Question about E=MC2...

[Singular Intellect]

I've tried googling up information on E=MC2, and that yields some useful applications of the formula.

But I do have a question I can't seem to find an answer to. Perhaps a dumb question, but I'd rather ask anyhow.

The method I've learned uses energy measured in Joules, speed in Meters, and mass in Kilograms. Example: Joules = Kilograms multipled by lightspeed in Meters squared.

My question is, by what method is it determined what unit of measure for any particular variable is? Example: Joules = Kilograms multipled by lightspeed in Kiloemters squared doesn't yield the same figure, even though the speed is same (just not the unit of measure).

Converting meters to kilometers for the speed of light obviously yields a different resulting number because one variable was reduced by a factor of a thousand, even though both units of measure mean the same thing. If all the variables are modified by the same relative factor the equation still works, but how did we come up with the "proper" units of measure in the first place?

Would much appreciate any answers or links to them! Or naturally any comments on being a dumbass for missing something obvious here...

[Darth Wong]

Joules are a compound unit, composed of other units multiplied together. In short, they are that way because they are defined that way. Obviously, if you throw an extra "kilo" in there, you have to adjust your results to compensate, because the unit does not have the extra "kilo" in it.

[SCRawl]

The method I've learned uses energy measured in Joules, speed in Meters, and mass in Kilograms. Example: Joules = Kilograms multipled by lightspeed in Meters squared.

Emphasis mine.

Nitpick, but speed is measured in metres per second (m/s). Distance (or displacement, if you prefer) is measured in metres. But yeah, other than that, what the Sith Lord said.

[darthbob88]

This may answer the wrong question,. but I believe the problem lies in multiplying units; the energy produced is in joules, correct? One joule is defined as one kilogram * one meter^2 per second^2, or mass in kilos multiplied by velocity, in m/s, squared. This is where the number you get changes, when you replace c as measured in m/s with c as measured in km/s, you change the output by a factor of 1E6, making the unit of the output megajoules rather than joules. I apologize if this is difficult to understand, I never was able to communicate as well as I'd liked.

[SCRawl]

This may answer the wrong question,. but I believe the problem lies in multiplying units; the energy produced is in joules, correct? One joule is defined as one kilogram * one meter^2 per second^2, or mass in kilos multiplied by velocity, in m/s, squared. This is where the number you get changes, when you replace c as measured in m/s with c as measured in km/s, you change the output by a factor of 1E6, making the unit of the output megajoules rather than joules. I apologize if this is difficult to understand, I never was able to communicate as well as I'd liked.

It makes perfect sense. I considered suggesting a dimensional analysis (the fancy term which means making sure the units all match up), but went with a dumbed-down nitpick, since DW already effectively answered the question. But you've spelled it out clearly enough, if I'm any judge of this sort of thing.

[General Trelane (Retired)]

My question is, by what method is it determined what unit of measure for any particular variable is?

There are two kinds of units: base units and derived units. Base units (such as kg for mass, m for length, and s for time) are arbitrarily defined. Derived units are, well, derived by multiplying/dividing base units. Examples:

Force: F = m*a = (kg)*(m/s^2)
Energy: E = F*d = m*a*d = (kg)*(m/s^2)*(m) or kg*(m^2)/(s^2)

These derived units expressed in base units are cumbersome, so for convenience, we define simpler substitutes. For force, we define the Newton as 1N = 1 kg*m/(s^2). For energy, we define the Joule as 1 J = 1 kg*(m^2)/s^2). But we can (and sometimes do) also use units of N*m for energy if it is convenient to do so. Likewise, there are other definitions of conveniece for energy (examples include kT and ergs).

The units we use are the ones that are most convenient for a given application. But to convert between units, you need to keep them straight. A consistent way to do this is to always remember the base units. In your example, if you use km/s instead of m/s, then you get:

kg*(km^2)/(s^2) * (1000m/1km)^2 = 1,000,000 kg*(m^2)/(s^2) or 1 MJ

This is what SCRawl called dimensional analysis, and it's very useful to make sure you don't screw up the magnitude (and doubly useful when converting between 'US Customary' and SI units).

[andrewgpaul]

It's also useful (and is the way I was taught it) to make sure your equation makes sense; the units on both sides should be the same. As an example, F = m*a[/i] makes sense because the units on both sides are the same (N = kg*m*s^(-2 ) ). F = m/a doesn't make sense, because that implies that N = kg*m^(-1)*s^2.