectly elastic and second, there are no outside forces acting on the system. Using these two equations and applying the principle of systems of equations, two variables can be solved for. If the mass of both the objects and their initial velocities are known, the final velocities can be solved for. Two-dimensional equations are quite a bit more complex because they involve angles, which require the use of trigonometry. However, using the Pythagorean Theorem and other trigonometric identities, the resulting angles of the two objects can be determined.
The compilation of the computer program requires a solid understanding of the math and physics involved in a collision. The program is designed to allow the user to input the masses and initial velocities of the two objects involved in the collision. The user also inputs the incoming angle of the objects if the collision is two-dimensional. The program will calculate and output the resulting velocities and angles of the two objects. After solving the two equations in terms of one variable, we discovered that the form resembled that of a quadratic equation: ax^2 + bx + c = 0. However, our equation looked like this: (m1m2 + m1^2)v1'^2 - (2m1^2v1 + 2m1m2v2)v1' + (m1v1^2m2 - m1^2m2^2 - 2m1v1m2v2) = 0, where v1' is the only variable and m1, m2, v1, and v2 are constants entered by the user. Using the quadratic formula, we were able to solve for the final velocity of the first object (v1'), and then substitute in to an earlier equation to find the final velocity of the second object (v2'). That was for a one-dimensional equation. For a two-dimensional equation, we modified the equations for conservation of momentum and energy to accommodate angles (using sine and cosine). We then re-solved the equations, and came up with an even longer and more complicated quadratic equation. The next step in our program is to calculate the results of a non-elastic or partial elastic collision. After these steps have been completed, we can modify our program to model the collisions and resulting courses of two objects in a bounded, two-dimensional plane.
In our model, the objects (of any mass) would be fired into a contained, frictionless region. The elasticity constants of the objects and the walls will affect the courses and velocities of the objects. The program will determine when and if the objects ever collide, and what the results of the collisions will be.
Andrew Webb, Nick Henry, Anthony Eastin
For questions about the Supercomputing Challenge, a 501(c)3 organization, contact us at: consult1516 @ supercomputingchallenge.org
New Mexico Supercomputing Challenge, Inc.
80 Cascabel Street
Los Alamos, New Mexico 87544