Elastic Collisions

As stated previously, there are two basic kinds of collisions in momentum: Elastic and Inelastic collisions. An elastic collision can be defined as a collision where both the momentum and the total kinetic energy before the collision are the same as the momentum and total kinetic energy after the collision. In other words, both momentum and kinetic energy are conserved in the collision. This means that there was no wasting force during the collision. Only in elastic collisions are both momentum and kinetic energy conserved.

When we solve problems in elastic collisions, we always start by saying that momentum before the collision is the same as momentum after the collision. Consider two particles, m1 and m2, with initial speeds v1 and v2. After the collision, both particles will still have the same masses, but there will be new velocities, v1' and v2'. We write our equation for conservation of momentum as:

and our equation for conservation of kinetic energy as:

but since 1/2 is common to very term, this may be re-written as:

Algebraically, we may end up with two equations and two unknowns, but we can plug through to solve our problems.

An example of an elastic collision would be a Newton's Cradle. The balls are hanging in a straight line. If you pull one back and make it hit the next ball, one ball at the other end will react just like the one on the first end. This
pattern will go on and on. If you pull two balls
back and make them strike the middle one, the two balls at the other end will fly out together and come back to hit the middle one. Because both momentum and energy are conserved, if two balls are released at one end, two balls will come flying out the other end with the same velocity as the original balls. What does not happen is that one ball comes out the other end with twice as much velocity. Although this would seem possible with conservation of momentum, it is precluded by the conservation of kinetic energy since the velocity is a square function (Remember: Kinetic Energy = 1/2 mv2.)

We can discuss elastic collisions in one-dimension and two-dimensions. The links to the applets at the end of this page demonstrate this further. A one-dimensional collision occurs when there is only motion along one axis, such as cars colliding on an air track.

A two-dimensional collision might occur if there is freedom of motion in the x and y axis, such as with a hockey puck.

If a hockey puck hits another puck that is at rest at an angle, both will be deflected at different angles. One cool thing to remember is that momentum is conserved in each axis, or momentum in the y- axis before the collision equals momentum in the y- axis after the collision, and similarly, momentum in the x- axis before the collision equals momentum in the x- axis after the collision.

Often, it is useful to see what someone else has to say on the topic; Check out these lessons from "The Physics Classroom" (Glenbrook High School). Each contains some practice problems at the end of the lesson.

Conservation Equations as a Guide to Problem Solving

Thinking About Conservation of Momentum

Conservation of Momentum in Explosions

Another work checking out:
http://hypertextbook.com/physics/mechanics/momentum-energy/

The NTNU (National Taiwan Normal University) Virtual Physics Laboratory provides excellent resources and I'm a big fan. Click here for an applet on conservation of momentum in one dimension, and here for an applet on conservation of momentum in two dimensions.

For Practice Problems, Try: From the University of Oregon, Go to this page and select and appropriate problem: http://zebu.uoregon.edu/~probs/mech/collid.html

 

For  Practice Problems, Try: Giancoli Multiple Choice Practice Questions (Select "Practice Problems) and try some.