# How is it possible to multiply a unit of mass by a unit of distance

#### einsteinian77

What exactly is the product mass and velocity. I can't seem to understand how one thing can be represented by the product of two completely different things. In other words, how is it possible to multiply a unit of mass by a unit of distance and time and end up with a unified answer.

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#### Quasaire

Albert Einstein's "Special Relativity" suggests that when mass is moving near light speed it becomes more massive. Since time and distance are related to speed (speed is the distance a object reaches in a certain time) and mass becomes more massive when near high speeds these entities that seem different are actually interchangable.

#### ahrkron

Staff Emeritus
Gold Member
Quasaire: welcome to the forums,

The use of quantities with mixed units has nothing to do with relativity (at least, not in this case). Also, mass and speed are not "interchangable".

Einsteinian77:

I don't have much time now, but here's a sketch of an answer: when characterizing a physical situation, there are quantities that seem to make more sense than others. For instance, when choosing a number for how much damage a piece of steel can do, you can choose only its mass (say, 10 Kg)... but then you realize that you will get quite different results if it is traevlling at 1cm/s than if it is at 100m/s. So you can decide to study the damages it produces on a wooden wall, let's say, as a function of the product (mass times speed). It makes sense, and it will show some nice dependence on a plot.

Many quantities have been tried this way. Some turn out to be more useful thant others, so we keep them and use them in more and more contexts.

You can also think of it from a different perspective: once force is defined as dp/dt (p=linear momentum), you automatically get some other combinations to be important, since:
1. many physical interactions depend on the integration of effects, or their rate of change,
2. These two concepts (integration and rate of change) are well represented by calculus' operations, and
3. Quantities related to others by integrations ot derivatives become automatically as important as the original ones. Sometimes more important, since they may be "conserved" under certain circumstances (energy, linear and angular momentum, etc.).

#### marcus

Gold Member
Dearly Missed
Re: ?=1/2md^2/t^2

Originally posted by einsteinian77
What exactly is the product mass and velocity. I can't seem to understand how one thing can be represented by the product of two completely different things. In other words, how is it possible to multiply a unit of mass by a unit of distance and time and end up with a unified answer.
To make these things easy to talk about, for starters it might help to
write the formula this way

?=1/2md^2/t^2

KE = (1/2) m V2

I think that by ? you may mean kinetic energy----the energy or work invested in a thing's motion.
And by d^2/t^2 you clearly mean the speed squared.

This KE formula is a good approximate formula that works at ordinary everyday speeds, but gets off at "relativistic" speeds----that it, at large fractions of the speed of light.

What you seem to be asking is how is it possible to multiply mass by square of speed and get energy. It is a question about types of quantities and a bunch of algebra rules that interconnect the various types.

There are half a dozen people always around PF who can explain this and also explain why the KE formula works. So I will leave it
for someone else and just give a hint to think about.

One way to describe a FORCE is as the force needed to give a certain mass some definite amount of acceleration
So you can actually write a force as
m d/t2
a certain mass m multiplied by an amount d/t2 of acceleration

But a very good way to think of an amount of work or energy is as force x distance-----the energy that goes into pushing with a certain force F for a certain distance d.

So if you multiply a certain force m d/t2
by a distance d, what do you get?

#### Quasaire

Originally posted by ahrkron
Also, mass and speed are not "interchangable".
Thanks for the welcome. Maybe interchangable is the wrong word but isn't it true that relativity says when objects approach light speed their mass increases? Consequently, isn't this saying (exclusively when at high speeds) mass and speed (near C) are related to each other in sort of a weird way?

#### HallsofIvy

Homework Helper
My father and I are related to each other. We are NOT interchangable!

In fact, one can reasonably say that all things are related in some way. Surely using the word "interchangeable" to mean "related" is one heck of a leap!

I can't seem to understand how one thing can be represented by the product of two completely different things. In other words, how is it possible to multiply a unit of mass by a unit of distance and time and end up with a unified answer.
Isn't that what we do anytime we do arithmetic? Dang! All that time wasted on arithmetic! Now, how do I calculate the area of this carpet? It certainly can't have anything to do with the length and width!

#### arcnets

Originally posted by ahrkron
It makes sense, and it will show some nice dependence on a plot.
Agree.
You could, of course, define any weird combination of quantities, like say m^5kg^7/s^(3/8). But in physics, only those make sense which remain constant in an experiment (or theory) where you vary some conditions, and keep other conditions unchanged.
Then you form a rule like "The quantitiy X remains constant if ..."

#### einsteinian77

Let me put it in the simplist terms possible. In 'arithmetic' we say that xy=x+x+x+....y amount of times, or xy=y+y+y+....x amount of times. This, in geometry, constructs a 2-dimensional figure. However, the new 2-dimensional figure is could be expressed only in terms of y, or it could be expressed only in terms of x.

However, in physics we take something like momentum and call it
mv. So my question is how do you multiply mass and velocity to end up with the arithmetic result. mv=m+m+m...v amount of times? That is an impossibility since velocity has no direct relation to mass in the arithmetic sense. Now do you understand my problem a bit more? If not Im still willing to post again

Last edited:

#### STAii

Well, mainly, all classical physics relations and equations come from real data and experiments.
So, for example, when u measure the acceleration of a mass when a certain force is exerted on it, and then measure the accelertion of the same mass when half the force is exerted, you will see that the acceleration dropped to half.
So, from starting points like this one, you can make proportional relations between variables and quantities.
Normally a group of proportionaly relations can be converted into an equation (containing a = symbol) by putting constants.
So this is how multiplication enters physics equation, i don't see a way to consider mv in physics as m+m+m+m ... v times.
actually, saying xy=x+x+x .. y times is only valid when y is an integer.

And BTW, units don't have always to be related, it is like u say :
20 cows * 10 sheep = 200 cow*sheep

This is mathematically right, but it doesn't make sense.

#### einsteinian77

I'm not saying that the proportions of physics equations are wrong, all im saying is that energy cannot be described by velocity alone nor mass alone. Therefore, multiplying mass and velocity seems to have no meaning. In my opinion, i believe an association of mass and velocity needs to be established before we can start to through them together in equations, otherwise you just get proportions.

#### marcus

Gold Member
Dearly Missed
Re: Re: ?=1/2md^2/t^2

What is Energy as a quantity if it is not force x distance?

What is Force as a quantity if it is not mass x acceleration?

These are the most basic meanings of elementary quantities.

It follows that Energy is equal to mass x acceleration x distance

and that is algebraically the same as mass x velocity2

The very units we measure energy in are equiv to a product of mass with the square of velocity which reflects the underlying idea of what energy (work or the potential to do work) is.

Do you have difficulty with any part of this?

Here's an exerpt from my post earlier in the thread

Originally posted by marcus

What you seem to be asking is how is it possible to multiply mass by square of speed and get energy. It is a question about types of quantities and a bunch of algebra rules that interconnect the various types.

...

One way to describe a FORCE is as the force needed to give a certain mass some definite amount of acceleration
So you can actually write a force as
m d/t2
a certain mass m multiplied by an amount d/t2 of acceleration

But a very good way to think of an amount of work or energy is as force x distance-----the energy that goes into pushing with a certain force F for a certain distance d.

So if you multiply a certain force m d/t2
by a distance d, what do you get?

#### einsteinian77

Nevermind my question since, no offense, no one seems to understand it.

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