### Graphing Calculator Solution for Elastic Collisions

In an elastic collision, both the momentum and the energy are conserved. A graphing calculator can be used to find the solution for elastic collision problems.

Write the equation for the conservation of momentum of the particles. Store this equation into "y1." This equation will yield a line.

Write two equations for the conservation of kinetic energy of the particles. Store the positive expression into "y2" and the negative solution into "y3." These equations will yield a circle.

The "line" will intercept the "circle" in two places. The two intercepts are the solutions to the elastic collision problem. One intercept will be equal to the original velocities of each particle and the other intercept will be equal to the final velocities of each particle.

Let's work an example: A 5 kg ball moving east at 10 m/s collides elastically with a 7.5 kg ball moving west at 6 m/s. What is the velocity of each after the collision?

Write the expression for the conservation of momentum. Remember, the sum of the initial momentum equals the sum of the final momentum. Let v1 and v2 represent the final velocities of the 5 kg ball and the 7.5 kg ball respectively.

(5 kg)(10 m/s) + (7.5 kg)(-6 m/s) = (5 kg)v1 + (7.5 kg)v2
or, 5 = 5v1 + 7.5v2

Let v1 represent y and v2 represent x. This equation can now be substituted into "y1="

v1 = 1 - 1.5 v2
or, y1 = 1 - 1.5 x

Now write an expression for the conservation of energy of the particles. Remember, the sum of the initial kinetic energy equals the sum of the final kinetic energy. And, kinetic energy is found by the following formula:

Ek = 1/2 m v2

1/2[ (5 kg)(10 m/s)2 + (7.5 kg)(-6 m/s)2 = (5 kg)v12 + (7.5 kg)v22]

or, 154 = v12 + 1.5v22

Let v1 represent y and v2 represent x. The positive solution to the equation can now be substituted into "y2=" and the negative solution to the equation substituted into "y3="

v1 = + (154 - 1.5v22)1/2
and, v1 = - (154 - 1.5v22)1/2
or, y2 = + (154 - 1.5x2)1/2
and, y3 = - (154 - 1.5x2)1/2

I am letting (xxx)1/2 represent a square root

Graph the three functions. The intersections represent the two solutions -- the original velocities of the objects and the other the final velocities of the objects. Use the trace key to verify that the solutions are (-6,10) and (6.8,-9.2). (Remember that the 10 kg ball was y and the 7.5 kg ball was x.) The first solution is discarded as it represents the original velocities. Using the graphing calculator, one determines that the final velocity of the 10 kg ball is 9.2 m/s, west, and that of the 7.5 kg ball is 6.8 m/s, east.