Last post, I finished the post cryptically, mentioning that there is a certain path that an object falling will take less time to travel than any other. Now, one of the solutions, as I mentioned, is just a straight fall. That is only one of the solutions, but only if the endpoint is directly underneath the starting point. What if the endpoint is to the side, what curve would the fastest path follow?
Another moon has been discovered going around Neptune. And like so many astronomical discoveries before, it has been captured before, in this case, in photos between 2004-2009 (same with Neptune, which has been recorded by Galileo, but not truly been discovered until much later). Read all about it here.
Last post, I talked about the fastest route between two mediums, and how light seems to follow that route when travelling from one medium to another. From what I just summarized, one would think light did that just because it is in a rush to get somewhere or something, but it is instead due to the wave nature of light. In classical physics, one can model how light reflects and refracts by applying something called Huygens Principle. Basically, a light wave at its edge forms a spherical wave at every point. Those spherical waves combine to form a single smooth wave. In the case of refraction, it looks like this:
Interestingly, the following model gives you the same math: Instead of the waves making more waves, let’s imagine instead that there are a chain of people holding hands and walking on the beach at constant speed. They suddenly reach the ocean, but at an angle. The person who reaches it first then bend the path, and then go forward. Otherwise, if the person wants to remain on the same straight path along with the others, the person must also move sideways. If you can’t picture it, it looks like this:
Now, let’s set the geometry up so that it shows the position of the line of people on the sand right before they go in the water and the line of people on the water right after they left land. Again, the math is the same, but I thought I could make it more intuitive. Notice that the length of the shore is the same for the triangle above and below it. That will be the key to creating the ratio for Snell’s law:
Before we go on, I have to mention that angle 1 in triangle BFC and angle 2 in triangle ADE are the angles given in Snell’s law. Then how do we know that angle 1 in ABC and angle 2 in AEF are the same respectively? ABF and BFC are similar triangles. So is AEF and ADE. The proof of that is not the subject for this post, so I might as well leave it as your homework assignment .
Now, since we want to find the ratio, we want to find which line is the same for the triangle ABF and AEF. Line l is the same. So we want to find something that looks like this:
Well, we can use trigonometry for that. For triangle ABF:
Doing the same for triangle AEF gets us:
As you can see equation 1=equation 2:
Plugging it back in equation 3 gets us:
Multiplying both sides by the speed of light c gets us the generic Snell’s law equation with n standing for index of refraction, which is equal c/v:
Cool, the light bending its path due to the wave slowing down results in a path that is naturally the fastest possible route!
Interestingly, if you use the math of Snell’s law and imagine stacking infinitely thin layers of mediums, the ones next to each other being only infinitesimally different in their travel velocity, you can solve another interesting problem that involves the fastest route from top to bottom by gravity as we shall see in the next part of this series! And no, just letting go off the object so it falls straight down is not the answer (although it is A answer).
This is a classical type of problem in physics. Imagine you are on the beach, and you throw a ball somewhere in the ocean. You the decide to have a race with someone who can get to the ball faster. Both of you run at the same speed, and when you enter the ocean, you slow down due to the water’s resistance. Therefore, the only thing that will affect who will win the race is what path each one of you takes. So the question is, which is the fastest path?
On Earth, one can indirectly find what the structure of inside the planet is by measuring the waves created by an earthquake. The Earth’s interior, having layers with different compositions, will refract and reflect those waves, and by measuring the wave all over the Earth, what can make a reasonable assumption as to what the Earth is like inside it. Unfortunately, we can’t exactly place seismographs in other planets. In the case of Saturn, though, there is a structure you can measure which will indirectly tell us what is going on inside the planet. It is the rings, which it turns out that while its shape is predominantly affected by Saturn’s moons, they alone don’t account for all the waves on it. The planet itself affects the rings, and one of the findings is that the inside of the planet is sloshing around. More details is in the link above.
This one is my favorite, Bramble Blast, which is a remix made for Super Smash Brawl of Stickerbrush Symphony from Donkey Kong Country 2:
This is the original Stickerbrush Symphony:
Like this (start from minute 13):
So, I was watching the let’s play for the visual novel Remember 11 when something to my interest popped up, which was science. In it, quantum mechanics is used as a sci fi plot device to explain all the mysterious things going on, and it goes in an extremely lengthy exposition in order to explain it. Really lengthy. As you can see, the explanation goes over several videos. While I understand it, how the heck are the general audience supposed to understand that!? Granted, it is in character for the person doing the extremely long exposition, and some of the mysteries do require a lot of explanation. At the same time, they didn’t have add all the jargons in the tips section that explains the explanation because now the explanation that explains the explanation doesn’t explain anything anymore because no one gets it. For example, the mention of quantum bits in the entanglement entry and going off tangent to another tips entry of quantum bits was completely unnecessary, and it was a poor explanation of what entanglement was. Heck, the entanglement entry wasn’t necessary because it was explained in the EPR paradox explanation. And why is there an entry about unitary transformation? Think about the audience! While not all the tips entries are bad, and not everything in quantum mechanics can be explained simply to the lay audience, many of them felt like they were trying to be sophisticated for no reason at all.
By the way, this book is a good example of how you explain quantum mechanics. Although I will admit it kind of is not fair as the book has more than 100 pages to explain everything, the book doesn’t try to bash your head in with the term ‘unitary transformation’.