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2.4. Momentum

Along with the Conservation of Energy, the Conservation of Momentum is up there with the most important fundamental ideas of the universe. As far as we know, momentum is always conserved in any collision or interaction. 

You will probably have encountered the idea of momentum at GCSE - it is a quantity of motion that is defined through its equation, i.e. an object's velocity multiplied by its mass. It's one of those topics that often sends my GCSE classes into a state of panic with some of the maths involved, but with a bit of practice and with a clear structure, it will become second nature. So for a bit of a light intro to the topic, I'd like to start with this video of some drunk tourists who have forgotten about the conservation of momentum (particularly the guy in the black tshirt).

... clearly if he gives himself a forward momentum by stepping off the boat, the boat must therefore have a backwards momentum as he steps off. Back to Physics class for you.

I've divided the section up as follows:

 

Conservation of Momentum

So to begin with, we need to recap what we already know about momentum. Momentum is a quantity of motion defined by the product of an object's mass and its velocity, given by the following equation:

 

p = mv 

Momentum is a vector quantity, therefore can be positive or negative depending on direction. Conservation of Momentum is the principle that states :

total momentum in a closed system is always conserved

In other words, in a collision (or explosion) the total momentum of all objects before the collision must equal the total momentum of all objects after the collision, in the absence of any outside forces.

We can use the idea of the Conservation of Momentum to solve various problems - Crash Course Physics has a nice video introduction to some of these problems.

So what do problems involving Conservation of Momentum (CoM) look like? Generally they come in two types:

  1.  Collisions - Two objects crashing together (or exerting some force on one another) and either sticking together (inelastic collisions) or bouncing apart (elastic collisions)

  2. Explosions - One object 'exploding' into two (or more) seperate objects. This does not need to be an 'explosion' in the loud, fiery sense; throwing a snowball and hitting a tennis ball are examples of explosions.

In a typical problem you will be given some combination of masses and velocities and asked to work out an unknown mass or velocity using the CoM. A simple example is shown below:

A 75 kg rugby travelling at 8 msˉ¹ collides with a 110 kg player travelling at 5 msˉ¹ in the opposite direction. Calculate their combined velocity after they collide and move off together. 

MomentumTackle.jpg

Conservation of momentum problems are initially a little tricky to get your head around. We know that the total momentum before a collision is equal to the total momentum after the collision. 

It can essentially be summarised by the following equation (the Sigma symbol means 'sum of')

Σpbefore   =   Σpafter   

Now, let's take a little look at this video from Physics Online to understand how to tackle these CoM problems.

Worked Example - an inelastic collision involving rugby players

Q. A 75 kg rugby travelling at 8 msˉ¹ collides with a 110 kg player travelling at 5 msˉ¹ in the opposite direction. Calculate their combined velocity after they collide and move off together. 

To go back to the rugby tackle example shown above, let's approach this problem by starting with a diagram - BEFORE the collision on the left, AFTER the collision on the right.  

MomentumRugby.png

Always start by stating the CoM, then go from there.

Σpbefore   =   Σpafter   

p1 + p2 = pF

m1v1 + m2v2 = (m1+m2) vF

(75 × 8) + (110 × -5) = (75 + 110) vF

600 - 550 = 185 vF

vF = 0.27 msˉ¹ (to the right)

N.B. We add the player's masses afterwards as they 'stick together' and become one combined object (i.e. an inelastic collision). Also note the negative sign for p2's initial velocity, as they are initially moving in opposite directions.

Now for a bit of practice; PHET have a nice simulation called Collision Lab (Flash required, try opening in Microsoft Edge, rather than Chrome).

Grab yourself a pen and paper, and try different initial masses and velocities. Have a practice using these CoM ideas to predict the final velocity of the objects as they move off together (inelastically)

Alternatively this Geogebra collisions simulation does similar.

phet-logo-trademarked.png

Video Lessons

Resources

IB Physics
Topic 2 Notes
IB-Physics.net
Chapter 2 Summary
IB Revision Notes
Isaac Physics
Momentum and CoM
Isaac Physics
Conservation of Momentum
Mr. G
2.4 Teaching Notes
2.4 Student Notes
Physics and Maths Tutor
Motion Definitions
Motion Key Notes
Motion Detailed Notes
Mechanics Flashcards
A Level Resources - content slightly different

Questions

Cambridge University Press
Topic 2: Add Qs
Topic 2: Add Qs MS
Topic 2: MCQs
CUP Website Link
Freely available online
Dr French's Eclecticon
Linear Momentum
Linear Momentum Solutions
Link to Dr French's Site
Extension: Pre-University Material
Grade Gorilla
2.4 (Momentum) MCQ
Topic 2 (Mechanics B) End Quiz
Quick IB Specific Mixed MCQs
Isaac Physics
Force and Momentum
Mr. G
2.4 Formative Assessment
Topic 2 Summary Qs
IB Specific Questions
Physics and Maths Tutor
Momentum (AQA 1)
Momentum MS (AQA 1)
Momentum (AQA 2)
Momentum MS (AQA 2)
A-Level Qs: overlapping content
Physics and Maths Tutor
Momentum (Edexcel 1)
Momentum MS (Edexcel 1)
A-Level Qs: overlapping content
Physics and Maths Tutor
MCQ Force, Energy, Momentum 1 (AQA 2)
MCQ Force, Energy, Momentum 1 MS (AQA 2)
MCQ Force, Energy, Momentum 2 (AQA 2)
MCQ Force, Energy, Momentum 2 MS (AQA 2)
A-Level Qs: overlapping content
 

Elastic and Inelastic Collisions

Collisions can either be classed as elastic or inelastic (at least they can in the IB, in reality they are somewhere in between these two extremes).

 

  • Elastic Collisions are collisions in which both momentum and kinetic energy is conserved.

    • Often these collisions will involve objects bouncing apart, such as snooker balls.​

  • Inelastic collisions are collisions in which only momentum is conserved. 

    • These problems will involve objects either sticking together (e.g. cars crashing) or explosions (e.g. a cannonball being fired).​

    • In an inelastic collision, some of the kinetic energy is transferred to heat and sound. In an explosion energy must be put in (e.g. chemical energy in gunpowder, or nuclear potential energy for decay of an alpha particle) to cause an increase in total kinetic energy.

For all collisions (elastic and inelastic) :       Σpbefore   =   Σpafter   

Only for elastic collisions :                            ΣEbefore   =   ΣEafter  

 

 

Worked Example - an inelastic collision involving rugby players

Q. For the rugby player example given above, calculate the kinetic energy lost in this collision.

ΔEK = ΣEK before - ΣEK after

ΔEK = ½m1v1² + ½m2v2² - ½(m1+m2)vF²

= [½×75×8²] + [½×110×(-5)²] - [½×(75 + 110)×0.27²

= 3770 J

Kinetic Energy has been lost in this collision therefore it must be inelastic.

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You can also simulate elastic and inelastic collisions in Collision Lab.

Adjust the slider to either inelastic or elastic and compare the behaviour (real life collisions will fall somewhere in between).

Video Lessons

Resources

IB Physics
Topic 2 Notes
IB-Physics.net
Chapter 2 Summary
IB Revision Notes
Isaac Physics
Collisions
Level 5/6 Beyond IB
Mr. G
2.4 Teaching Notes
2.4 Student Notes
Physics and Maths Tutor
Motion Definitions
Motion Key Notes
Motion Detailed Notes
Mechanics Flashcards
A Level Resources - content slightly different

Questions

Cambridge University Press
Topic 2: Add Qs
Topic 2: Add Qs MS
Topic 2: MCQs
CUP Website Link
Freely available online
Dr French's Eclecticon
Linear Momentum
Linear Momentum Solutions
Link to Dr French's Site
Extension: Pre-University Material
Grade Gorilla
2.4 (Momentum) MCQ
Topic 2 (Mechanics B) End Quiz
Quick IB Specific Mixed MCQs
Isaac Physics
Conservation of Momentum
Mr. G
2.4 Formative Assessment
Topic 2 Summary Qs
IB Specific Questions
 

Impulse and Ft Graphs

Impulse in Physics means exactly the same thing as change in momentum (N.B. Not rate of change of momentum, students always get this idea wrong).

impulse ≡ change in momentum

If I have an initial momentum of 1600 kgmsˉ¹ and I come to a stop over 5 seconds, my change of momentum/ impulse is 1600 kgmsˉ¹ (i.e. it doesn't matter how quickly the deceleration happens).

Our wordy definition of Newton's 2nd Law has already told us that it is actually Force that equals the rate of change of momentum, or in other words. 

impulse.png

If we look at the units here, we see some equivalency - change in momentum = force x time, therefore:

1 kgmsˉ¹ ≡ 1 Ns

This equation is a GCSE favourite, when looking at safety features such as crumple zones and seatbelts. By increasing the time taken for the collision, these features are able to reduce the force acting.

When a Force/ time graph is plotted, the area between the graph and the x-axis will give us the total change in momentum, or impulse. 

impulse graph.png
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A couple of Geogebra simulations to get to grips with these Force/ time graphs.

  1. Constant Force : Remember that the area of these graphs gives us the change in momentum.

  2. Changing Force: Here we have a changing force, in this case the average force equals half the maximum force.

Video Lessons

Resources

IB Physics
Topic 2 Notes
IB-Physics.net
Chapter 2 Summary
IB Revision Notes
Isaac Physics
Impulse
Mr. G
2.4 Teaching Notes
2.4 Student Notes
Physics and Maths Tutor
Motion Definitions
Motion Key Notes
Motion Detailed Notes
Mechanics Flashcards
A Level Resources - content slightly different

Questions

Cambridge University Press
Topic 2: Add Qs
Topic 2: Add Qs MS
Topic 2: MCQs
CUP Website Link
Freely available online
Grade Gorilla
2.4 (Momentum) MCQ
Topic 2 (Mechanics B) End Quiz
Quick IB Specific Mixed MCQs
Mr. G
2.4 Formative Assessment
Topic 2 Summary Qs
IB Specific Questions
 

Additional Resources

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Definitions and Key Words : Chapter 2

A set of Quizlet flashcards of the key words and definitions for this chapter is provided here. 

IB Questions

A question by question breakdown of the IB papers by year is shown below to allow you to filter questions by topic. Hopefully you have access to many of these papers through your school system. If available, there may be some links to online sources of questions, though please be patient if the links are broken! (DrR: If you do find some broken links, please contact me through the site)

 

Questions on this topic (Section 2) are shown in red.

Use this grid to practice past IB questions topic by topic. You can see from the colours how similar the question topic breakdown is year by year. The more you can familiarise yourself with the IB question style the better - eventually you will come to spot those tricks and types of questions that reappear each year.