Leojames26

You're right, the average energy per particle is just part of the story. In any gas, some particles will have much higher energy than others because of the Maxwell-Boltzmann distribution. It's those higher-energy particles that might actually manage to fuse, while the rest won't contribute much to fusion at all. That's why in practical scenarios like fusion reactors, we aim to push the overall energy so high — to increase the chances of these successful high-energy collisions. It’s quite a challenge to get enough particles up to those needed energy levels.

That’s a really interesting question! Fusion, like what happens in the sun, requires an incredible amount of energy because you're essentially forcing atomic nuclei to come together, which naturally repel each other due to their positive charges. To get to the point where fusion can happen, you need to overcome the Coulomb barrier, which is the energy barrier due to the electrostatic force between the positively charged nuclei. For hydrogen atoms, specifically, this happens at extremely high temperatures—like millions of degrees Kelvin. In the core of the sun, where fusion happens naturally, temperatures are around 15 million degrees Kelvin. The minimum temperature needed to start fusion in hydrogen (like in a hydrogen bomb or in stars) is roughly in the range of 10-15 million degrees Kelvin. To put it in terms of energy, you’re looking at needing around several keV (kilo electron volts) of energy per particle. For fusion in laboratory conditions, like in a tokamak or other fusion reactors, the conditions are even more extreme due to the lower density compared to the sun. So, below these extreme temperatures (and corresponding energies), the hydrogen atoms won’t fuse—they’ll just get excited or ionized, which means electrons will leave the atoms and you’ll have a plasma, but not fusion. The energy levels required for ionization are much lower than those needed for fusion. In short, the energy required for fusion is incredibly high, much higher than just causing excitation or ionization. It’s why we can have plasma at much lower energies, but actually achieving fusion is a whole other challenge.

You’ve got a function y(x)y(x)y(x) over a range from x0x_0x0 to x1x_1x1, and you want to find the chance that yyy will be a certain value or within a certain interval if you randomly pick a value of xxx. To figure that out, you would: Look at the range of yyy values you care about. Find out which xxx values give you those yyy values. See how big that xxx range is. Then, just divide that range by the total xxx range (from x0x_0x0 to x1x_1x1). That’s your probability. It’s basically asking, “Out of all possible xxx values, how many give me a yyy that’s in the interval I care about?” If you’ve got more specifics about the function or the interval, it could make this clearer, but that’s the general approach!

This is an interesting thought experiment! When you’re dealing with two different voltages powering the same circuit in parallel, things can get pretty tricky. Normally, in an electrical circuit with two generators at different voltages, you'd run into serious issues like one generator trying to drive current into the other, which could cause damage or malfunction. Now, if we replace the generators with moving magnets, it becomes a bit more abstract, but the same principles would generally apply. The interaction between the magnetic fields and the induced voltages could create some unpredictable behaviors. If the magnets are far enough apart, their effects might not interfere with each other as much, but the overall system is still likely to be unstable, just like with different voltage sources. In real circuits, this setup would be avoided because of the risks, but in your hypothetical scenario, it’s a fascinating way to explore the complexities of electromagnetic induction and the challenges of parallel circuits with differing inputs. Just remember, even though it’s a thought experiment, the underlying physics doesn’t change—different voltages in parallel typically lead to complications, whether they’re coming from batteries or magnets.

Alright, so here’s the deal. The author’s solution is actually correct, even though it might seem counterintuitive at first. The key is that the strategy isn’t just about guessing randomly or assuming B will be bigger or smaller than A—there’s a bit more to it. When A is negative (A < 0), B is more likely to be bigger than A, and vice versa when A is positive. This tilts the odds in your favor slightly, but not in a huge way. That’s why the long-term success rate isn’t 50%, but rather 75%. As for the whole "one and a half quadrants" thing, imagine plotting points on a graph where A is on the x-axis and B is on the y-axis. When A < 0, B is more likely to be bigger, which covers one and a half out of two quadrants where A < 0. This gives you that 3/4 success rate. So yeah, the author nailed it. You’re right that for two completely independent draws, it would be 50%, but here the strategy adds a bit of an edge. Hope that clears it up!

That’s an awesome idea! Using GM algae to harness photosynthesis could really work, especially near a star. The concept of turning the ship’s exterior into something like a giant leaf is super clever. It’s a natural, self-sustaining system that could definitely complement other life support systems. Keep those creative ideas coming!

Hey, this is a really cool idea! I love how you’re thinking about ways to reclaim oxygen in space—it’s definitely an important challenge. So, what you’re suggesting—breaking down CO2 into carbon and oxygen—sounds interesting. The basic idea is solid, but yeah, you’re right that it would take a ton of energy to pull off. Compressing CO2 into a liquid, cooling it down, and then splitting it into carbon and oxygen would be pretty intense, especially in space where energy is super limited. The part about separating the CO2 into carbon and oxygen sounds awesome, but it’s really tough to do with current technology. It takes a lot of effort to break those molecules apart, and right now, we don’t have a super efficient way to do it. But honestly, it’s awesome that you’re thinking about this. Who knows? With the way technology keeps advancing, ideas like yours could totally inspire new solutions. Keep brainstorming—space exploration needs all the creative ideas it can get!

Hey there! I totally get where you’re coming from—it’s easy to get tangled up in all the math and rules when it comes to triangles and forces, especially when you’re dealing with something as specific as the Triangle of Forces. So, you’ve got your right-angle triangle, and you’re calculating the forces based on the sides. That makes sense, but you’re absolutely right to question the results you’re getting. The key thing you’re missing here is the direction of the forces, which the angles you mentioned definitely affect. Here’s the deal: when you’re working with forces, it’s not just about multiplying the length of the side by the force along that side. The direction the force is acting in (which is where those angles come into play) is super important. In this case, because the forces are changing direction, you can’t just multiply the side length by the force and call it a day. For example, when you shift from AB to BC, the force is no longer acting purely horizontally, right? It’s at an angle. So, to get the accurate force along BC, you’d need to consider that angle using trigonometric functions, like sine or cosine, depending on what you’re trying to find. The book might be simplifying things a bit or leaving out some of these details, which is why you’re getting those unexpected results. In short, yep, the angles should definitely be part of the math here. Without them, you’re missing a crucial piece of the puzzle. Hope that clears things up a bit! Let me know if you need any more help with this—these kinds of problems can definitely be a headache, but once you get the hang of how the angles play into it, it starts to make more sense.