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.

This is a representation of conserving momentum with vectors. It shows a two snapshots of before and after they collide. In the before snapshot, two balls with different masses denoted as m one and m two where both balls are moving down and towards each other with some momentum at some angle theta one and theta two. In the after snapshot, the two balls are moving down and away from each other at a new and different momentum denoted as momentum a for mass one and momentum b for mass two at new angles theta a and theta b respectively. There are two triangles that shows the change in momentum before and after the collision of the two masses. It is noted below that this diagram shows momentum vectors, which are parallel to their respective velocity vectors. Note momentum one plus momentum two is equal to momentum a plus momentum b and the change in momentum before is equal to the change in momentum after as conservation of momentum demands.

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

OpenStax Reading


Openstax Section 8.6  |  Collisions of Point Masses in Two Dimensions

Openstax College Textbook Icon

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?

An image of two masses moving toward each other with mass one moving to the right and mass two moving to the left with another arrow that shows the trajectory of both moving after the collision at some angle theta.

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

BoxSand's Conceptual 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.

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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...

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

 

Equations, definitions, and notation icon Concept Map Icon
Key Terms Icon Student Contributed Content Icon

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.

https://www.youtube.com/watch?v=RLQrStX1oqc

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.

Phet Interactive Simulations Icon

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

The Physics Classroom Icon

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

Simulation Icon

Demos


Hockey Pucks Colliding

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

 For additional demos involving this subject, visit the demo repository

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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.

The Physics Classroom Icon

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 $\frac{\Delta p}{\Delta t}$. 

Other Content Icon

Handy resource for 1D Conservation of Momentum.

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Other Resources

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

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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


Physical Representations describes the physical phenomena of the situation in a visual way.

 

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

An image of a race car of mass m moving to the right with a velocity v

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

An image of an object of mass m at some position from the origin moving at some velocity vector at some angle. The velocity vector is broken down into its x and y components and visually represented as vector arrows.

Classic Elastic collision

An image of three snapshots of a collision of two balls of mass m one and m two. In the first snapshot titled before the mass m one is moving to the right at a constant velocity v one and mass two start starts lower and moves upward and to the right at a velocity v two. In the second snapshot during the elastic collision, both masses touch in their current trajectory with the same velocity depicted in the before snapshot. In the after shapshot it shows the mass one traveling at the trajectory of mass two, at some angle with a constant velocity two and vice versa for mass two.

Classic Inelastic collision

An image of three snapshots of a collision of two balls of mass m one and m two. In the first snapshot titled before the mass m one is moving to the right at a constant velocity v one and mass two start starts lower and moves upward and to the right at a velocity v two. In the second snapshot during the inelastic collision, both masses touch in their current trajectory with the same velocity depicted in the before snapshot. In the after shapshot it shows the both the masses connected and moving at a new velocity in a new angle with the velocity denoted as v three.

 

Mathematical


Mathematical Representation uses equation(s) to describe and analyze the situation.

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.

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

A representation with the words conservation of momentum (co.mo) on the top. There is an equation that shows that if the impulse of the system is zero, the summation of the initial momentum is equal to the summation of the final momentum. This is also written in words below with the note that says remember that the impulse is equal to the sum of the forces external to the system multiplied by the change in time, therefore the impulse could be zero if either the summation of the external forces or the change in time is zero.

 

Graphical


Graphical Representation describes the situation through use of plots and graphs.

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.

. This is a graph of force in newtons over time in seconds. The graph increases from zero in force over some time, has a constant force over some time, and then decreases back to zero.

 

Descriptive


Descriptive Representation describes the physical phenomena with words and annotations.

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


Experimental Representation examines a physical phenomena through observations and data measurement.
If a badly tied up load fell out of a car while traveling at a constant velocity, assuming a short time period so we don't lose much energy to friction, you could measure the velocity of your car before and after the drop, and from that figure out the momentum of the dropped load, provided you knew both the masses.

Lab on Conservation of Momentum in 2-D

https://www.youtube.com/watch?v=6L4RA5VopY0