Hi all, it’s Mason with another blog post! (Do I need to keep saying that? Who knows! I don’t)
This is basically a post where I nerd out and just explain some really interesting things! I hope you enjoy!
This is really cool idea that I first saw in Calculus, but it’s probably possible to first see it somewhere else. In short, it’s a fancy way to add things up, but it’s so much cooler than that!
Imagine, for a moment, that you’d like to find the area under some fancy looking curve:
Unfortunately, this shape is not simple. We have formulas for the areas of simple shapes – circles, triangles, rectangles, that kind of thing. This shape, however, is complex, so we have no simple formula for it.
So, we settle for an approximation of the shape’s area. Let’s do that by putting some rectangles in our weird area of interest – rectangles are easy to draw, and their area is easy to calculate. Area = Base * Height:
We can see that this is a decent approximation, but there are definitely areas of our shape that aren’t covered by our rectangles, and sometimes the rectangles cover more area than our original area does. We can improve our approximation by using a larger amount of smaller rectangles to approximate that same area:
We can see that this is a better approximation than the last – the amount of our area that’s not covered by rectangles is smaller, and the area that the rectangles cover that our shape doesn’t is also smaller. Wow! Using more, smaller rectangles worked well to improve our approximation – let’s do it again:
This approximation is even better than the last! Clearly, the more rectangles we use, the better the approximation becomes.
In fact, if we use an infinite amount of infinitely small rectangles, we get the exact area under the curve! I think that’s really cool! We take a relatively simple shape, a rectangles, and we throw infinity in the mix, and we have a powerful tool that we can use to calculate the area of complex shapes!
The coolest part is that this process isn’t only useful for calculating areas of shapes – this idea, called integration, is widely used in engineering and science (even if it’s usually done on a computer). If we know the power output of a power plant over time, we can add up the power usage over every infinitely small element in time to find the total energy output of the power plant. If we have a complicated object with varying density, we can add up the density of each infinitesimal piece of the object to find its overall mass. The possibilities are endless!
I learned Special Relativity in my first physics class at Mudd, and I thought it was really cool!
It’s an entire branch of physics that comes out of 2 seemingly harmless assumptions:
- The laws of physics are the same for everyone, regardless of whether you are moving.
- The speed of light in a vacuum is the same for everyone, regardless of whether you are moving.
The first assumption should make sense – if you’re making scientific observations, your movement relative to other people should not change how you see the laws of physics. i.e. The world should work in the same way, regardless of whether you’re standing on the ground or in a moving car.
The second assumption is actually not so obvious. For a long time, in fact, scientists thought the exact opposite. To understand how weird this is, consider the following setup:
Imagine you set up a fancy science experiment to measure the speed of light. You get a really nice meterstick, and a really nice stopwatch, and a really nice flashlight. You turn on your flashlight, and you measure how fast the light comes out of it, and you get 670,616,629 mph. Fun!
Now let’s say you strap your flashlight on top of a car moving at 100 mph and repeat your process of measuring the speed of light. You might expect that the light gets a little ‘boost’ from the fact that it’s strapped on top of a moving car. So, you’d expect you’d measure the speed of light to be 670,616,629 + 100 = 670,616,729 mph. This is a reasonable expectation – this is how things would work if you were to throw a normal object like a baseball off the top of your moving car. However, this isn’t how light works at all! It turns out, light always moves at that same speed for everyone, no matter how hard scientists try to refute it!
This is a weird discovery all on its own, but things get even weirder. If you assume the speed of light has this unusual property, of always moving at the same speed, then you can use math to show the following bizarre but true phenomenon:
- Time Dilation
- Length Contraction
- Leading Clocks Lag
- Conservation of Mass Isn’t a Thing!!
I know what you’re thinking – What’s Time Dilation, you ask? What’s Length Contraction? What does Leading Clocks Lag mean? And what are you talking about, Conversation of Mass isn’t true????
Well, come to HMC and you’ll find out your freshman year 😉 (or you can try to find online resources yourself, like MIT Open Course Ware)
1st and 2nd Law of Thermodynamics
These laws talk about energy!
The first law is also known more commonly as Conservation of Energy, and says that energy is not created or destroyed. Energy can take many different forms, which makes this even more interesting! There is energy in sunlight, in lightning bolts, in chemical bonds, in individual atoms, in moving objects, in electromagnetic fields, and so much more!
Energy takes so many different forms, and generally looks different in all those forms, so it is not at all obvious that energy is conserved. For example, consider a lightning bolt that strikes the ground. There is potential energy associated with the position of electric charges before the bolt actually happens. As the bolt travels through the air, there’s energy released as sound, light, and heat. It’s absolutely baffling to me, that there is some quantity that is associated with all these different things that is conserved! There’s this mathematical thing called energy associated with the BOOM you hear from a lightning bolt, and the FLASH you see as it strikes, and the heat generated in the air as it flies, and it’s conserved! Wow!
The 2nd Law of Thermodynamics states that the entropy of the universe is always increasing, or that the entropy of an isolated system can never decrease. Entropy is a weird idea to explain, and I don’t really understand it well enough to try and explain it to y’all, but I know that the 2nd Law has very important implications! For example, the 2nd Law tells us that there’s no such thing as a perfectly efficient machine. If you have a machine that takes in energy, and uses it to do something, the machine will never be able to extract all the energy it was given – there will always be some wasted energy. Another consequence of this is that perpetual motion machines do not exist – if you want a machine that moves forever, you’ll need to constantly input energy.
This is a very shallow exploration of some very cool ideas, and that is intentional! I hope I’ve inspired some kind of interest in you to go out and learn more!
Suppose you have a list of ordered numbers, something like this:
1, 6, 13, 59, 70, 81, 103, 120, 457
We say that each number in the list has an ‘index’. 1 has an index of 1, 6 has an index of 2, 13 has an index of 3, 59 has an index of 4, and so on. The index is just a fancy word for describing the order of the numbers in a list. Let’s say that you are given this list of numbers, and you know you want the index of the number 80. How might you go about finding it?
Perhaps the most straightforward way to do this might be to look through the list one-by-one. Look at the first number. Is that 70? If so, then the index is one. If not, then look at the next number. Is that 70? If so, then the index is two. You would continue on this straightforward, linear search until you found 70, then you’d write down that the index is 6, and then you’d be done.
This method works, and will get the job done, but what happens if you’re given a REALLY long list? What if, instead of giving you a list of 9 numbers, I gave you a list of 9,000,000? Then, you’d potentially have to go through every single number in the list, and do up to 9,000,000 checks! You might be thinking to yourself: “well that’s just the price you have to pay to sort through more numbers,” but you don’t have to go through all that!
Imagine, instead of searching through the list from beginning to end, one-by-one, we took an entirely different approach, and leveraged the fact that we’ve been given numbers in order. Here’s what we’ll do:
We’ll start by looking at the middle number. Then, we’ll ask ourselves: Is this number in the middle what we’re looking for? If it is, great! Done! If it’s not, then is it bigger or smaller than what we’re looking for? If it’s bigger, then we only need to look at the half of the list after our number, and if it’s smaller, then the opposite. Then, on our next guess, we do the same thing, and get rid of half our numbers again! Here’s a graphic to help you visualize that:
We call this method Binary Search, and it works so well because cutting something in half over and over makes it really small really quickly! I think this is really cool, because it really hits at something deep – sometimes, there are just more efficient ways to do things, and if you’re clever, you can solve hard problems easily! Here’s another graphic, just for funsies!
If you’d like a nice way to visualize this and/or an alternate explanation, Khan Academy has some really cool stuff too! Learning is wonderful!
Thanks for reading y’all! If you thought any of these ideas were interesting, I highly encourage you to seek them out on your own! There’s a lot of great, free learning resources out there!
– Mason “i rly just lectured for 1730 words ” Acevedo