Conservation of Momentum in 2 dimensions follows the same fundamental equations and principles as in 1 dimension. The main difference is that you must use a component analysis and conserve momentum in in both x and y directions.

Check out the Space Blog for a primer on the subject.

https://www.youtube.com/watch?v=4IYDb6K5UF8

## Pre-lecture Study Resources

Read the BoxSand Introduction and watch the pre-lecture videos before doing the pre-lecture homework or attending class. If you have time, or would like more preparation, please read the OpenStax textbook and/or try the fundamental examples provided below.

### BoxSand Introduction

## Momentum | Conservation of Momentum

The impulse momentum theorem states that the change in momentum on a system is equal to the average net force multiplied by the change in time.

$\Delta{}\vec{p} = \Sigma{}\overline{\vec{F}}_{external}\Delta t$

If the right-hand-side of this equation is zero or approximately zero, than the left-hand-side must also be very small. When this condition is met, the change in momentum of a system is zero. This often is the case when collisions occur if you include the objects colliding into your system. If you consider two billiard balls colliding, individually they have the force from the other billiard ball acting on them, which in turn changes their momentum during the collision. If you include both balls in your system though, and draw a dotted line around both, then the force between them becomes an internal force and the net external force is very small during the collision. The normal force from the table cancels out the gravitational force. The force from friction on the table may be considered a net external force while the balls interact but this happens so quickly that $Delta t$ is very small. Thus the impulse acting on the system is approximately zero and the net momentum does not change. This can mathematically be applied with the following.

if $\Sigma{}\overline{\vec{F}}_{external} \Delta t \approx 0$, then $\sum{\vec{p}_i} = \sum{\vec{p}_f}$

The appropriate physical representation to analyze the interaction is a vector operation of addition of momentum, like that shown below.

Remember momentum is a vector quantity, so $\sum{\vec{p}_i} = \sum{\vec{p}_f}$ implies more than one equation, in principle it is three equations, one for each direction or component of momentum.

$\sum{{p_i}_x} = \sum{{p_f}_x}$

$\sum{{p_i}_y} = \sum{{p_f}_y}$

$\sum{{p_i}_z} = \sum{{p_f}_z}$

## Key Equations and Infographics

### BoxSand Videos

## Required Videos

## Suggested Supplemental Videos

### OpenStax Reading

### Fundamental examples

3. Consider the image of a collision below. The blue object, $m_1 = 7 \, kg$ is initially traveling with a speed of $3 \, m/s$. The red object, $m_2 = 4 \, kg$ is initially traveling with a speed of $5 \, m/s$. If the final velocity of $m_1$ is $\langle 1.73 \, , \, -1.00 \rangle \, m/s$, what is the magnitude and direction of $m_2$ after the collision?

CLICK HERE for solutions.

## Post-Lecture Study Resources

Use the supplemental resources below to support your post-lecture study.

### Practice Problems

Recommended example practice problems

BoxSand's Quantitative Practice Problems

BoxSand's Multiple Select Problems

- Openstax has practice problems toward the end of each section, Website Link
- Swanson Physics 10 problem worksheet on Momentum, PDF Link
- Science Notes classical conservation of momentum problem, 2 Trainz, Website Link
- Albert.io quiz on conservation of momentum, Website Link

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

## Additional Boxsand Study Resources

### Additional BoxSand Study Resources

### Learning Objectives

## Summary

The goal is for students to identify when conservation of momentum can be assumed and analyze the system in both the physical and mathematical representations.

## Atomistic Goals

Students will be able to...

- Identify collisions in physical phenomena.
- Define that a quantity is conserved when the change in that quantity is zero.
- Identify whether the forces are internal or external to a system and if the net external force is zero.
- Show that momentum is conserved for systems where the net external force is zero.
- (UPMF) Justify that momentum conservation can be assumed when the impulse on the system is negligible.
- Draw an appropriate physical representation including the initial and final momentum vectors and a wise coordinate system.
- Draw a vector operation diagram with initial, final, and change in momentum vectors.
- Apply a 1-D momentum analysis in the mathematical representation when appropriate.
- Construct, in the mathematical representation, a conservation of momentum vector equation in 2-D.
- Determine when momentum is conserved in one direction but not another.

### YouTube Videos

Note how momentum is conserved along each direction.

https://www.youtube.com/watch?v=dbx_8tDgO50

2-D Momemtum Example - Simultaneous equation solving.

### Simulations

This Phet demonstrates the collisions of balls. You can control how elastic the collisions are. It is suggested that you turn on both the momentum vectors and the momenta diagram. When you're comfortable with the introductory simulation click the tab at the top and switch over to the advanced one.

This smashing simulation from The Physics Classroom will help understanding conservation of momentum in inelastic collisions.

For additional simulations on this subject, visit the simulations repository.

### Demos

Hockey Pucks Colliding

https://www.youtube.com/watch?v=ralAQ49Kj28

For additional demos involving this subject, visit the demo repository

### History

Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Physics Fun

Oh no, we haven't been able to post any fun stuff for this topic yet. If you have any fun physics videos or webpages for this topic, send them to the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Other Resources

The Physics Classroom takes you through the basics of 1D Conservation of Momentum- try the embedded animations for extra help.

Wyzant's section on Linear Momentum does a good job at connecting the previous content of impulse and momentum. wherever you see an equation that looks like $\frac{dp}{dt}$ just know this is another representation for a change in the property p with respect to property t, which you are more familiar with as $\frac{\Delta p}{\Delta t}$.

Handy resource for 1D Conservation of Momentum.

### Other Resources

This link will take you to the repository of other content related resources.

## Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

### Assumptions

1. Before starting Conservation of Momentum analysis, always check to make sure we can make the assumption that the change in impulse is 0 or approximately 0. This is often a simplifying case for physical situations that have all involved forces inpart very quickly, and with some generality, can be chalked up to two cases: Collisons and Explosions.

2. For elastic collisons, assume that no energy is lost in the collision. However, in inelastic collisons (when things move together at the beginning or end), energy is NOT conserved because it is usually added or lost to deformation or friction.

3. The 2D momentum decomposed into perpindicular directions (usually the standard x and y axes), are completely independent of one and other, in much the same way as when we did 2D kinematics.

### Algorithm

1. Read and re-read the whole problem carefully.

2. Visualize the scenario. Mentally try to understand what the object(s) is(are) doing and what forces are acting on it (them).

3. Define your system of interest.

4. Confirm that there are no net external forces acting on the objects included in the system.

5. Define a coordinate system.

6. Determine initial and final conditions.

a. If any objects are sticking to one another, they share the same velocity.

b. If objects are not sticking to one another, they have their own unique velocity.

7. Write out the conservation of momentum component wise for the system you defined. Be careful to consider the negatives that arise based off of your coordinate system definition.

8. Simplify the conservation of momentum equations as best as possible by plugging in any known values.

9. Rearrange equations to solve for the required quantity.

10. Evaluate your answer, make sure units are correct and the results are within reason.

### Misconceptions & Mistakes

- Conservation of momentum does not mean that the momentum of each object is conserved individually. It means that the system as a whole conserves momentum- the momentum can be transferred- and most likely will be.
- Momentum is a vector, but it is
**not**a force, so it should**not**show up on a FBD. - Momentum is conserved at all points of time when there are no net external forces, while it is common to compare right before and right after a collision, those are not the only times that can be compared. Sometimes it is advantageous to compare more than 2 different times to use all information you know.

### Pro Tips

- Think about all the information you have in the system and pick the times (2 or more) that you want to use conservation of momentum on.
- It can be helpful to write out all the knowns and unknowns for all of the times that you can in the system and then decide which ones to use conservation of momentum on.
- Do not forget about drawing pictorial representations. If you draw all the known momentum vectors to scale, often times you can determine the direction of unknown momentum vectors if enough information is given.

### Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

### Physical

Below we can see a car of mass (m) with some 1D velocity moving horizontally to the right.

Here, an object with mass (m) is moving with 2D motion and therefore has both vertical and horizontal velocity components.

Classic Elastic collision

Classic Inelastic collision

### Mathematical

Impulse = change in momentum

$ \vec{J} = \Delta \vec{P} = \sum \vec{F}_{ext} \Delta t $

Momentum = mass * velocity

Remember that momentum is a vector quantity comprised of a velocity vector multiplied by a scalar mass quantity.

$\vec{p} = m \vec{v}$

### Graphical

As shown in the mathematical representation for momentum, the only way the momentum of an object may change is either by changing the mass or the velocity. Here we have a force vs time graph. Force is related to velocity through acceleration, recall Newton's Second Law. Therefore, we may analyze a force vs time graph to determine the change in momentum of an object with respect to time.

.

### Descriptive

Often, insurance agents will use Conservation of Momentum combined with the descriptions individuals from both sides of car collisions to determine who is telling the truth. Often skid marks, the distance and damage between vechiles, and each side of the story can tell the truth in disputed auto accident cases. Solve the problem below for an example:

Your friend is going to court to determine fault in an automobile accident in Paris, Texas. She thinks that the other car was speeding and she was not. She is looking to you for help to prove her assertion. During discovery, the following evidence is presented, none of which is disputed by the other driver:

- Weather records indicate that there was no rain the day of the collision
- Your friend was traveling north just before the collision and the other car was traveling east
- The two cars got stuck together and remained so after the collision while skidding to a stop
- The speed limit on both roads is 45 mph
- The accident occurred on level ground
- Measurements of skid marks created after the collision indicate that the cars skidded 50 feet at an angle of 30 degrees north of east before stopping
- According to driver's manuals, your friend's car weighs 2400 lbs and the other car weighs 2000 lbs. Each driver weighs 150 lbs and there was no significant cargo
- A physics textbook indicates that the coefficient of kinetic friction for rubber on dry pavement is 0.80

What can an analysis of the physics of this collision and aftermath tell the court regarding the speeds of each of the vehicles before they collided?

Lets analyze a rocket launch. At time $t1$ our rocket has some initial mass $m_{1}$ and an initial velocity $\vec{v}_{1}$, giving it a momentum of $\vec{p}_{t1} = m_{1} \vec{v}_{1}$. After some later time $t2$, the rocket has used up some fuel and the mass of the rocket has changed to $m_{2}$, while the velocity has not changed, giving the rocket a different momentum of $\vec{p}_{t2} = m_{2} \vec{v}_{1}$.

### Experimental

Lab on Conservation of Momentum in 2-D