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### Course: AP®︎/College Physics 1>Unit 6

Lesson 4: Center of mass and two-dimensional collisions

# 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}}=\frac{{m}_{1}{x}_{1}+{m}_{2}{x}_{2}+\text{…}}{{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.
$\begin{array}{rl}{p}_{xi}& ={p}_{xf}\\ \\ \\ {m}_{1}{v}_{xi}+{m}_{2}{v}_{xi}& ={m}_{1}{v}_{xf}+{m}_{2}{v}_{xf}\end{array}$
Also, the total momentum in the $y$-direction will be the same before and after the collision.
$\begin{array}{rl}{p}_{yi}& ={p}_{yf}\\ \\ \\ {m}_{1}{v}_{yi}+{m}_{2}{v}_{yi}& ={m}_{1}{v}_{yf}+{m}_{2}{v}_{yf}\end{array}$

For deeper explanations, see our video introducing center of mass.
To check your understanding and work toward mastering these concepts, check out the exercise on predicting motion using the center of mass.

## Want to join the conversation?

• Why,(if a boat and a man are an isolated system), when the man walks away from the shore, does the boat move towards the shore? I don't see this in the video.
• Lets say a man is standing on a boat which is floating and is at rest. If the man walks to one end of the boat, the boat will move in the opposite direction in order to conserve momentum. Due to this effect, the center of mass of the man-boat system will stay stationary. Hope this helps!
• I would like to know why there is no change in the center of mass when there is change in the acceleration of internal particles of a system ?
• how does the c.m of planets revolving be calculated.the double planet revolution and also the planet revolving around their star
• here, the two stars together form a system on which there is an external force due to the star. Hence the motion of the CM of the two planets would be same as if the net force is acting on the CM
• What are some examples of center of mass motion?
• Imagine throwing a ball in the air. Notice how the ball has different points each with a small mass[divide the ball into multiple small balls stuck together] but we do no calculate its motion by calculating the motion of each small part[or ball]. Instead we use the motion of its point of centre of mass to find its motion.
• I struggled with this question in the assignment about the centre of mass of planets orbiting around each other and their respective stars. Where would be the centre of mass?
• This might be 3 years old but I'll go ahead and answer anyways.
The center of mass of the system with the 2-plants is the middle.
Since they are both orbiting each other we can assume that they are rotating around a center point in-between themselves.
The question asks for the acceleration of the system.
Since they weigh the same and are orbiting each other, the movement is equal between the two and we can ignore their movement.
The only acceleration is now from the star's gravity.
So the acceleration is inwards towards the star.
(1 vote)
• In one of the problems, a hockey player hits the puck away from him and the center of mass of the hockey player-puck system has the same velocity as the hockey player originally did. Why isn't it different because the center of mass would change because the distance of the hockey puck is changing from the player?
(1 vote)
• what are same examples of center of mass motio
(1 vote)
• Well, one example could be a volleyball player spiking a volleyball right over the net, or another could be spear hitting the center of a target.
(1 vote)
• Hey .. Let's say a solid cube with two metals mixed in a 1:1 ratio, each having a different density, and the particles are arranged in a way that one type of metal is adjacent to the other. Where would the Centre of Mass located at ...
(1 vote)
• I didn't really understand what it means for the system to be closed. It is mentioned that the velocity of the centre of mass remains the same as long as the system is closed.
(1 vote)
• what is the state motion after a pure inelastic motion?