# Center of mass and two-dimensional collisions review

Review the key concepts, equations, and skills for the center of mass and two-dimensional collisions, including how to understand center of mass motion.

## Key terms

Term (symbol)Meaning
Center of massAverage position of all parts of the system, weighted by mass. The velocity of a system’s center of mass does not change if the system is closed.

## Equations

EquationSymbolsMeaning in words
$x_{\text {CM}} = \dfrac{m_1x_1+m_2x_2+...}{m_1 + m_2}$$x_{\text {CM}}$ is the center of mass, $m_1$ and $m_2$ are masses, and $x_1$ and $x_2$ are the position of the massesCenter of mass is the sum of each mass times its position, divided by total mass

## How to find the center of mass

A symmetric object’s center of mass is at the center.
The center of mass for a two object system with one large and one small mass will be closer to the large mass.

## Center of mass and motion

The velocity of the system’s center of mass does not change, as long as the system is closed. The system moves as if all the mass is concentrated at a single point.
If we throw a tennis racquet, the racquet rotates around its center of mass. However, the center of mass itself does not rotate; instead it will make a parabolic path, as if it was a point particle.
Likewise, for an exploding projectile, the center of mass will continue on the parabolic trajectory. The final location will be at the weighted distance between the masses.

## How to analyze momentum in two-dimensional collisions

For a collision where objects are moving in $2$ dimensions (e.g. $x$ and $y$), the momentum is conserved in each direction independently as long as there are no external net forces in that direction.
The total momentum in the $x$-direction will be the same before and after the collision.
\begin{aligned}p_{xi}&=p_{xf}\\ \\\\ m_1v_{xi} + m_2v_{xi} &= m_1v_{xf} + m_2v_{xf} \end{aligned}
Also, the total momentum in the $y$-direction will be the same before and after the collision.
\begin{aligned}p_{yi}&=p_{yf}\\ \\\\ m_1v_{yi} + m_2v_{yi} &= m_1v_{yf} + m_2v_{yf} \end{aligned}