You can produce colored flames by adding various metal salts to the burning materials. That's what's done in fireworks. These metal salts decompose when heated so that individual metal atoms are present in the hot flame. Thermal energy in the flame then excites those atoms so that their electrons shift among the allowed orbits or "orbitals" and this shifting can lead to the emission of particles of light or "photons". Since the orbitals themselves vary according to which chemical element is involved, the emitted photons have specific wavelengths and colors that are characteristic of that element.
To obtain a wide variety of colors, you'll need a wide variety of metal salts. Sodium salts, including common table salt, will give you yellow light—the same light that's produced by sodium vapor lamps. Potassium salts yield purple, copper and barium salts yield green, strontium salts yield red, and so on. The classic way to produce a colored flame is to dip a platinum wire into a metal salt solution and to hold the wire in the flame. Since platinum is expensive, you can do the same trick with a piece of steel wire. The only problem is that the steel wire will burn eventually.
Your solution should work nicely—the pulley and weight system should protect your cable from breaking because the weights should maintain a constant tension in the line. As the tree swings back and forth, the weights should rise and fall while the tension in the cord remains almost steady. Obviously, if the rising weights reach the pulley the cord will pull taut and break, so you must leave enough hanging slack.
However, if the tree's motion is too violent, even this weight and pulley system may not save the cable. As long as everything moves slowly, the tension in the cord should be equal to the weight of the weights. But if the tree moves away from the house very suddenly, then the tension in the cord will increase suddenly because the cord must not only support the weights, it must accelerate them upward as well. Part of the cord's tension acts to overcome the weights' inertia. Just as a sudden yank on a paper towel will rip it free from the roll, so a sudden yank on your cable will rip it free from the weights. If sudden yanks of this type cause trouble for you, you can fix the problem by coupling the cord to the weights via a strong spring. On long timescales, the spring will have no effect on the tension in the cord—it will still be equal to the weight of the weights. But the spring will stretch or contract during sudden yanks on the cord and will prevent the tension in the cord from changing abruptly either up or down. The spring shouldn't be too stiff—the less stiff and the more it stretches while supporting the weights, the more effectively it will smooth out changes in tension.
As far as the weight of the weights, that depends on how much curvature you want in the cable supporting the feeders. The more weight you use, the less the cable will sag but the more stress it will experience. You can determine how much weight you need by pulling on the far end of the cable with your hands and judging how hard you must pull to get a satisfactory amount of sag.
To understand the two bulges, imagine three objects: the earth, a ball of water on the side of the earth nearest the moon, and a ball of water on the side of the earth farthest from the moon. Now picture those three objects orbiting the moon. In orbit, those three objects are falling freely toward the moon but are perpetually missing it because of their enormous sideways speeds. But the ball of water nearest the moon experiences a somewhat stronger moon-gravity than the other objects and it falls faster toward the moon. As a result, this ball of water pulls away from the earth—it bulges outward. Similarly, the ball of water farthest from the moon experiences a somewhat weaker moon-gravity than the other objects and it falls more slowly toward the moon. As a result, the earth and the other ball of water pull away from this outer ball so that this ball bulges outward, away from the earth.
It's interesting to note that the earth itself bulges slightly in response to these tidal forces. However, because the earth is more rigid than the water, its bulges are rather small compared to those of the water.
If a whistle's tube is relatively narrow, its pitch is determined primarily by its length and by how many of its ends are open to the air. That's because as you blow the whistle, a "standing" sound wave forms inside it—the same sound wave that you hear as it "leaks" out of the whistle. If the whistle is open at both ends, almost half a wavelength of this standing sound wave will fit inside the tube. Since a sound's wavelength times its frequency must equal the speed of sound (331 meters per second or 1086 feet per second), a double-open whistle's pitch is approximately the speed of sound divided by twice its length. For example, a whistle that's 0.85 centimeters long can hold one wavelength of a sound with a frequency near 19,500 cycles per second—at the upper threshold of hearing for a young person. If the whistle is closed at one end, the air inside it vibrates somewhat different; only a quarter of a wavelength of the standing sound wave will fit inside the tube. In that case, its pitch is approximately the speed of sound divided by four times its length. However, if you blow a whistle hard enough, you can cause more wavelengths of a standing sound wave to fit inside it. A strongly blown double-open whistle can house any half-integer number of wavelengths (1/2, 1, 3/2, or more), emitting higher pitched tones as it does so. A strongly blown single-open whistle can house any odd quarter-integer number of wavelengths (1/4, 3/4, 5/4, or more).
I'm afraid that you confuse the hypothetical with the actual. While people have hypothesized about superluminal particles called tachyons, they have never been observed and probably don't exist. This speculation is based on an interesting but apparently non-physical class of solutions to the relativistic equations of motion. Although tachyons make for fun science fiction stories, they don't seem to have a place in the real world.
What a great idea! Mylar is DuPont's brand of PET film, where "PET" is Poly(ethylene terephthalate)—the same plastic used in most plastic beverage containers (look for "PET" or "PETE" in the recycling triangle on the bottom). PET isn't a particularly inert plastic and you shouldn't have any trouble gluing to it. To form a rigid structure, you need either a glassy plastic backing (one that is stiff and brittle at room temperature) or a stiff composite backing. I'd go with fiberglass—mount the Mylar in a large quilting or needlepoint frame, coat the back of the Mylar with the glass and epoxy mixture, invert it, weight it with water, and let it harden. Mylar doesn't stretch easily, so you'll get a very shallow curve and a very long focal length mirror. While the mirror will probably have some imperfections and a non-parabolic shape, it should still do a decent job of concentrating sunlight.
While I'm not an expert on geysers and would need to visit the library to verify my ideas, I believe that they operate the same way a coffee percolator does. Both objects involve a narrow water-filled channel that's heated from below. As the temperature at the bottom of the water column increases, the water's stability as a liquid decreases and its tendency to become gaseous steam increases. What prevents this heated water from converting into gas is the weight of the water and air above it, or more accurately the pressure caused by that weight. But when the water's temperature reaches a certain elevated level, it begins to turn into steam despite the pressure. Since steam is less dense than liquid water, the hot water expands as it turns into steam and it lifts the column of water above it. Water begins to spray out of the top of the channel, decreasing the weight of water in the channel and the pressure at the bottom of the channel. With less pressure keeping the water liquid, the steam forming process accelerates and the column of water rushes up the channel and into the air. Once the steam itself reaches the top of the channel, it escapes freely into the air and the pressure in the channel plummets. Water begins to reenter the channel and the whole process repeats.
Your comparison between the limitless counting numbers and the limited speeds in the universe is an interesting one because it points out a fundamental difference between the older Galilean/Newtonian understanding of the universe and the newer Einsteinian understanding. The older understanding claims that velocities can be added in the same way that counting numbers can be added and that there is thus no limit to the speeds that can exist in our universe. For example, if you are jogging eastward at 5 mph and a second runner passes you traveling eastward 5 mph faster, then a person watching the two of you from a stationary vantage point sees the second runner traveling eastward at 10 mph. The velocities add, so that 5 mph + 5 mph = 10 mph. If the second runner is now passed by a third runner, who is traveling eastward 5 mph faster than the second runner, then the stationary observer sees that third runner traveling eastward at 15 mph. And so it goes. As long as velocities add in this manner, objects can reach any speed they like.
At this point, you might assert that velocities do add and that objects should be able to reach any speed. But that's not the case. The modern, relativistic understanding of the universe says that even at these small speeds, velocities don't quite add. To the stationary observer, the second runner travels at only 9.9999999999999994 mph and the third runner at only 14.9999999999999988 mph. As you can see, when two or more velocities are combined, the final velocity isn't quite as large as the simple sum. What that means is that the velocity you observe in another object is inextricably related to your own motion. This interrelatedness is part of the theory of relativity—that observers who are moving relative to one another will see space and time somewhat differently.
For objects traveling close to the speed of light, the failure of velocity addition becomes quite severe. For example, if one spaceship travels past the earth at half the speed of light and the people in that spaceship watch a second spaceship pass them at half the speed of light in the same direction, then a person on earth will see the second spaceship traveling only four-fifths of the speed of light. As you can see, relativity is making it difficult to reach the speed of light. In fact, it's impossible to reach the speed of light! No matter how you combine velocities, no observer will ever see a massive object reach or exceed the speed of light. The only objects that can reach the speed of light are objects without mass and they can only travel at the speed of light.
So while the counting numbers obey simple addition and go on forever, velocities do not obey simple addition and have a firm limit—the speed of light. The additive counting numbers are an example of a mathematical group that extends infinitely in both directions, but there are many examples of groups that do not extend to infinity. The group that describes relativistic, real-world velocities is one such group. You can visualize another simple limited group—the one associated with walking around the surface of the earth. No matter how much you try, you can't walk more than a certain distance northward. While it seems as though steps northward add, so that 5 steps north plus 5 steps north equals 10 steps north, things aren't quite that simple. Eventually you reach the north pole and start walking south!
Terminal velocity is the result of a delicate balance between two forces—an object's downward weight and the upward drag force that object experiences as it moves downward through the air. Terminal velocity is reached when those two forces exactly balance one another and the object experiences a net force of zero, stops accelerating, and simply coasts downward at a constant velocity. Since the upward drag force increases with downward speed, there is generally a velocity at which this balance occurs—the terminal velocity.
But while a parachutist can't change her weight, she can change the relationship between her downward speed and the upward drag force she experiences. If she rolls herself into a compact ball, she weakens the drag force and ultimately increases her terminal velocity. On the other hand, if she spreads her arms and legs wide so as to catch more air, she strengthens the drag force and decreases her terminal velocity. Popping open her parachute strengthens the drag force so much that her terminal velocity diminishes almost to zero and she coasts slowly downward to a comfortable landing. So to answer your question—two twin parachutists will descend at very different terminal velocities if they adopt different profiles or if only one opens a parachute.
To keep soda carbonated, you should minimize the rate at which carbon dioxide molecules leave the soda and maximize the rate at which those molecules return to it. That way, the net flow of molecules out of the soda will be small. To reduce the leaving rate, you should cool the soda—as long as ice crystals don't begin to form, cooling the soda will make it more difficult for carbon dioxide molecules to obtain the energy they need to leave the soda and will slow the rate at which they're lost. To increase the return rate, you should increase the density of gaseous carbon dioxide molecules above the soda—sealing the soda container or pressurizing it with extra carbon dioxide will speed the return of carbon dioxide molecules to the soda. Also, minimizing the volume of empty bottle above the soda will make it easier for the soda to pressurize that volume itself. The soda will lose some of its carbon dioxide while filling that volume, but the loss will quickly cease.
One final issue to consider is surface area: the more surface area there is between the liquid soda and the gas above it, the faster molecules are exchanged between the two phases. Even if you don't keep carbon dioxide gas trapped above soda, you can slow the loss of carbonation by keeping the soda in a narrow-necked bottle with little surface between liquid and gas. But you must also be careful not to introduce liquid-gas surface area inside the liquid. That's what happens when you shake soda or pour it into a glass—you create tiny bubbles inside the soda and these bubbles grow rapidly as carbon dioxide molecules move from the liquid into the bubbles. Cool temperatures, minimal surface area, and plenty of carbon dioxide in the gas phases will keep soda from going flat.
As for pouring the soda over ice causing it to bubble particularly hard, that is partly the result of air stirred into the soda as it tumbles over the ice cubes and partly the result of adding impurities to the soda as the soda washes over the rough and impure surfaces of the ice. The air and impurities both nucleate carbon dioxide bubbles—providing the initial impetus for those bubbles to form and grow. Washing the ice to smooth its surfaces and remove impurities apparently reduces the bubbling when you then pour soda of it.