# Momentum

#### dshnay

10+ Year Member
IS momentum always conserved wether it's elastic or inelastic collision?

#### BloodySurgeon

Moderator Emeritus
10+ Year Member
IS momentum always conserved wether it's elastic or inelastic collision?
For collisions yes, but when there is an external force like gravity, friction, etc. momentum is not conserved (i.e. pendulum, spring, waves)

OP
D

#### dshnay

10+ Year Member
Surgeon,

you've just awakend me.
How are we supposed to solve for problems where momenum is not conserved and it's an inelastic collision?

#### BloodySurgeon

Moderator Emeritus
10+ Year Member
Surgeon,

you've just awakend me.
How are we supposed to solve for problems where momenum is not conserved and it's an inelastic collision?
The total momentum of the involved bodies is conserved, regardless whether the collision is elastic or inelastic. m1vi1 + m2vi2 = (m1+m2)vf

#### TawMus

10+ Year Member
5+ Year Member
since there are no nonconservative forces in action in a collision reaction, does that mean mechanical energy is also conserved??

OP
D

#### dshnay

10+ Year Member
sorry,

I shouldnt have mentioned collision. What I've meant but did not type was how to find momentum as you've mentioned earlier when gravity is involved or other forces.
Thanks

#### TypeSH07

##### Junior Member
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15+ Year Member
For collisions yes, but when there is an external force like gravity, friction, etc. momentum is not conserved (i.e. pendulum, spring, waves)
The energy in pendulum, springs, and waves are conserved though aren't they??

#### BloodySurgeon

Moderator Emeritus
10+ Year Member
The energy in pendulum, springs, and waves are conserved though aren't they??
Yes the Total Energy is conserved.

#### TawMus

10+ Year Member
5+ Year Member
Energy and momentum are both conserved in all waves and occillations???

#### physics junkie

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Momentum is ALWAYS conserved in all situations and in classical as well as quantum mechanics. It can be mathematically proven using noether's theorem that space translational symmetry implies conservation of momentum. You don't need to know this but it's my way of intimidating you into believing me.

To quote hyperphysics(great website, http://hyperphysics.phy-astr.gsu.edu/Hbase/hph.html#hph )

A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision. An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision. Any macroscopic collision between objects will convert some of the kinetic energy into internal energy and other forms of energy, so no large scale impacts are perfectly elastic. Momentum is conserved in inelastic collisions, but one cannot track the kinetic energy through the collision since some of it is converted to other forms of energy.
Summary
Elastic - momentum and kinetic energy are conserved
Inelastic - momentum and total(not kinetic) energy are conserved

#### TawMus

10+ Year Member
5+ Year Member
Momentum is ALWAYS conserved in all situations and in classical as well as quantum mechanics. It can be mathematically proven using noether's theorem that space translational symmetry implies conservation of momentum. You don't need to know this but it's my way of intimidating you into believing me.

To quote hyperphysics(great website, http://hyperphysics.phy-astr.gsu.edu/Hbase/hph.html#hph )

Summary
Elastic - momentum and kinetic energy are conserved
Inelastic - momentum and total(not kinetic) energy are conserved
Agreed, but we are actuallly talking about conservation of energy and momentum in waves and oscillations...

#### BloodySurgeon

Moderator Emeritus
10+ Year Member
Are you sure about that? I had at least 5 questions that asked what is conserved in a pendulum and it says total energy and not momentum. It further elaborated that whenever there is an external force on an object, momentum is not conserved (I don't understand it, but I memorized it). Also momentum is a vector and the direction of velocity changes as we swing from one side to another.

You are physics junkie and I know you understand this stuff probably better than I do, but could you please verify this again. I don't mind admitting that Im wrong, I don't want to come out with the wrong information, both for me and others.

#### BloodySurgeon

Moderator Emeritus
10+ Year Member
Dude, that article is so hard to read.... I can barely comprehend what he is talking about..
yea, because it aint and article... its a discussion forum like this but for professors. It just compares waves to particles and since particles have masses E=mc^2 it can therefore have momentum. And if waves have momentum then they can have conservation of momentum.

#### TawMus

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5+ Year Member
Can they also have conservation of mechanical energy??

#### BerkReviewTeach

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Are you sure about that? I had at least 5 questions that asked what is conserved in a pendulum and it says total energy and not momentum. It further elaborated that whenever there is an external force on an object, momentum is not conserved (I don't understand it, but I memorized it). Also momentum is a vector and the direction of velocity changes as we swing from one side to another.

You are physics junkie and I know you understand this stuff probably better than I do, but could you please verify this again. I don't mind admitting that Im wrong, I don't want to come out with the wrong information, both for me and others.
I think Physics Junkie is referring to collision processes, but I won't put words in his mouth. Depending on your reference frame, momentum can be thought of as conserved for any collision. The general rule to follow is that momentum is conserved in the absence of a net external force. Consider a falling object hitting another falling object. If we consider only the two objects, then momentum is not conserved, because they are experiencing an acceleration due to gravity that increases their momentum during the collision process. However, if we also take the Earth into consideration along with the two balls, then there is no net external force, and thus momentum is conserved. For practical MCAT purposes, we typically disregard the upward acceleration of the Earth in such a case.

As far as your pendulum example goes, you are absolutely correct that momentum is not conserved for the pendulum bob during its path. That is because there is no collision and we are not considering anything else besides the bob. As it swings, it changes it's speed and direction, so it must experience a change in momentum. Again, however, Physics Junkie may be looking at the pendulum as a part of a much larger system.

#### physics junkie

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Agreed, but we are actuallly talking about conservation of energy and momentum in waves and oscillations...
Oh, well that's plenty more interesting. In waves and oscillations energy is lost over time due to heat, friction, etc. They fall under the category of dissipative systems(http://en.wikipedia.org/wiki/Dissipative). Noether's theorem doesn't always apply for those types of systems since they may or may not be described by a lagrangian or hamilitonian. In other words the total energy of the system depends on the time at which you measure it.

Simple pendulums can be described by lagrangians but momentum conservation doesn't hold because the system is dissipative. Waves? Well, electromagnetic waves they hold because the energy density of an electromagnetic wave is given by it's Poynting vector and E/c = mc which has units of momentum. Phonons/matter waves? No idea. It probably depends on the model used and how many degrees of freedom are allowed and so forth.

In the end momentum and energy are always conserved for the universe.

There was actually an interesting thread on physicsforums http://www.physicsforums.com/showthread.php?t=66697&highlight=tether+ball where it's shown that angular momentum is not conserved when you hit a tether ball. This is because there is a non-central force acting on the system though that destroys the symmetry.

#### physics junkie

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I think Physics Junkie is referring to collision processes, but I won't put words in his mouth. Depending on your reference frame, momentum can be thought of as conserved for any collision.
Correct. I was only trying to answer the original poster's question. I apologize for not specifying that I was referring to closed systems when I said momentum is always conserved. Open systems are a different ball game. I jumped the gun by not reading the thread. Not to stroke your ego but I am consistently impressed by your replies, BRT. Every time I look through your posts critically for an error I never find one. It is exceedingly rare to find a person can explain physics conceptually without a loss of rigor.

bloodysurgeon said:
Are you sure about that? I had at least 5 questions that asked what is conserved in a pendulum and it says total energy and not momentum. It further elaborated that whenever there is an external force on an object, momentum is not conserved (I don't understand it, but I memorized it). Also momentum is a vector and the direction of velocity changes as we swing from one side to another.
Force is defined as delta-momentum/delta-time. In calculus that's expressed F=dp/dt where p is momentum. If you integrate force over time then you get the change in momentum and thus it can't be conserved. Let's say there is no force, F=0. That means that dp/dt = 0. If dp/dt = 0 then p is a constant because when you take the derivative of a constant you get zero. Saying momentum is a constant is equivalent to saying it is conserved! So momentum is conserved unless outside forces are involved...when outside forces are involved energy changes with time and it's a dissipative system.

If there is an outside force, let's say 5 newtons. then you have F=5N. dp/dt = 5N. Then if you rearrange this as 5dt = dp and integrate both sides you get p=5(tf-ti). This means the change in momentum is equal to 5(tf - ti) where tf is the ending time and ti is the starting time. In this situation momentum isn't conserved.

That should clear up why momentum isn't conserved in a system when you have an external force. A closed system is one where you have no external forces acting.

BRT, I just had an interview to teach for kaplan today. I suppose that makes us rivals? #### TypeSH07

##### Junior Member
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Correct. I was only trying to answer the original poster's question. I apologize for not specifying that I was referring to closed systems when I said momentum is always conserved. Open systems are a different ball game. I jumped the gun by not reading the thread. Not to stroke your ego but I am consistently impressed by your replies, BRT. Every time I look through your posts critically for an error I never find one. It is exceedingly rare to find a person can explain physics conceptually without a loss of rigor.

Force is defined as delta-momentum/delta-time. In calculus that's expressed F=dp/dt where p is momentum. If you integrate force over time then you get the change in momentum and thus it can't be conserved. Let's say there is no force, F=0. That means that dp/dt = 0. If dp/dt = 0 then p is a constant because when you take the derivative of a constant you get zero. Saying momentum is a constant is equivalent to saying it is conserved! So momentum is conserved unless outside forces are involved...when outside forces are involved energy changes with time and it's a dissipative system.

If there is an outside force, let's say 5 newtons. then you have F=5N. dp/dt = 5N. Then if you rearrange this as 5dt = dp and integrate both sides you get p=5(tf-ti). This means the change in momentum is equal to 5(tf - ti) where tf is the ending time and ti is the starting time. In this situation momentum isn't conserved.

That should clear up why momentum isn't conserved in a system when you have an external force. A closed system is one where you have no external forces acting.

BRT, I just had an interview to teach for kaplan today. I suppose that makes us rivals? So basically in collisions momentum is conserved but in things such as pendulum swings and waves only energy is??

#### Kaustikos

##### Archerize It
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So basically in collisions momentum is conserved but in things such as pendulum swings and waves only energy is??
I think that what you should understand from this is that the momentum is conserved as long as there is no outside force to change things - no varying force.

So the pendulum has a disipitative force which causes momentum to not be conserved.

So, my question then is; can momentum NOT be conserved if a collision were to occur between free-falling objects? Because of the applied force of gravity?

#### physics junkie

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\Also momentum is a vector and the direction of velocity changes as we swing from one side to another.
I forgot to mention, in a situation like a pendulum it would be best to consider if conservation of angular momentum applied--not linear momentum. Angular momentum is given by L=mvr, or the vector product form is L=r X p=mvr*sin(theta) where theta=pi/2 so sin(pi/2)=1. You can have conservation of angular momentum even though the vector describing the trajectory of the mass is changing by choosing a vector that stays the same throughout the system. So for angular momentum you have the r-vector that points from the center of the circular path of the mass out to the mass and a p-vector that points tangent to the velocity of the mass at all times. When you use the right-hand rule and multiply these two vectors you get a new vector that runs through the center of the circle and points either up or down(you get to choose). This new vector is the vector we use to describe the angular momentum so that the vector stays the same.

In the pendulum system the way the angular momentum vector would change looks like this(Gravity would be applying the torque force but this diagram doesn't represent that situation well because this diagram wasn't made to describe a free swinging pendulum but just to show how if you apply a torque, in this case gravity, that angular momentum won't be conserved): Notice how the green vector points on the same path(but changes whether it points up or down as the mass changes direction), only magnitude. That's because angular momentum isn't conserved in a pendulum. If the mass kept swinging around and no torque was applied it would stay the same magnitude and direction.

If you were rotating a tire while sitting in a chair the angular momentum vector pops out the center of the circle like this: In part B of the image the guy is spinning in the chair as well.

#### physics junkie

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So basically in collisions momentum is conserved but in things such as pendulum swings and waves only energy is??
Momentum is conserved in elastic and inelastic collisions. Momentum is not conserved in pendulum swings because there is a torque force acting on the system(gravity). Momentum is conserved in electromagnetic waves such as light. The way you can ask yourself if momentum is conserved or not is to ask yourself, "is there any outside force or torque acting on the system?" If the answer is yes then momentum is not conserved.

#### BloodySurgeon

Moderator Emeritus
10+ Year Member
Physics Junkie you must really love physics to spend all that time explaining it. Are you or were a physics major?

#### physics junkie

10+ Year Member
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Physics Junkie you must really love physics to spend all that time explaining it. Are you or were a physics major?
I got my degree in biophysics so it was the equivalent of picking up a bio minor and a physics minor. I've been self-studying physics and mathematics for a while though(most of the stuff I'm interested in is at the graduate level) and answering questions here is just another way to keep myself sharp and share my passion. The sense of satisfaction I get from figuring things out is one of the greatest highs.

#### TawMus

10+ Year Member
5+ Year Member
Ok, I get when momentum is conserved but still not sure when energy is conserved... Is energy always conserved unless work is being performed on the system?? and if so , can an outside force like gravity be considered "work"?? Thanks.

#### Kaustikos

##### Archerize It
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Ok, I get when momentum is conserved but still not sure when energy is conserved... Is energy always conserved unless work is being performed on the system?? and if so , can an outside force like gravity be considered "work"?? Thanks.
Energy in a system is not always conserved. Energy of the universe is...
So if the system includes gravity, then energy is conserved. It depends on how you define the "system".