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
it’s the other way around.
If you multiply a mass unit by a speed-squared, you get a unit that’s energy.
In metric, if you multiply 1 kg * (1 m/s)^2, you get a 1 kgm2/s2, or a joule. The joule was chosen specifically to be a unit of energy that made immediate sense and a clean result from that kind of multiplication.
But there are other ways you might want to think about energy, like having to do with temperature. Kilocalories are another way to think about energy — how much energy do you need to raise the temperature of 1kg of water by 1 degree centigrade? If you ask “If I turn this microgram of stuff into energy, how much will it raise the temperature of this 1000 liters of water?” then you’re going to need to do more math after you run e=mc2 to convert from joules to kilocalories.
I’ll explain the neatness the same way i explained it to my students. Although they tend to have much neater equations with units.
When we put our numbers into an equation, it’s not just the digits we include, we also bring the units along! So it’s energy = mass * velocity^2. We plug in the same types of units whether it’s metric or Imperial.
In the metric system, we would have an energy unit of a joule, which is equal to a kilogram-meter^2 per second^2, or a kilogram-(meter per second)^2. The kg is a unit of mass, and the m/s is a unit of velocity. The joule was made to fit neatly with the other units, so the calculations are super simple!
In the imperial system, it’s harder. The energy unit is ft-lbf (foot-pound force if you want to be specific), which works just fine if we are using feet and pounds (force)… but we are using mass in our equation. So we would have to do something like slug * (ft/s)^2. We can see that this equation is just as simple as the metric! And it actually is as simple as that, the slug-(ft/s)^2 is equal to the ft-lbf. The key problem: when are you given the mass of something in slugs? So you’ll need to remember to convert your pounds into slugs.
A bigger problem comes up when you are given bigger or smaller units. If it becomes kilometers per second, we just have to change the decimal place. But if it’s miles per second, we now have to account for a conversion of (5280 ft per 1 mile)^2.
Sure thing!
Isn’t it wild how metric units just fit perfectly into E=mc^2? Even though they weren’t planned for it, they’re like a puzzle piece that just clicks into place. It’s all because the metric system uses grams for mass and meters per second for speed, and when you square that speed, bam, you get energy in joules. But for imperial units like pounds and feet, the equation doesn’t play nice without a bunch of extra conversion factors. So, metric gets the gold star for simplicity when it comes to Einstein’s famous equation.
Well when you really start to break the variables down it’s not so simple. That’s how most equations are, just more equations hidden behind variables.
I could be wrong here, but I think the actual answer OP is looking for is… *because we’ve specifically designed the equation so it works that way*. The E part of the equation is measured in Joules, and Joules are defined as (kg)( m^2 / s^2), which happens to correlate to mc^2. In other words, the equation works because really it’s just saying that “Joules=Joules”. Now, if you wanted to measure it in kilocalories, you would certainly need a coefficient…if my half asleep math is correct, it would be E=(0.000239)mc^2.
You could express mass as stones, and the speed of light in nanoparsecs per microfortnight, and you’d still get a valid answer for energy… it just wouldn’t be in joules.
Explains each part in great detail
Edit: For instance, the squared part comes from experiments where a ball was dropped into sand or clay or something from different heights. The expectation was that 2x velocity at impact would = 2x crater depth, but reality showed it was 2x velocity at impact = 4x crater depth, 4x velocity at impact = 16x crater depth, etc. It’s part of how objects in motion behave
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