# Day 66: Collisions in the Center of Mass Frame

*AP Physics C:* Based on the results of our video analysis of collisions, we know that both momentum and kinetic energy are conserved in elastic collisions. So I as warm-up, students worked on the above problem on whiteboards.

After some flexing of algebra muscles and a messy simultaneous equation, I ask if they’d like to see a short cut….

Based on the video analysis yesterday, we also saw that the velocity of the center of mass (yellow) remains constant.

And for elastic collisions, the carts pre- and post-collisions velocities *relative to the center of mass* were equal and opposite.

So we applied those concepts to the above problem to generate an easier solution:

- Find the velocity of the center of mass.
- Find the initial velocities of the blocks in the center of mass frame.
- The final velocities of the blocks in the center of mass frame are equal and opposite to the velocities in #2.
- Translate the velocities in #3 back into their actual velocities.

*NGSS Science and Engineering Practices:
#5. Using mathematics and computational thinking*