Momentum paradox: Why can't we write it as p=m+v ?

[akashpandey]

So as we know momentum has a formula p=mv right ?
But why we can't write it as p=m+v ?
The real question is why we multiply both mass and velocity quantity
And not add them ?

 

[Ibix]

Because observation tells us that $mv$ is a useful quantity, and $m+v$ makes no sense. You can't add quantities that have different dimensionalities.

 

[akashpandey]

Because observation tells us that $mv$ is a useful quantity, and $m+v$ makes no sense. You can't add quantities that have different dimensionalities.

Ok so as we know multiplication is repeated addition..
How would explain momentum according to multiplication definition ?
Does this mean we are scaling velocity by multiplying mass.

 

[akashpandey]

Or more simply i can put the question like
What does multiplication mean? If we can't add two different types of quantities, then how can we multiply two different types of quantities, like p=mv ?

 

[Ibix]

You can't add quantities with different units because it makes no sense. $m+v$ is the mathematical expression of a question like "what's 1kg more than 10m/s", which makes no sense at all.

You can multiply quantities with different units because the units also multiply. That's possible because quantities with units are already a multiplication - for example 3m is just saying "three times the length of a metre rule" in a compact way. And constructing compound units such as m/s is also clearly meaningful - it's the mathematical expression of a question like "how many times the length of a metre rule did an object travel in the time it took my clock to tick once".

(I'm aware that modern SI defines units in a slightly more sophisticated way than "here's a lump of metal of a defined length". It doesn't make a difference to the point at hand.)

 

[akashpandey]

You can't add quantities with different units because it makes no sense. $m+v$ is the mathematical expression of a question like "what's 1kg more than 10m/s", which makes no sense at all.

You can multiply quantities with different units because the units also multiply. That's possible because quantities with units are already a multiplication - for example 3m is just saying "three times the length of a metre rule" in a compact way. And constructing compound units such as m/s is also clearly meaningful - it's the mathematical expression of a question like "how many times the length of a metre rule did an object travel in the time it took my clock to tick once".

(I'm aware that modern SI defines units in a slightly more sophisticated way than "here's a lump of metal of a defined length". It doesn't make a difference to the point at hand.)

Yeah got your point..
But definition of multiplication doesn't go hand in hand with this momentum thing.. that's all i am saying.

 

[akashpandey]

You can't add quantities with different units because it makes no sense. $m+v$ is the mathematical expression of a question like "what's 1kg more than 10m/s", which makes no sense at all.

You can multiply quantities with different units because the units also multiply. That's possible because quantities with units are already a multiplication - for example 3m is just saying "three times the length of a metre rule" in a compact way. And constructing compound units such as m/s is also clearly meaningful - it's the mathematical expression of a question like "how many times the length of a metre rule did an object travel in the time it took my clock to tick once".

(I'm aware that modern SI defines units in a slightly more sophisticated way than "here's a lump of metal of a defined length". It doesn't make a difference to the point at hand.)

m/s is how man times the length of a meter rule object traveled in one tick of my clock.

So how would mathematically define this "kg.m/s" ?

 

[Dragon27]

What kind of unit would this hypothetical physical quantity have and what can it be used for? So we take 1 kilogram and 1 meter per second and add them and get 2 kg+m/s? What if we choose different units? Momentum will scale in the obvious manner, 1 kg·m/s = 1000 g·m/s, whether we multiplied 1kg by 1m or 0.5 kg by 2m/s, but your added quantity will depend on the values we adding up, so if we add 1kg and 1m/s it will be 2 kg+m/s = 1001 g+m/s, but if we add 0.5 kg and 1.5 m/s we will get 2 kg+m/s = 501.5 g+m/s, which can't be derived independently from the original 2 kg+m/s without knowing the separate values that went into it. We'll have to keep track of this values all the time (kind of like that: 1kg+1m/s=1000g+1m/s) so there was no point in adding the numbers in the first place.
The multiplied value holds its own as an independent value under the change of units.

 

[Lnewqban]

The real question is why we multiply both mass and velocity quantity

The magnitude of an object's mass multiplies the magnitude of its vector velocity without changing its direction.

An object's momentum is a vector quantity, which has a magnitude and a direction.

Please, see:
https://en.m.wikipedia.org/wiki/Scalar_multiplication

https://en.m.wikipedia.org/wiki/Momentum

 

[Stephen Tashi]

m/s is how man times the length of a meter rule object traveled in one tick of my clock.

Physical units can have more than one interpretation. One example of the interpretation of m/s is as a velocity, but there can be different examples.

So how would mathematically define this "kg.m/s"

That question is not addressed by mathematics. For example, trigonometry does not tell us that the hypoteneuse of a triangle must be a ladder leaning against a wall. Properties of a ladder leaning against a wall might be analyzed by using trigonometry, but the mathematics of trigonometry does not tell us that only one particular application of trigonometry exists.

 

[A.T.]

Ok so as we know multiplication is repeated addition..

How do you write √2 * π as repeated addition?

 

[PeroK]

Summary:: Why we multiply mass and velocity ?

So as we know momentum has a formula p=mv right ?
But why we can't write it as p=m+v ?
The real question is why we multiply both mass and velocity quantity
And not add them ?

If you think of a mass $M$ as being composed of a lot of smaller masses, $m$, all moving together at some velocity $v$. Let's say 100 small masses make up the larger mass.

If you add mass and velocity, the the momentum of each component would be $p = m + v$ and the momentum of the whole thing would be $P = 100(m + v) = 100m + 100v$.

But, if you calculate the momentum of the large mass directly you get $P = M + v = 100m + v$. And you get a different answer.

Adding two quantities in this way makes no sense.

Whereas, if you multiply quantities, then we have $p = mv$, and $P = 100mv = Mv$, and it all makes sense.

 

[Frigus]

Summary:: Why we multiply mass and velocity ?

So as we know momentum has a formula p=mv right ?
But why we can't write it as p=m+v ?
The real question is why we multiply both mass and velocity quantity
And not add them ?

I think when people started studying motion,they didn't knew that their is quantity momentum which remains conserve but when they started doing maths they found that this quantity mv remains same and they gave name momentum to it.
It would be like saying something like why cucumber is cucumber not orange.

 

[A.T.]

Whereas, if you multiply quantities, then we have $p = mv$, and $P = 100mv = Mv$, and it all makes sense.

It makes sense formally, but it also has to be useful somehow, which it is because it's conserved.

 

[sophiecentaur]

But definition of multiplication doesn't go hand in hand with this momentum thing.. that's all i am saying.

Take a step backwards. When did you first learn the definition of Multiplication as repeated additions? I suspect it was when you were less than ten years old. Since then, the more advanced concepts of Maths crept into your awareness - even if you never did an advanced Maths Analysis course and you just do it mechanically. However, once you go beyond integers in the application of that definition then you have to introduce implied multiplication and insert that in your 'multiple additions' definition. It has to pull itself up by its own bootstraps - especially when the numbers are not even rational.

The 'rules' about what you are allowed to do about dimensions are pretty straightforward and are actually pretty intuitive. You can multiply different quantities and get a meaningful answer. Amp Hours as the capacity of a battery make sense because you can use the same capacity battery for twice as long by taking half the current etc. etc. etc.. But what would adding possibly 10A to 5H mean? 15 'Somethings'? Equivalent to 1A plus 14H?? Clearly not.

Once you have established the principle of what does and what dean;t make sense then you OP question is answered.

 

[Dale]

Yeah got your point..
But definition of multiplication doesn't go hand in hand with this momentum thing.. that's all i am saying.

The repeated addition thing is not the definition of multiplication. It is the easiest way to teach multiplication to little children who only know addition. The repeated addition thing only even works for multiplication by integers.

 

[Meir Achuz]

How could a joke get so many answers?

 

[PeroK]

The repeated addition thing is not the definition of multiplication. It is the easiest way to teach multiplication to little children who only know addition. The repeated addition thing only even works for multiplication by integers.

That's how mutlipication is defined for positive integers. Multiplication of the rationals and the reals is defined by a systematic generalisation.

 

[Frigus]

The repeated addition thing is not the definition of multiplication. It is the easiest way to teach multiplication to little children who only know addition. The repeated addition thing only even works for multiplication by integers.

My whole life was a lie.

 

[Dale]

My whole life was a lie.

Well, I may be wrong about that. I cannot find an independent definition of real multiplication. So don’t question your life quite yet!

 

[Paul Colby]

Well, in the grand scheme of things using dimensional real numbers to quantify things in the physical sciences is really a rather remarkable process. One could argue all day how "intuitive" it is but in the end the real reason is it's observed to work really well.

 

[PeroK]

Well, I may be wrong about that. I cannot find an independent definition of real multiplication. So don’t question your life quite yet!

Addition and multiplication of the rationals is, of course, defined by:
$$\frac a b + \frac c d = \frac {ad + bc}{bd}, \ \ \ \frac a b \times \frac c d = \frac {ac}{bd}$$
Where $a, b, c, d$ are integers, with $bd \ne 0$.

Multiplication of the Real numbers depends on how you construct them from the rationals. One approach is to define a real number as an equivalence class of sequences of rationals. Then you have to show that if you take two such real numbers, then the product can be well-defined by multiplying the sequences term by term etc.

As has been mentioned before constructing the real numbers from the rationals is a non-trivial exercise.

 

[martinbn]

Multiplication of the rationals is, of course, defined by:
ab×cd=ad+bcbd
Where a,b,c,d are integers, with bd≠0.

Multiplication of the Real numbers depends on how you construct them from the rationals. One approach is to define a real number as an equivalence class of sequences of rationals. Then you have to show that if you take two such real numbers, then the product can be well-defined by multiplying the sequences term by term etc.

As has been mentioned before constructing the real numbers from the rationals is a non-trivial exercise.

Well, that is addition. multiplactions is $\frac{ac}{bd}$.

 

[PeroK]

Well, that is addition. multiplactions is $\frac{ac}{bd}$.

So it is! I better correct it.

 

[anorlunda]

Even with repeated addition, you are still not adding things with different units.

1 apple x 3 means 1 apple + 1 apple + 1 apple. You are not adding the 3 to the 1.

50 mph *2 hours = (50 mph *1 hour) + (50 mph * 1 hour) = 50 miles + 50 miles = 100 miles

 

[Dale]

Ok, I am convinced.

Of course, multiplication of units cannot be identified with either multiplication of reals or integers. It has its own rules and it is just one of those rules that you can multiply disparate units but not add them.

 

[akashpandey]

How do you write √2 * π as repeated addition?

So tell me how else you define multiplication ?

 

[kuruman]

Summary:: Why we multiply mass and velocity ?

So as we know momentum has a formula p=mv right ?
But why we can't write it as p=m+v ?
The real question is why we multiply both mass and velocity quantity
And not add them ?

The short answer to your question is "because we multiply acceleration by mass to get the net force." Consider the following observation (please don't try it at home - it's a thought experiment).

Put your hand flat on a table. Drop on it a 0.5-kg book from a height of 1 meter. You will feel no pain. Now drop a 10-kg lead brick on your hand from the same height. You will fell a lot of pain and probably will have to go to the hospital with a broken hand. The speed $v$ of both objects just before contact with your hand is the same. Why the difference in pain level?

Answer: Each object is stopped by your hand in pretty much the same time interval $\Delta t$, therefore the acceleration is pretty much the same. However, the net force required to stop the book is much less than the one required to stop the book because the lead brick is 20 times more massive. The more force your flesh and bones are required to exert, the more painful the sensation; if that force exceeds a certain limit, your bones will break.

BTW, I would not question why write $F_{net}=ma$ and not $F_{net}=m+a$ lest Sir Isaac turn over in his grave.

 

[akashpandey]

How could a joke get so many answers?

I guess no question is dumb enough to ask ?

Say you need to explain it to 10 years old child and he only nows that multiplication is adding a number repeated time.
So how could you explain him that p=mv and p is not m+v.

 

[Dale]

So how could you explain him that p=mv and p is not m+v.

p=mv by definition. Even a 10 year old knows that words have definitions.

Adding m+v is against the rules. Even a 10 year old knows that math has rules.

When the inevitable “why” question is asked the answer is that a bunch of scientists and mathematicians agreed to use those definitions and rules because it gave them useful results for the real world.

 

[Frigus]

I think this will solve your problem,
Suppose their is a quantity m/v and you named it jake,suresh or any other name and one day someone comes to you and ask why jake or suresh is m/v not mv.
Then how will you answer that?

 

[Paul Colby]

So how could you explain him that p=mv and p is not m+v.

Has anyone pointed out that the net $mv$ in a collision is conserved while the net $m+v$ isn't?

 

[akashpandey]

I think this will solve your problem,
Suppose their is a quantity m/v and you named it jake,suresh or any other name and one day someone comes to you ask why jake or suresh is m/v not mv.
Then how will you answer that?

I just don't want to accept the that this is definition that's how it is.
I want to know the real meaning or experiment why it was define that way and not the other.
So just saying this is the definition and accepting it what school does.
Sorry i mean no disrespect to any member of community.
If you guys this question dumb or something else just ignore it.

 

[jbriggs444]

So tell me how else you define multiplication ?

Try this.

One standard approach has already been described in rough outline.

You start with multiplication of the ordinary positive integers defined in terms of repeated addition. Optionally you add a rule for multiplication by zero.

You extend this to signed integers. There are several ways to do this, but they all yield equivalent results. Perhaps the simplest is to simply say by fiat that positive times positive = positive, positive times negative = negative and negative times negative = positive.

You extend this to rational numbers as in #22 (equivalence classes of ordered pairs of signed integers).

You extend this to real numbers using equivalence classes of Cauchy sequences of rationals. Or Dedekind cuts.

 

[Nugatory]

So tell me how else you define multiplication ?

Without mathematical rigor, but at a level that is useful to a child who ready to move past integers and the "repeated addition" definition:

$a\times b$ is the area of a rectangle with sides of length $a$ and $b$.

This works as a lie to children because every naif has an intuitive notion of length and area so there's no need to explore the difficulty of formally defining either. Likewise we gloss over the problem of extending the rational numbers to the reals by blithely stating that $\sqrt{2}\times\pi$ is a rectangle with sides of those lengths.

 

[PeroK]

I just don't want to accept the that this is definition that's how it is.
I want to know the real meaning or experiment why it was define that way and not the other.
So just saying this is the definition and accepting it what school does.
Sorry i mean no disrespect to any member of community.
If you guys this question dumb or something else just ignore it.

If you took a job at $20 an hour and worked 10 hours and the manager said "that's $20 + 10 = $30". Would you go away happy with your $30? If not, why not?

 

[jbriggs444]

If you took a job at $20 an hour and worked 10 hours and the manager said "that's $20 + 10 = $30". Would you go away happy with your $30? If not, why not?

I want to work for 36 billion microseconds.

 

[akashpandey]

If you took a job at $20 an hour and worked 10 hours and the manager said "that's $20 + 10 = $30". Would you go away happy with your $30? If not, why not?

No i know i will add 20$, 10 times that's the basic definition of multiplication

Wel i guess i understood that multiplying with a number is repeated addition cause they are of same type.

But when it comes to two different type of quantity we cannot say that one quantity is added to other multiple times

 

[russ_watters]

Yeah got your point..
But definition of multiplication doesn't go hand in hand with this momentum thing.. that's all i am saying.

Sure it does. You're just having trouble with momentum because it is more abstract.

For example, there's multiplication-by-addition problems you probably do every day: drive 60 mph For 1 hour each way too and from work. How long did you drive and how far did you go?

One soda costs $3 and you are about to buy your third. How much have you spent and will you spend?

Wel i guess i understood that multiplying with a number is repeated addition cause they are of same type.

No they aren't. One has units of dollars, another hours, and the relation combines them.

 

[Frigus]

This works as a lie to children

So until now I was living in a lie.
From now onwards I am even doubting even if I have studied anything right.

 

[Nugatory]

So until now I was living in a lie.

That's OK, you're in good company - essentially every adult who has managed to avoid college-level abstract number theory and therefore accepts "two plus two equals four" as a truism.

 

[russ_watters]

I just don't want to accept the that this is definition that's how it is.
I want to know the real meaning or experiment why it was define that way and not the other.
So just saying this is the definition and accepting it what school does.
Sorry i mean no disrespect to any member of community.
If you guys this question dumb or something else just ignore it.

It's not dumb, but it it is a fundamental misunderstanding of what science and math are. Science is a way to understand the world around us and math is a tool/language to describe what we see.

mv is a quantity that scientists hundreds of years ago discovered was often conserved, and that's useful. There isn't, nor does there need to be any deeper meaning to it than that.

You may also want to look into the history of the development of momentum vs conservation of energy. We've had several discussions of it. There was a lot of feeling in the dark for relationships between velocity and mass, and debate over the value of the two relations. At the time they were competing as much as complimentary.

 

[Dale]

I just don't want to accept the that this is definition that's how it is.

That is silly. There is no other reason for definitions. You need to accept definitions just because “that’s how it is” or you will be unable to communicate.

Once you have agreed to use the definition then you can ask things like why is this concept useful? But a definition is just an arbitrary convention with no further justification needed

 

[akashpandey]

That's OK, you're in good company - essentially every adult who has managed to avoid college-level abstract number theory and therefore accepts "two plus two equals four" as a truism.

Are you mocking me ?

 

[etotheipi]

But why we can't write it as p=m+v ?

It is not dimensionally homogenous but maybe more alarmingly you are adding a scalar to a vector... $m + \vec{v}$... that can't be good!

Maybe we might ask why velocity is to the first power? $mv^1$ and $mv^2$ are meaningful, and we associate the first one with the idea of momentum and the second with the idea of kinetic energy, but I have no idea if $mv^3$, $mv^4$, etc. are of any use. Perhaps it's a complete fluke, and those two combinations just happened to play nicely. In any case, those two useful ones are themselves just approximations in th low speed regime of the relativistic formulae which are a little more ugly, so $mv$ is not even the full story for momentum!

 

[Paul Colby]

It is not dimensionally homogenous but maybe more alarmingly you are adding a scalar to a vector... m+v→... that can't be good!

Actually, in Clifford Algebras this kind of thing is permitted and meaningful for some value of meaning. And of course not meaningful here. The real answer to the OPs question is, as others have pointed out, in the application of algebra to physical theories. Why is the quantity $mv$ an algebraically useful concept in physics.

 

[A.T.]

I want to know the real meaning or experiment why it was define that way and not the other.

There are plenty of experiments that show the conservation of "it" and not "the other". You can look up simple collisions for a start.

 

[etotheipi]

Actually, in Clifford Algebras this kind of thing is permitted and meaningful for some value of meaning.

That is very cool, I have not heard of this! I have found an introduction to the topic here so that should make for interesting reading... thanks

 

[jbriggs444]

Are you mocking me ?

No. The response about 2+2=4 was a good humored jab at the fact that something seemingly obvious and well accepted by the general public can nonetheless become dark, mysterious and counter-intuitive once mathematicians have worked to explore the full generality of things.

It was clearly not intended to mock yourself, @Hemant, the general public or even mathematicians. The parable of the blind men and the elephant comes to mind: the same thing can appear differently depending on which aspect one considers.

 

[sophiecentaur]

I just don't want to accept the that this is definition that's how it is.

There are some aspects of Maths that you either just have to accept or go to the trouble of learning Maths to a much greater depth. You can't just expect a quick hand waving explanation of 'why' we can use Maths the way we do.
Your comment is a bit like saying that you don't want to watch TV until you are capable of designing and building the whole chain from camera to display. You just have to treat things as black boxes if you haven't the time or inclination to get in deep.

Why is the quantity mv an algebraically useful concept in physics.

Extremely useful and, in fact the conservation of momentum is one of those principles that 'works, whatever', which is pretty rare.
It has to be remembered that "mv" is not a full description of momentum. For a start, that formula doesn't work for massless particles like photons. So it's not at all obvious how a mass times a velocity suddenly becomes Planck's constant divided by a wavelength when a photon is involved in a collision. Getting too precious about mv won't get one very far.