# How are momentum and kinetic energy related?

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A body with relative motion (transitional not vibrational motion) to the Earth's frame of reference, got going in a certian direction by having a force applied to it, different than the forces being applied to its surroundings. This force had a vector quality associated with it. A magnitude, and a direction. Momentum is thusly outfitted. But kinetic energy is stripped of its direction and is a scalar quantity only. Just an amount, with no direction. There might be definitional reasons for this, but my speculation here is that the two, kinetic energy and momentum are related, and are not unrelated concepts. That is, when energy is added to an object or mass, by a force, it is added in a particular direction, like the cue hitting the cue ball and sending it off in a particular direction, the cue ball transfers that energy to the object it hits, with a vector included.

The energy is stored in the moving mass and transferred to the item it contacts, and there is a similarity to the direction in which the force is applied to the object hit, that can be traced back along the route of the moving object to the point at which a force in that direction, first acted upon it.

So why are kinetic energy and momentum two diffent concepts, when they are so closely related, conceptually?

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Momentum = m.v (mass x velocity)

K.E. = 1/2 m.v^2 (half the mass x velocity squared)

They are both defined by mass and velocity.... so yes, they are related by half v.

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They both relate to motion, so they are not completely unrelated. As scalars, KE = p^2/2m for single massive objects where relativistic corrections can be ignored. However there are systems in which momentum is conserved and kinetic energy is not. KE is but one form of energy, while momentum is momentum, and is only changed by an external force. Momentum is conserved, for example, in an inelastic collision, but KE is not.

Also, KE is also not a vector. It has no direction. Two balls of equal momentum headed toward each other need not have the same kinetic energy, so while the net momentum of the system is zero, the kinetic energy is not.

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That is, when energy is added to an object or mass, by a force, it is added in a particular direction, like the cue hitting the cue ball and sending it off in a particular direction, the cue ball transfers that energy to the object it hits, with a vector included.

Ok so they are related, but they are also different.

It is possible to transfer momentum, without energy transfer.

Take for instance the impact of a stream of water on a blank wall.

The water impact transfers all its forward momentum as a (pressure) force exerted on the wall.

Hower since the wall does not move this force does no work and no energy is transferred.

Particles bouncing off a surface are another example of this.

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But as an "impulse" that comes from one direction, and is heading in one direction, can be conceptually considered as a thing, or a packet of energy, or a wave, or a particle, if this packet of energy is embodied in the mass, by virtue of its motion, does not the motion of the mass indicate a direction that the energy is taking?

In discussions of energy, or potential energy and kinetic energy, a mass' position, in respect to gravity, for instance, matters. If a bowling ball is about to roll off a shelf, it is going to eventually apply a force on the floor, not the ceiling. The kinetic energy of the falling ball, is going in the same direction as its momentum.

SwansonT,

Kinetic energy might not stick as closely to a mass as its momentum does, but the energy has to be accounted for, during an impact, even though it may not do work, and even though it may change its form...into waste heat for instance. But while a body is in motion, its momentum represents a force that will act on a stationary inelastic mass it might run into. There is a good chance that the force that got the mass moving, had a directional component which the moving body is still carrying, and which will be transferred to the inelastic body which is contacted. In this, the mass is a carrier of a force, with the directional characteristics of the force that got it moving in the first place. How does one decide which parts of this impulse is KE and which parts is momentum?

Regards, TAR

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Quote:"Kinetic energy might not stick as closely to a mass as it's momentum does,..."

Yes it does... both are directly proportional to the mass of the object.

Quote"which part is KE and which part is momentum?"...

The mass doesn't 'carry a force' as you put it... it has momentum (mv) and KE (1/2mv^2)... it imparts a force (an impulse) on another object during a collision and momentum and energy can be transferred.

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How does one decide which parts of this impulse is KE and which parts is momentum?

You don't; this is another reason why they are separate things. Impulse is all momentum, by definition. KE is tied in with work.

Impulses are forces exerted over a time, changing the momentum. Work is force exerted over a distance. The commonality is a force, but momentum and KE each have a separate additional part, that are not inherently related.

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But as an "impulse" that comes from one direction, and is heading in one direction, can be conceptually considered as a thing, or a packet of energy, or a wave, or a particle, if this packet of energy is embodied in the mass, by virtue of its motion, does not the motion of the mass indicate a direction that the energy is taking?

No that is the opposite of what I said.

The stream of water comes from one direction, carrying its energy and momentum with it.

The stream hits the wall and transfers all the momentum but none of the energy.

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In discussions of energy, or potential energy and kinetic energy, a mass' position, in respect to gravity, for instance, matters. If a bowling ball is about to roll off a shelf, it is going to eventually apply a force on the floor, not the ceiling. The kinetic energy of the falling ball, is going in the same direction as its momentum.

The discussion was whether KE was a vector — it isn't.

PE wasn't part of the conversation, but it isn't either, since it only depends on relative location. It has a sign, but not a direction.

Every time you say anything along the lines of "kinetic energy … is going in the same direction as its momentum" one of Schrödinger's cat's kittens dies.

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In my water stream example the water looses all its 'sense' of its former direction as it streams off against the wall in all directions.

It could have come from lots of different places.

Here is another example.

Consider a pendulum.

It has lots of kinetic energy as it swings through the low point from left to right.

It also has lots of momentum towards the right.

It also has zero potential energy at this point in its swing.

When it swings back the other way it has exactly the same kinetic energy,

But its momentum is exactly opposite from the left to right swing.

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SwansonT,

OK, I forgot there was an issue determining both a particle's postion and its momentum.

I will have to rethink why I think KE should be more than a scalar.

studiot,

I was thinking pressure washer, or cleaning my patio off with the hose set to "jet". I can direct the dirt and the built up water toward the low corner of the patio by positioning the stream in such a way as it pushes the dirt and water in the same direction as it is coming out the nozzle.

Regards, TAR

OK, maybe the pendulum shows me.

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I was thinking pressure washer, or cleaning my patio off with the hose set to "jet". I can direct the dirt and the built up water toward the low corner of the patio by positioning the stream in such a way as it pushes the dirt and water in the same direction as it is coming out the nozzle.

Yes you can move the dirt, but not (hopefully ) the patio concrete.

All that shows is that in some cases momentum but no energy is transferred, in others both are transferred.

Is that a problem?

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SwansonT,

OK, I forgot there was an issue determining both a particle's postion and its momentum.

I will have to rethink why I think KE should be more than a scalar.

Heisenberg has nothing to do with this. It's purely a classical physics issue.

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Studiot,

No, not a problem per se. But KE and Momentum are not mutually exclusive items. They are used to figure different things about a moving item. My concern is that the momentum of a moving object carries a directional component and when the momentum is transferred to another item during a collision it causes the other item to accelerate in the direction the first item was moving, as the first item decelerates or perhaps accelerates in the opposite direction. The kinetic energy is also traveling with the first mass, and transfers to item number two. Just an amount though, without a direction.

Why does the direction of an impulse not matter? How is that desk prop with the steel balls touching each other analized in terms of momentum vs KE when the left ball is dropped on its neighbor and the far right ball takes on the momentum. The middle balls never move, they never change their momentum, but some KE travels through, in a certain direction.

Regards, TAR

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No, not a problem per se. But KE and Momentum are not mutually exclusive items. They are used to figure different things about a moving item.

I'm not sure what you mean by mutually exclusive.

Yes they are different and they account for different properties, but sometimes and only sometimes there is a direct connection.

This is rather like mass and volume.

For a solid an increas in volume means an increase in mass.

But for a gas an increase in volume may simply mean a decrease in pressure.

Both roughly indicate the something about the amount of matter and how it will behave in specific circumstances.

Why does the direction of an impulse not matter? How is that desk prop with the steel balls touching each other analized in terms of momentum vs KE when the left ball is dropped on its neighbor and the far right ball takes on the momentum. The middle balls never move, they never change their momentum, but some KE travels through, in a certain direction.

I assume you mean what we call Newton's Cradle?

Raise the left hand ball and let it drop back.

The dynamics runs like this:

You have two objects, one ball and a group of three balls.

At the moment of impact the 3ball has zero momentum and KE and the 1ball has a certain KE and a certain momentum.

At impact you now have a single object a group of four balls.

This object has the total original momentum and KE.

If you watch very carefully the group does in fact start to move, but the impulse travels through the balls at the speed of sound. That is hundreds of times faster than the original ball was travelling, and of course the 4ball will travel at 1/4 the 1ball speed given the same momentum. So the difference is enhanced by the regrouping of the balls.

The centre two balls have something to restrain them but there is nothing beyond the right outer ball so it carries off the momentum and KE, leaving a group of three behind.

Momentum is thus preserved.

You should analyse this with momentum first to get the new velocity and then work out the KE at the new speed.

What happens to the 'lost energy' you ask?

Well you hear some of it in the clack sound and some of it remains distributed in elastic deformation of the 4ball until again the fact that the outer ball is unrestrained and allows it to accept the KE from the elastic recovery and end up with all the cake.

That's the simplified version.

Why does the direction of an impulse not matter? How is that desk prop with the steel balls touching each other analized in terms of momentum vs KE when the left ball is dropped on its neighbor and the far right ball takes on the momentum. The middle balls never move, they never change their momentum, but some KE travels through, in a certain direction.

Remember an impulse is a (large) force applied for a short time. Only momentum can be converted to force not energy, and it take force to accelerate the stationary balls.

That is why we take the momentum balance as the primary and work the KE out to suit.

During the exchange the energy doesn't disappear it suffuses all the balls (without direction) as elastic stored energy and is returned to supply the motion KE.

You can be very complicated and calculate all the exchange forces as each ball deforms elastically in turn and you will get the same answer.

Edited by studiot
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KE and Momentum are not mutually exclusive items.

Nobody has said they are.

Why does the direction of an impulse not matter?

It does matter, if you are using impulse as defined in physics. If you aren't then it's possible nobody knows what you are asking.

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Do you understand the difference between velocity and speed, tar ?

Momentum is mass x velocity, while kinetic energy is 1/2 mass x speed squared.

As swansont and studio have stated, a mass can bounce off a wall, or a pendulum can swing in the opposite direction, and have a negative momentum ( opposite direction ) while retaining the same ( nearly ) kinetic energy.

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Remember an impulse is a (large) force applied for a short time.

In this case. It can also be a small force applied for a long time.

I wonder if reading the definition of impulse will help tar get to grips with this: http://en.wikipedia.org/wiki/Impulse_%28physics%29

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OK, maybe a little clearer. Speed is just a scalar, Velocity is vector, with a direction. I think I got misdirected by a KE formula I saw somewhere, that had a v in it. And still a bit cloudy on the impulse thing, as it seems an impulse can have a definite direction in which it is acting, and a mass does not seem to have to move, to carry the impulse.

I am currently, mentally considering a solenoid with a 100 meter, inelastic shaft attached inline to the end in which direction actuation would take the solenoid, and with a hundred meter, inelastic wire, attached to its back end. During activation of the solenoid, a force, or impulse will travel along the shaft and act upon an item placed at the distant butt end of the 100 meter shaft. Similarly a force during activation will pull on the wire and be transferred to the other end of the wire (which is attached to a small loose mass which with move in the direction of activation)

What is the speed of the impulse traveling along the shaft? Speed of sound? Speed of light(magnetic force, strong force, weak force)? Speed of the soleniod? How about the speed of the impulse along the wire? How does the momentum get from the solenoid end of the shaft to the far end of the shaft, in that particular direction, if no mass moves more than the throw of the solenoid?

Is there not a directional component of the impulse that belongs with the analysis? How does one satisfy themself with a center of mass formula, when the important parts of what is happening are happening 100meters away from each other?

Regards, TAR

Edited by tar
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What does that have to do with the OP?

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I am sorry to complicate matters for you tar, but Strange's discourse on Impulse is incomplete.

Impulse is intended for impact forces, not for disturbances travelling in elastic media, and subject to different laws of propagation.

A simple example ( I seem to be trying to find a lot of these)

Think of a block of sponge and and block of wood of similar density placed against a wall.

Now apply an impulse ( with a hammer or billiard ball ?) to the sponge and to the wood and consider how the 'push' is transmitted to the wall.

It is the reaction force from the wall on the two blocks that prevents bodily movement of the block under the impacts.

There are two stages.

The first stage the impactor strikes and the block deforms locally and elastically under the strike.

The sponge is deformation much slower and softer so the impulse is less.

The rate of deformation depends on the characteristics of both the impactor and the block material, but is much slowerr than the speed of sound in the blocks.

This is the mechancial impulse.

The local is transmitted through the blocks as an elastic wave disturbance from molecule to molecule until the last molecule at the other end pushes against the end wall.

This is not called an impulse and travels at the speed of sound in the material of the block.

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Impulse is intended for impact forces

Maybe you need to read the definition of impulse, as well.

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Maybe you need to read the definition of impulse, as well.

Perhaps I was just being kind to Wikipedia?

Defining impulse as the indefinite integral of Fdt is meaningless.

Impulse is properly defined as the definite integral

$I = \int_{t2}^{t1}F.dt$

The point is that it has a begining and and end.

The disturbance travelling through the elastic media does not necessarily have either.

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Impulse is properly defined as the definite integral

$I = \int_{t2}^{t1}F.dt$

As the Wikipedia article says. (But thanks for admitting you were wrong, previously.)

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As the Wikipedia article says. (But thanks for admitting you were wrong, previously.)

I'm sorry? here is a screenshot from the link you gave.

How is the Wiki integral the same as mine?

Edited by studiot

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