Understanding Kinetic Energy and Momentum: Explained with a Collision Example

[someone]

Hi ppl..

Can someone please explain the following problem to me?

Assuming a 3kg ball traveling at 4m/s hits and joins with a static 1kg ball (a perfect join with no loss of energy).

Momentum before collision = p = mv = 12kgm/s

Using conservation of momentum final velocity of total 4kg mass is 3m/s

The problem I am trying to resolve is the total kinetic energy before the collision = Ke = 1/2 m(v)2 = 24J. I do not understand how the total kinetic energy after the collision calculates to be 18J if no energy is lost when the balls join.

Are my calculations incorrect, or is a perfect collision impossible?

Thanks heaps,
Someone

 

[jcsd]

Your calculations are correct, this shows that the situation is impossible as energy like momnetum muct be conserved.

 

[pmb]

Originally posted by someone
Hi ppl..

Can someone please explain the following problem to me?

Assuming a 3kg ball traveling at 4m/s hits and joins with a static 1kg ball (a perfect join with no loss of energy).

Momentum before collision = p = mv = 12kgm/s

Using conservation of momentum final velocity of total 4kg mass is 3m/s

The problem I am trying to resolve is the total kinetic energy before the collision = Ke = 1/2 m(v)2 = 24J. I do not understand how the total kinetic energy after the collision calculates to be 18J if no energy is lost when the balls join.

Are my calculations incorrect, or is a perfect collision impossible?

Thanks heaps,
Someone

There is no reason for it to be conserved. Kinetic energy is not a conserved quantity. Total energy is the conserved quantity. Only in completely elastic collisions will kinetic energy be conserved.

In this case the kinetic energy is changed into thermal energy. When the items collide and stick together the resultant object will heat up a little.

Pete

 

[jcsd]

pmb, the OP stated that no energy was lost duing the collsion which shows that this is an impossible scenario.

 

[pmb]

Originally posted by jcsd
pmb, the OP stated that no energy was lost duing the collsion which shows that this is an impossible scenario.

Thanks. I didn't notice that (just woke up).

At least he has a complete answer as to why it can't happen.

Pete

 

[Chi Meson]

The OP say "no loss of energy". It does not say "no loss of Kinetic Energy." So Pete's post is valid.

If the author of the problem intended to say "there is no loss of kinetic energy" then the author forgot that only a perfectly elastic (a "boingy" collison) can conserve KE, and the problem is invalid.

In the real world, both thermal energy and sound are produced, and the amount of thermal energy plus sound energy must equal the lost kinetic energy.

 

[pmb]

Originally posted by Chi Meson
The OP say "no loss of energy". It does not say "no loss of Kinetic Energy." So Pete's post is valid.

Questions like this are interactive. The person may not have said what they were thinking or had something in mind that we didn't get.

We'll have to wait and see if the person clarifies.

Pete

 

[jcsd]

Originally posted by Chi Meson
The OP say "no loss of energy". It does not say "no loss of Kinetic Energy." So Pete's post is valid.

If the author of the problem intended to say "there is no loss of kinetic energy" then the author forgot that only a perfectly elastic (a "boingy" collison) can conserve KE, and the problem is invalid.

In the real world, both thermal energy and sound are produced, and the amount of thermal energy plus sound energy must equal the lost kinetic energy.

Well no loss of enrgy is a given, due to the conservation of energy, but as written ther is only one way to take it; as the system retaining it's kinetic energy.

 

[pmb]

Originally posted by jcsd
Well no loss of enrgy is a given, due to the conservation of energy, but as written ther is only one way to take it; as the system retaining it's kinetic energy.

Hence the person's conundrum. For some reason many people don't take thermal energy into account when they think about things colliding and sticking together. That was probably something which the person overlooked. Hopefully they are clear on this now.

Pete

 

[Chi Meson]

I just wanted to say that I was agreeing with everyone; I only intended to clarify the source of confusion for Someone.

Was that a paradox? An oxymoron?

 

[someone]

Thanks all for answering

i have no problem if some of the kinetic energy is lost during the collision but the out standing problem that i originally had still remains…

ie. It must be possible for 2 objects to collide and for the losses due to the collision to vary depending on the nature of the material. Conservation of momentum dictates that there must be only one final velocity therefore the final kinetic energy must also be fixed. If this is correct it dictates that there will always be a loss kinetic energy of a fixed amount at the collision.

This does not seem correct to me, different collisions must have different losses. Hence the problem.

1) If there is no collision loss of energy there must be a problem
2) If there is energy lost the calculations dictate that it must be constant and this also appears to be a problem as different collisions must loose different amounts of energy.

Hope this clarifies,
Someone

 

[pmb]

It must be possible for 2 objects to collide and for the losses due to the collision to vary depending on the nature of the material. Conservation of momentum dictates that there must be only one final velocity therefore the final kinetic energy must also be fixed. If this is correct it dictates that there will always be a loss kinetic energy of a fixed amount at the collision.

Absolutely. Kinetic energy is lost in non-elastic collisions. Energy is not lost. However momentum is always conserved.


E.g. Suppose that you ram two identical balls of clay together at the same speed. I.e. they have the same exact mass and are traveling in equal and opposite velocities. Suppose they stick together upoin collision. Then since the total momentum must be conserved the resultant ball of clay will not move. Momentum is then conserved and kinetic energy will not be conserved.


re - "1) If there is no collision loss of energy there must be a problem"

No. There is no problem.

re - "2) If there is energy lost the calculations dictate that it must be constant and this also appears to be a problem as different collisions must loose different amounts of energy."


There is no energy lost. There is only kinetic energy which is lost. But the energy remains constant. There is an increase in thermal energy of the objects.

Try to think of each object like a system of particles. It should be clearer then as to the physics. Each particle in each system adds to the total momentum. However the systems my undergo internal rearrangement such that the bulk motion of the objects suffer a change in internal energy. The change in internal energy then results in less bulk motion energy. Does than make sense?

Pete

 

[turin]

Why do you think that different materials should have different energy losses. The energy loss may have a different profile (the transfer chooses the different forms with different proportions), but the overall loss, when you add up the loss in each form, should, like you said, be identical in every completely inelastic collision given the same mass for each object and the same initial velocities for each object.

 

[pmb]

Originally posted by turin
Why do you think that different materials should have different energy losses. The energy loss may have a different profile (the transfer chooses the different forms with different proportions), but the overall loss, when you add up the loss in each form, should, like you said, be identical in every completely inelastic collision given the same mass for each object and the same initial velocities for each object.

Who are you addressing this question to?

Pete

 

[someone]

Thanks for the reply Pete..

I should have made my description clearer..

re - "1) If there is no collision loss of energy there must be a problem"

No. There is no problem.

What i mean here is if there is no loss of energy, that is KE, converted to thermal etc.. then there is a problem because if momentum is conserved then the final speed is determined irrespective of the nature or material or the collision. If this is true then the calculations dictate that a fixed amount of kinetic energy must always be lost to thermal etc. and that just doesn't seem correct.

Just for clarification i realize that no total energy is lost what i mean by lost is KE converted (lost) to thermal at the collision.

Thanks again,
Someone

 

[Chi Meson]

Originally posted by someone

What i mean here is if there is no loss of energy, that is KE, converted to thermal etc.. then there is a problem because if momentum is conserved then the final speed is determined irrespective of the nature or material or the collision. If this is true then the calculations dictate that a fixed amount of kinetic energy must always be lost to thermal etc. and that just doesn't seem correct.

Just for clarification i realize that no total energy is lost what i mean by lost is KE converted (lost) to thermal at the collision.

Thanks again,
Someone

I just happen to teach this very point about 4 times every year.

The nature of the collision is dertermined by the elasticity of the materials involved. The more elastic (like a spring) a substance is, the less kinetic energy is turned into heat.

In a collision of two objects, it is not true to think that conservation of momentum will determine the final velocities of any single object. The only situation where this is essentially true is the kind where the two objects are stuck together (perfectly inelastic). This happens because the materials are NOT like springs, but more like balls of clay; that is, very very, so not elastic.

IF two perfectly elastic balls struck each other, total KE is conserved AND total momentum (p) is conserved. The balls will bounce off each other (assuming one is not a lot more massive than the other), one going east, one going west. Since momentum is a vector, the net p is from a vector sum, where one of the p is "negative."

If two balls of clay strike each other, they stay stuck. The single velocity of the two balls will make the KE and p easier to calculate, but this velocity is obviously nothing like either velocity of the elastic collision.

 

[turin]

Originally posted by pmb
Who are you addressing this question to?

Pete

Sorry, Pete. I was addressing it to someone.

 

[someone]

Thanks for sticking with this one Pete.

I understand your replies that account for the loss of kinetic energy to thermal. But that doesn’t address the problem I’m having.

Let me try to explain.

In the original example the velocity and mass are fixed hence the total kinetic energy and momentum are fixed. When they join together, in order to conserve momentum the final velocity (although it is different from the initial velocity) has also to be fixed. Therefore the final kinetic energy has also to be fixed. Calculations determine that the final kinetic energy is less than the initial by a fixed amount. Now it’s easy to say that that lost kinetic energy has been converted to thermal and I’m sure that in most real world situations that is correct.

The problem is therefore this:

The loss of KE is always a “constant” amount irrespective of the type of material.
1) That means that no matter what type of material joins the losses are the same…. That doesn’t seem correct because losses, for example due to friction, are highly dependent on the type of material.
2) It is impossible for two objects to join together in a lossless connection. This doesn’t seem correct either.

To Quote Pete
re - "1) If there is no collision loss of energy there must be a problem"

No. There is no problem. [the calcs say that there ‘is’ a loss of KE]

re - "2) If there is energy lost the calculations dictate that it must be constant and this also appears to be a problem as different collisions must loose different amounts of energy."

[sorry for being sloppy, when I say energy lost I should really say KE converted to thermal]

There is no energy lost. There is only kinetic energy which is lost. But the energy remains constant. There is an increase in thermal energy of the objects.

Try to think of each object like a system of particles. It should be clearer then as to the physics. Each particle in each system adds to the total momentum. However the systems my undergo internal rearrangement such that the bulk motion of the objects suffer a change in internal energy. The change in internal energy then results in less bulk motion energy. Does than make sense?

[yes this I understand, see above]
End Pete Quote.

 

[pmb]

The loss of KE is always a “constant” amount irrespective of the type of material.
1) That means that no matter what type of material joins the losses are the same…. That doesn’t seem correct because losses, for example due to friction, are highly dependent on the type of material.
2) It is impossible for two objects to join together in a lossless connection. This doesn’t seem correct either.

What is it you mean by "fixed"?

In a given inelastic collision there is a loss of kinetic energy which will depend on the material and the shape etc. If you assume that there is a given amount of kinetic energy lost then you're ruling out this being a general property and have reduced this to a problem in which a specific amount of kinetic energy is lost.

And by shape I mean that the collision may loose energy by distorting the material. Work has to be done to disort the material and that work and distortion will depend on the geometry of that which is coliding.

For example: Let two steel plates collide face on. There will be no disortion of the plates for the most part. However if the plates hit each other on the ends then the plates can bend and that takes energy to do that.

Pmb

 

[turin]

Originally posted by someone
The loss of KE is always a “constant” amount irrespective of the type of material.
1) That means that no matter what type of material joins the losses are the same…. That doesn’t seem correct because losses, for example due to friction, are highly dependent on the type of material.
2) It is impossible for two objects to join together in a lossless connection. This doesn’t seem correct either.

If you're going to think real world, then the equations you're using doing exactly work; they only model ideal collisions (ideally inellastic). Lumps of clay are very close to this idealization; billiard balls are not, and rubber erasers are somwhere in the middle, but none of these real world collisions will exactly fall into the idealized categories. It looks like you have intuitively stumbled on the limitation of the equation to ideally inelastic collisions.

btw: not all of the loss in translational KE is thermal. It could be sound (I forget who first mentioned this, sorry), rotation, or probably something else that I'm not thinking of. Also, the idealized model you use assumes that the system is closed, but radiated heat and sound violate this closure. In the real world it is sometimes very difficult to remove the environment to approxiamte the system as closed.

 

[someone]

Thanks for all your contributions.

By fixed i mean specified in the original problem.

In summary.. by all my calculations, because the final velocity is fixed because of the law of conservation of momentum the loss of kenotic energy is always constant in the example given without any factor involver which relates to the type of material this to me doesn’t seem right.

 

[Chi Meson]

The only case in which a final velocity is "fixed" by the law of conservation of momentum is the case where the two objects stick together. The only reason that they stick together is because of the material they are made from.

Different materials will "bounce" with something between 0% and 100% elasticity, resulting with varying final velocities.

 

[krab]

Originally posted by someone
In summary.. by all my calculations, because the final velocity is fixed because of the law of conservation of momentum the loss of kenotic energy is always constant in the example given without any factor involver which relates to the type of material this to me doesn’t seem right.

I'm curious to know why you think this simple prediction from physics "doesn't seem right". To me, it's no different than saying that if I drop a mass M from height h, and it doesn't bounce, the energy it absorbs is always the same, irrespective of its material.

 

[someone]

Chi as specified in the original problem the two masses do stick hence the final velocity is fixed hence the loss of kinetic energy is fixed.

krab, the part i find unusual is that i would expect masses of two different types of material eg. clay to clay or steel to steel is to stick together and generate more or less loss, clay for eg. i would expect to stick less efficiently than steel, with contact adhesives. Maybe I’m wrong and all sticking actions absorb the same energy but this certainly isn't the case with for eg. friction.

Thanks for all your contributions.

 

[Chi Meson]

Someone,

I understand your conundrum, and it shows that you are thinking deeper into the concept than most do.

Suffice it to say that there is no contrdiction and you will probably be comfortable with it pretty soon.

 

[krab]

Originally posted by someone
the part i find unusual is that i would expect masses of two different types of material eg. clay to clay or steel to steel is to stick together and generate more or less loss, clay for eg. i would expect to stick less efficiently than steel, with contact adhesives. Maybe I’m wrong and all sticking actions absorb the same energy but this certainly isn't the case with for eg. friction.

Here's something that maybe will help. If the two particles are clay, each will deform a certain amount, thus absorbing the lost energy. If one is steel, and the other clay, and the collision is still completely inelastic (BTW, "inelastic" is the technical term for collisions that stick), then the result that the amount of absorbed energy is still the same and the fact that the steel deforms negligibly, predicts that the one clay piece must deform to absorb all the energy missing from the final kinetic energy of the joined particles, and hence will deform more than each does in the clay-clay case. This is how physics is used in analyzing collisions, for example between vehicles.

You can test this sort of thing experimentally. Take a ball of clay and drop from a given height onto another ball of clay. Measure the deformation suffered by each ball. Now reform one of the clay balls, and drop onto a steel ball. Measure how much it deforms. I predict it deforms twice as much as in the first case.

 

[someone]

Thanks krab i understand all of that.. but the one example that you didn't mention is the steal to steal stick. The equations indicate that steal to steal is just as lossy as clay to clay, that's the part that doesn’t seem right to me. I would expect clay to clay collision to be very lossy but steal to steal to much less so as it would be in an elastic collision.

P.S. The steal to steal collision in the real world mould of course require a thin film of some type of contact adhesive but deformation of material would be nothing like it would be for clay.

Thanks again,
Someone

 

[ahrkron]

Originally posted by someone
Thanks krab i understand all of that.. but the one example that you didn't mention is the steal to steal stick.

That's because you will need something else in between to absorb precisely the energy to be lost.

The equations indicate that steal to steal is just as lossy as clay to clay

Yes, as long as you find a way to have them stick together upon collision. As you mentioned, you will need to provide a third object ("some type of contact adhesive"). Think about it: not any adhesive layer will do. If too thin, its deformation will not be enough to absorb all the needed energy, and the two steel spheres bounce off each other (although with a reduced speed, compared to that which they would have had without adhesive).

You can imagine yourself repeating the experiment, each time applying some more adhesive, until they stick. When is the adhesive layer "good enough" to hold the steel pieces together upon collision?precisely when it is thick enough to absorb the right amount of energy, (given its other physical properties).

 

[someone]

Thanks all for your contributions, I need to study this subject further.

 

[turin]

It seems as though some of us are ignoring the fact that the systems in questions are not closed. Two clay balls make a sound when they collide; this is kinetic energy lost in the collision, and it is not absorbed, it is lost. I presume that there are other forms of energy loss, but sound is the only one that comes to mind. When two metal rods collide, there is probably a loud clang, which probably indicates even more energy leaving the system. For sure, this amount of energy loss itself is insufficient, so you will need a bit of adhesive. But the lost kinetic energy does not all get absorbed by the colliding bodies.