I might just be stupid because math is not my strong suit, but it seems like the fact that metric units are able to convert between mass and energy so cleanly is astounding. Especially since the metric system was invented so long before relativity, meaning these units were obviously not designed with this in mind.
What’s am I missing here? Is it possible to write an equivalent equation for imperial units?
In: Mathematics
You are just a little confused about the difference between units and physical quantities.
I think it will be easier to understand if we think about a simpler formula, like the one for speed (velocity) for example.
Speed equals distance traveled over time:
v=d/t
That is the general formula using physical quantities. No units are assigned to either the velocity, distance or time.
When applied to something in reality each of these physical quantities will have a specific value.
Lets say you like to listen to “Cotton Eyed Joe” over and over and while walking home from school you notice that you pass exactly 10 street lights in the same time it takes to listen to the song one time.
That means that you are walking with a velocity of 10 sl/ceJ (10 street lights per cotton eyed Joe). So 10 street lights are the distance (d) and cotton eyed Joe is the time (t) in our first formula
But that same velocity is also 3.24 knots which is 3.24 nautical miles per hour. Where nautical miles is d and hour is the t in our first formula.
Or 0.1 km/min where every minute you travel 100 meters.
Or 37.3 mph where every hour you travel 37.3 miles.
The units can be what ever you like, what matters is their real life quantity.
That formula (and all formulas) still works regardless of the units.
e is typically measured in joules. But an equivalent unit to a joule is a kg*m^2 /s^2.
If you measure mass in kg and c in m/s then you can see how the units of the formula make sense.
Say you instead wanted to measure c in feet per second. Now the output of the formula will be in kg*ft^2/ s^2. It’s a unit that nobody uses but it’s still valid. The results when you actually plug in numbers to the formula will still be correct. Just in a different unit.
You can invent a totally new unit of mass. Call it z. If you use it in the formula you’ll get a correct answer in units of z*m^2 /s^2.
Basically all formulas like these are fundamentally true. We define the units in ways that are most convenient to us.
Units in the metric system are based on a handful of fundamental properties: time (second), mass (kilogram), distance (metre), temperature (°C or K, they change at the same rate so if it’s temperature difference then it doesn’t matter which; absolute temperature, kelvin, needs to be used in an energy calculation though), amounts of stuff (moles) and a couple of others.
Other properties have units that are derived from these, for example Force = mass x acceleration so 1 newton = 1 kgms^-2. Energy = force x distance so 1 joule = 1 kgm^2s^-2. Pressure = force / area so 1 pascal = 1 kgm^-1s^-2 and so on.
E = mc^2 is a fundamental property of the universe, determining how much energy we can release from matter (or how much energy we need to put in to create matter). We do this all the time in chemical reactions, though the amounts involved are so miniscule that we approximate to the law of conservation of mass (e.g. burning 1 mole, 16g, of methane in 64g oxygen releases about 890kJ of energy, at the cost of less than 10ng of mass: a rounding error in a total system mass of 80g).
Using different units wouldn’t change the fact that E = mc^2, merely the numbers involved due to conversion factors.
The equations of basic physics do too, to almost a freaky extent.
There has been some fudging though, making units intentionally that would align.
And sometimes they had to throw some numerical constant in there to make equations of state work, Gas constant, Planks constant, etc.
Much of physics is understanding ratios. Some of this is equal to several of that. Now toss in some modifiers and constants, and you get a relationship you can show as an equation.
Squeeze air and it gets hot. On the way, the volume went down as the temperature went up. Keep the volume and increase the pressure: air gets hot. Keep track of the moving parts, and the stationary parts, and you’ll wind up with PV=nRT
Lather, rinse, repeat for the rest of the universe, and you get things like the Gravitational Constant, Stefan-Boltzman’s Law, Ohm’s Law, and how to properly cook a cheeseburger.
That last one is still undergoing peer review at my holiday party this 4th of July. I’ll let you know how it turns out.
The constant for the speed of light are simplified to “c” to make the relationship work. If it wasn’t written as c squared, it would be “8.98755179 × 10^(16) m^(2) / s^(2)”
Edit to add: this is how most equations are resolved in physics and chemistry. pv=nrt is another one. pressure x volume = number of moles x constant x temperature. There are others such as dealing with specific heat of a material. Each material has a unique coefficient. Same goes for friction. Materials each have a coefficient of friction. There are many scientists who conduct lab tests to determine these constants as accurately as possible. The resulting equations are neat by design by calculating the constants that make the equation work.
You are missing the concept of base vs derived units.
There are base units/parameters like mass, distance, time, temperature, count, ect.
And then derived units/parameters like force = mass × accelerated (second time derivative of distance).
However, units like kg•m/s^2 are unwieldy so we condense them into parameter specific units like Newtons and Joules.
Formulas like E = mc^2 or Pe = m•g•h work because the units on each side are required to agree. If you try using E = mc^2 you need to use m in kg and c in m/s to get an output in J. If you instead used lbs and ft/s then your energy will be for lbs(mass)•ft/s which doesn’t cleanly convert to a more sensible energy unit.
Can I ask a related question?
I get the mass and energy equivalency. In some manner e can be converted to m and visa versa.
What I DON’T understand is how c relates to those other two factors.
Why is c any part of the equation? And did Einstein really mean c squared literally?
Why is c a required component of the formula?
>What’s am I missing here? Is it possible to write an equivalent equation for imperial units?
It’s already done.
It’s `e=mc^2`.
It describes the relationship between energy and mass. Units don’t matter.
You don’t actually weigh more if your scale measure in pounds.
The metric units, any unit, is just a word we use for a phenomenon on nature.
Einstein’s formula says that energy equals the mass of an object times the seepd of light, times the speed of light. It does not even have anything to do with math if you look at it this way.
Aliens who do not have any of our systems will observe the same thing and will agree with Einstein.
Scientists use a system called SI-Units. You know this as the metric system but technically the terms are not the same.
The unit for Energy is called Joule (J).
But you can also tell by the formula that Joule so not really it’s own thing.
If einsteisn formula is true and we use SI units, then we can translate it to 1J = 1kg * (m²/s²)
The reason for this is that kg is the SI unit for mass, while meter per second is the SI unit for velocity.
The SI units are designed in a way that this is easy and intuitive. But you could easily replace kg with g, but then you would have to add 1000 to the equation unless you want to change the definition of how much energy 1 Joule is supposed to represent.
I don’t know how to convert the SI units to imperial system but you can easily do it.
It would translate to
1 Joule = 2lbs (assuming 2 lb = 1kg) * (1550*inches²/s²)
1m² = 1550 square inches is what I got from Google.
Energy can also be described as N*m (newton times meter) or W*s (Watts times second). This makes sense since there are many ways to convert and describe energy.
How can Einstein’s formula be tested? For example by measuring a radioactive particle and measuring it again after it split up and measuring the particles it splits into. If you add all those up there is still some mass missing. The missing mass is the energy that got emitted. Same happens with chemical reactions. The mass will not be the same as the mass of each atom that got combined, it can be slightly heavier if there is some binding energy.
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