Momentum vs Kinetic Energy in classical physics

[Prem1998]

I've added 'in classical physics' in the thread title because all the differences between them that I found on the internet involved relativistic physics. It was something like both momentum and kinetic energy being components of a four-momentum or something like that. But I cannot understand those concepts.
These two quantities were defined before there was any relativistic physics. In classical physics, both of them are a quantitative way to express the amount of motion contained in a body. So, I need an intuitive explanation of why do we need to define both of these two quantities.

 

[Vanadium 50]

Why do we need position and velocity? They are different things.
Why do we need momentum and energy? They are different things.

 

[Prem1998]

Why do we need position and velocity? They are different things.
Why do we need momentum and energy? They are different things.

Well, that doesn't help at all. I don't think momentum and kinetic energy are analogous to position and velocity at all.
Both momentum and kinetic energy quantify the 'amount' of motion present in a body. Then, what's the difference between the two? What's the difference between mv and 1/2mv^2 if they're both two ways to measure motion and give it a number?

 

[Merlin3189]

Essentially your big error is to say, "Both momentum and kinetic energy quantify the 'amount' of motion present in a body." Speed (or velocity) alone quantifies motion. (I don't know whether to say speed or velocity, because I'm not sure whether your quantity of motion is a scalar or a vector.)

As Vanadium says, momentum and KE are used because they measure different things.

For example, if you take simple classical kinematics with two colliding objects, it is found that momentum and mass are conserved in such collisions. It is not found that KE or velocity is conserved. They are different things.

Mass, velocity, momentum and KE are all related. If you know any two of them, you can calculate the others. But each one has its own significance.

Two objects with equal KE have the same capability to do work, or required the same amount of work to gain that KE. This is not true of two objects with equal mass, or with equal velocity, or with equal momentum.
Two objects with equal mass require the same force to give them a certain acceleration. This is not true for objects with equal velocity, or with equal momentum, or with equal KE.
Two objects with equal velocity take the same time to move a given distance. This is not true for two objects with equal mass, or equal momentum, or equal KE.

(For clarification I am using "or" here in its exclusive sense. If two objects have any two of these things equal, like equal momentum AND equal KE, then obviously all four quantities must be equal - trivial .)I don't like the idea of using analogies here, because I don't know what aspect of these we are trying to find an analogy for! But another situation where there are four related quantities which tell us completely different things, is DC electricity. There is voltage, current, resistance and power. If you know any two of these, then you can calculate the others. YOU could say, all four just quantify the amount of electricity, or something. So why do we need voltage and current? But just as with m, v, p and e above, they are really four different things that have a quantifiable relationship.
We could scrap any two of them and still do all the calculations we want. Let's say you wanted to scrap resistance and power. Then we can still describe conductors as having a property called, say, the current to voltage ratio (call it R for Ratio, maybe?) It could have units of Siemens or Amps per Volt. We can describe the heat produced in the conductor again in terms of the two things we kept, voltage and current and use units such as volt amps. or Watts.
But since they are four different things, all of interest to us in different ways, why not use all four of them?

 

[Dale]

In classical physics, both of them are a quantitative way to express the amount of motion contained in a body.

It would really help if you would stick to standard definitions. "The amount of motion contained in a body" is not a standard definition of momentum and it is flat out wrong for energy. Kinetic energy is only one form among many different forms of energy, most of which do not in any way quantify an amount of motion.

The basic reason that we define any word is because the concept is useful and it is more efficient to assign it a name than to repeatedly describe it. It is much easier to say "kinetic energy" than it is to say "half of the mass times the square of the speed". Since the quantity shows up a lot it is useful to define the word.

I need an intuitive explanation of why do we need to define both of these two quantities

As far as "intuitive" goes, I think that your best bet is Noether's theorem, which gives one underlying framework for understanding momentum, energy, charge, angular momentum, and many other concepts. It is one of the foundations of modern physics, both classical and quantum mechanical.

It basically says that for every symmetry of a physical law, there is a corresponding conserved quantity. We expect the physical laws now to be the same as the physical laws later, that is the symmetry that leads to energy conservation (not just KE). We also expect the laws of physics to be the same here as they are there, that is the symmetry that leads to momentum conservation.

 

[Nugatory]

Then, what's the difference between the two?

One is a vector and one is a scalar. And that's not just a quibble - it's easy to construct a system in which the momentum is zero and the kinetic energy is non-zero.

Another way of appreciating the difference between the two is to try solving a few elementary collision problems: for example, two given masses with given initial speeds collide and stick together, or collide and rebound elastically.

 

[PeroK]

Well, that doesn't help at all. I don't think momentum and kinetic energy are analogous to position and velocity at all.
Both momentum and kinetic energy quantify the 'amount' of motion present in a body. Then, what's the difference between the two? What's the difference between mv and 1/2mv^2 if they're both two ways to measure motion and give it a number?

Let me give you a different answer. First, I think it's actually a very good question. It seems to me that they are similar and, from elementary classical physics alone, it is mysterious why both are so relevant, each in its own different way. Yet they are clearly related (at least for a single particle).

There are lots of examples of why we need both. But, until I learned Special Relativity, I couldn't help feeling there was something missing here - or, at least, something unexplained.

When I did learn SR, the unification of KE and momentum in the energy-momentum four-vector was one of those lightbulb moments. And, an approximation for the KE of a particle with a low velocity much less than the speed of light is:

##(\gamma -1)mc^2 \approx \frac{1}{2}{mv^2}##

It's not an exaggeration to say that that is what made me decide to learn some physics!

You may have to wait until you can master SR or Noether's theorem before you see the light, but classical physics is still worth mastering first, even if it doesn't have all the answers.

 

[A.T.]

So, I need an intuitive explanation of why do we need to define both of these two quantities.

Because energy and momentum are conserved quantities, which is useful to make quantitative predictions.

 

[Prem1998]

Essentially your big error is to say, "Both momentum and kinetic energy quantify the 'amount' of motion present in a body." Speed (or velocity) alone quantifies motion. (I don't know whether to say speed or velocity, because I'm not sure whether your quantity of motion is a scalar or a vector.)

As Vanadium says, momentum and KE are used because they measure different things.

For example, if you take simple classical kinematics with two colliding objects, it is found that momentum and mass are conserved in such collisions. It is not found that KE or velocity is conserved. They are different things.

Mass, velocity, momentum and KE are all related. If you know any two of them, you can calculate the others. But each one has its own significance.

Two objects with equal KE have the same capability to do work, or required the same amount of work to gain that KE. This is not true of two objects with equal mass, or with equal velocity, or with equal momentum.
Two objects with equal mass require the same force to give them a certain acceleration. This is not true for objects with equal velocity, or with equal momentum, or with equal KE.
Two objects with equal velocity take the same time to move a given distance. This is not true for two objects with equal mass, or equal momentum, or equal KE.

(For clarification I am using "or" here in its exclusive sense. If two objects have any two of these things equal, like equal momentum AND equal KE, then obviously all four quantities must be equal - trivial .)I don't like the idea of using analogies here, because I don't know what aspect of these we are trying to find an analogy for! But another situation where there are four related quantities which tell us completely different things, is DC electricity. There is voltage, current, resistance and power. If you know any two of these, then you can calculate the others. YOU could say, all four just quantify the amount of electricity, or something. So why do we need voltage and current? But just as with m, v, p and e above, they are really four different things that have a quantifiable relationship.
We could scrap any two of them and still do all the calculations we want. Let's say you wanted to scrap resistance and power. Then we can still describe conductors as having a property called, say, the current to voltage ratio (call it R for Ratio, maybe?) It could have units of Siemens or Amps per Volt. We can describe the heat produced in the conductor again in terms of the two things we kept, voltage and current and use units such as volt amps. or Watts.
But since they are four different things, all of interest to us in different ways, why not use all four of them?

Your DC current analogy was good. Voltage is as good as current to determine how powerful a stream of electrons is and give it a number.
So, none of these quantities are really fundamental and defining quantities is just about giving a name to a quantity which is convenient to solve a particular problem even if that could be solved by already defined quantities.

 

[A.T.]

defining quantities is just about giving a name to a quantity which is convenient

Yes

to solve a particular problem

Not one particular problem, but many cases.

 

[robphy]

In Tom Moore's Six Ideas text, he presents "changes in kinetic energy" in the following way.

In an interaction, there is momentum exchanged. Consider a tiny net-impulse received by the object ##d\vec p##.
So, the momentum vector of the object changes to ##\vec p + d\vec p##... it may have changed direction, changed magnitude, or both.
However, the "magnitude of the momentum vector" need not change if ##d\vec p## is perpendicular to ##\vec p##.

The [net work done] "change in kinetic energy" ##dK=\frac{\vec p_{new}\cdot \vec p_{new} }{2m}-\frac{\vec p\cdot\vec p}{2m}=\frac{1}{2m}\left( \left(p^2+2\vec p\cdot d \vec p+d\vec p\cdot d \vec p\right) - p^2 \right)=\frac{1}{2m}\left( 2\vec p\cdot d\vec p+dp^2 \right)\approx \vec v\cdot d\vec p##, where we neglected the small ##dp^2## term. With this last expression, if ##\vec v\cdot d\vec p=0##, the kinetic energy was unchanged by the interaction.

 

[David Lewis]

Quantity of motion is the phrase Newton used to refer to what we now call momentum.
The big difference with kinetic energy is that KE goes up as the square of speed.

Newton believed kinetic energy was directly proportional to speed.
This was later proved through a series of experiments by Émilie du Châtelet to be incorrect.

Quantity of electricity is usually called charge (SI unit coulombs).

 

[hackhard]

Both momentum and kinetic energy quantify the 'amount' of motion present in a body. Then, what's the difference between the two? What's the difference between mv and 1/2mv^2 if they're both two ways to measure motion and give it a number?

product mv was always found to be const when net force was 0 . so they gave a special name to it
sum of f.s = mv^2 /2 so right side given special name since it appears frequently