# Hard Disk Collision

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Conservation laws play a fundamental role in physics and most physicists would argue that they are more fundamental then Newton's Laws. In the Hard Disk Collision model, conservation of energy and momentum are used to compute the trajectories of two balls aced on by a strong but brief collision force. The hard disk (impulse) approximation assumes that the interaction acts instantaneously along a line connecting the particle centers when the the particles come into contact. Because the particles do not interact except for this brief instant, we need only consider kinetic energy when computing the total energy of the system.

Etotal = (1/2) m1 v12 + (1/2) m2 v22

The total momentum of the system is also easy to compute and is the sum of the momentum of the two particles.

ptotal = m1 v1 + m2 v2

The bold typeface indicates that momentum and velocity are vectors so the above equation becomes to two equations when written xy component form. Newtonian mechanics predicts that for an isolated system total momentum is always conserved and total energy is conserved if the particles do not have hidden degrees of freedom such as spin or heat capacity. Collisions that conserve energy are said to be totally elastic because the balls bounce off of each other.

The analysis of a physics problem can be difficult or easy depending on the frame of reference. Satisfying the conditions of conservation of energy and momentum leads to a system of coupled linear and quadratic equations that is an algebraic mess. However, the collision dynamics becomes trivial if this same collision is viewed from a reference frame that is moving with the center of mass of the particles. The particles meet at the center of mass and reverse direction. Moving to the center of mass reference frame does not changed the physics, only the mathematics that is needed to solve the problem has changed. The center of mass is shown a a cross in the simulation.

Note: The model shown assumes equal masses. Because this Launcher package is a tutorial, readers are asked to modify the model to include arbitrary values of m1 and m2.

## References:

The Free Fall in Cartesian Coordinates model is designed to teach Ejs modeling. Right click within the simulation to examine this model in the Ejs modeling and authoring tool. See:

- "Modeling Physics with Easy Java Simulations" by Wolfgang Christian and Francisco Esquembre, The Physics Teacher, November 2007, 45 (8), pp. 475-480.

The Easy Java Simulations (EJS) documentation can be downloaded from the ComPADRE Open Source Physics collection and from the Ejs website.

### Note:

This simulation was created by Wolfgang Christian using the Easy Java Simulations (Ejs) modeling tool. You can examine and modify this simulation if you have Ejs installed by right-clicking within a plot and selecting "Open Ejs Model" from the pop-up menu. Information about Ejs is available at: <http://www.um.es/fem/Ejs/>.