How can I better understand dimensional analysis
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As we know, we can think of a distance as speed times time. In this video, I want to show you how to use this pretty simple formula, how you can use this pretty simple formula to understand that units in algebra can and can be viewed as variables when we are manipulating an equation be really practical. We just have to make sure that we get the result in a unit that makes sense. For example, if we have given a speed, let's assume a speed of 5 meters per second, plus a time of 10 seconds. Then with this information we could use this formula. Then we say the distance is equal to the speed of 5 meters per second times the time, i.e. times 10 seconds. And the great thing is: we can treat these units like variables. treat like variables. That would be the same - we have to multiply everything, so it doesn't matter in which order we calculate it. therefore it doesn't matter in which order we calculate it. This is the same as 5 times 10, 5 times 10 times meters per second, times meters per second, times seconds, times seconds. If we used these units like variables, If we used these units like variables, then we could say that we have seconds divided by second, i.e. seconds in the denominator multiplied by seconds in the numerator. And that cuts out. And 5 times 10 is of course 50. And 5 times 10 is of course 50. 50 are left, and the unit that remains for us is meters, 50 meters. That's great. That works with the units. If we treat the units as variables, we end up with the unit for the distance covered, meters. Then you might say, well, that's great, but worrying about it with such a simple formula seems a bit over the top. But what I wanted to show you is that even with a formula as simple as distance equals speed times time, what I did was very useful and what I did is called dimensional analysis. It's useful for something as simple as distance equals speed times time, but in chemistry, physics, and engineering, you can apply it to much more complicated formulas. If you do a dimensional analysis, it will ensure that the units are correct. this ensures that the units are correct. Let's do a slightly more difficult example. Let's do a slightly more difficult example. Let us assume that our speed is again 5 meters per second, and the time is now given in hours instead of in seconds. The time is equal to 1 hour. Now let's try to apply the formula. The distance is equal to 5 meters per second, 5 meters per second times time, so times 1 hour. What do we get then? 5 times 1, 5 times 1, is 5. But remember, we have to treat the units in terms of algebra. We do a dimensional analysis. So 5, we have meters per second times hours, times hours, you can also say 5 meters hours per second. Well, that doesn't look so good ... This is not a unit that we know and that would make sense. This is not a common unit for a route, so let's see what cuts out. Now you might say, well, if we can't get rid of the hours, if we can only express it in seconds, then that cuts out and there remains meters, which is a unit of distance traveled that we know. So how do we do that? We'd like to multiply that by something that has hours in the denominator and seconds in the numerator, basically seconds per hour. How many seconds is there in an hour? Well that's 3600 ... I'll do this in a different color. An hour has 3600 seconds, you can also say 3600 seconds for every hour. If you multiply that now, these hours are shortened by these hours, these seconds by these seconds, and what remains is 5 times 3600. And that is? 5 times 3000 would be 15,000, 5 times 600 is another 3000, so that's 18,000. The remaining unit is meters. The remaining unit is meters. 18,000 18,000 18,000 18,000 meters. We're done with that. Now we have found an expression for the distance traveled in a unit that we know. If you walk 5 meters per second for an hour, you have walked 18,000 meters. But we can go further here in our dimensional analysis. What if we don't like the answer in meters and we want one in kilometers? Then what could we do? 18,000 meters. 18,000 meters. If we could multiply that by something that had meters in the denominator, meters in the denominator, and kilometers in the numerator, then meters would get out and we'd keep kilometers. Well, how can we multiply this without changing the value? Basically, we want to multiply by 1, we want to keep equivalents in the numerator and denominator. So, 1 kilometer is the same, is equivalent to 1000 meters. You could also say that we just multiply by 1, one kilometer divided by 1000 meters. 1 kilometer is 1000 meters, so this is equivalent to 1. If we multiply, meter is shortened by meter, and we get 18,000 divided by 1000, which equals 18. And so the only unit we end up with is kilometers , and we're done. We expressed our distance traveled in kilometers instead of in meters.
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