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Cory, here's a different way of thinking about gravity assists that may help: First is my short answer for readers in a hurry: What is really going on is a giant game of pool, with fast-moving planets acting as massive cue balls that impart some of their energy when they whack into tiny spacecraft. Since you can't bounce a spacecraft directly off the ...

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The problem is what Konstantin Tsiolkovsky discovered 100 years ago: as speed increases, the mass required (in fuel) increases exponentially. This relation, specifically, is $$\Delta v=v_e\ln\left(\frac{m_i}{m_f}\right)$$ where $v_e$ is the exhaust velocity, $m_i$ the initial mass and $m_f$ the final mass. The above can be rearranged to get $$... 61 Can photons push the source which is emitting them? Yes. If yes, will a more intense flashlight accelerate me more? Yes Does the wavelength of the light matter? No Is this practical for space propulsion? Probably not Doesn't it defy the law of momentum conservation? No In fact that last question is the key one, because photons ... 56 One word: inertia. When you're riding a bike on a level gradient you just need to give it a push to get going, then you can coast for quite a while before friction and air resistance slow you down. The human body doesn't have wheels that can store kinetic energy, so while running you have to give a good kick to get going, and then another kick to keep going ... 42 When light is propagating in glass or other medium, it isn't really true, pure light. It is what (you'll learn about this later) we call a quantum superposition of excited matter states and pure photons, and the latter always move at the speed of light c. You can think, for a rough mind picture, of light propagating through a medium as somewhat like a ... 39 At the physics 101 level, you pretty much just have to accept this as an experimental fact. At the upper division or early grad school level, you'll be introduced to Noether's Theorem, and we can talk about the invariance of physical law under displacements in time. Really this just replaces one experimental fact (energy is conserved) with another (the ... 34 For the photons that make up light to exist they have to be travelling at the speed of light. This means that to store them you have to put them in a container where they can move around at the speed of light until you want to let them out. You could build the container out of mirrors, but no mirror we can build is 100% reflective, or indeed can be 100% ... 34 Newton's second law As you probably know, Newton thought that energy is linearly proportional to velocity. The second law's original formulation reads: "Mutationem motus proportionalem esse vi motrici impressae" = "any change of motion (velocity) is proportional to the motive force impressed". This law, which nowadays is wrongly interpreted as: F = ma ... 33 Energy is in fact conserved, even in gravitational slingshots. After the slingshot, the velocity of the spacecraft may indeed change, which means its kinetic energy will also change. If this happens, the energy increase (or decrease) will be made up by a commensurate decrease (or increase) in the kinetic energy of the planet. In plain English: The planet ... 32 The extra force in a lever comes from the distribution of energy over a smaller distance of action. We can look at it very generally from the perspective of conservation of energy, or the lossless transmission of work through a machine. When we move something against a resistance, with a fixed force, we are putting in energy and doing work. This work is ... 27 TL;DR: This answer arrives at roughly the same conclusion as Kyle Kanos', i.e. in addition to payload considerations, the difficulty lies in stuffing a small rocket with a mass of fuel exceeding to the mass of the rocket itself. This answer, however, is more rigorous in how the \Delta v budget is treated. Developing a relationship between rocket and ... 25 Your examples are a bit misleading. For example you say: We can store cold (ice),heat (i.e. hot water bag) But we can only store heat temporarily, just as we can only store light temporarily. Your ice pack will eventually heat up and your hot water bottle will eventually cool down, just as light stored between two mirrors will eventually escape. ... 23 A longer popular text why energy conservation becomes trivial (or violated) in general relativity is e.g. here: http://motls.blogspot.com/2010/08/why-and-how-energy-is-not-conserved-in.html To summarize four of the points: In GR, spacetime is dynamical, so in general, it is not time-translational-invariant. Because it's not, one can't apply Noether's ... 23 We treated this a while back at University... First of all, I assume you mean global cancellation, since otherwise the energy that is missing at the cancelled point simply is what is added to points of constructive interference: Conservation of Energy is only global. The thing is, if multiple waves globally cancel out, there are actually only two ... 22 The topic of "Energy Conservation" really depends on the particular "theory", paradigm, that you're considering — and it can vary quite a lot. A good hammer to use to hit this nail is Nöther's Theorem: see, e.g., how it's applied in Classical Mechanics. The same principle can be applied to all other theories in Physics, from Thermodynamics and Statistical ... 20 The surface of any fluid has an associated energy-per-unit-area, known as the surface energy, a.k.a. surface tension. This energy is not a property of the fluid alone, but of the fluid and the medium it is in contact with. In your case you would have associated surface energies for the water-air interface, e_{wa}, as well as for the water-paper interface, ... 18 Contrary to what is stated in many textbooks, energy-momentum conservation alone cannot explain the behavior of Newton’s cradle. For N balls we have two equations and N final velocities to calculate. Hence, conservations laws can do the job only for N=2. This means that if we want to give an explanation of the cradle behavior based on conservation laws, we ... 18 MSalters already said "yes". I would like to expand on that by computing the change. Let's take a 10 kg cannon ball, made of lead. Heat capacity of 0.16 J/g/K means that in dropping from 1000 K to 100 K it has lost 10000\cdot 900 \cdot 0.16 \approx 1.4 MJ. This corresponds (by E=mc^2) to a mass of 1.6 \cdot 10^{-11} kg or one part in 6\cdot 10^{11}. ... 17 Many of us have ridden bicycles at some time in our lives. and in fact this mode of transportation has become markedly more popular recently as a result of the energy shortage. Each morning at my own university, Duke, people can be seen riding machines with masses of 10 to 20 kilograms and struggling to reach one of the ... 16 Yes, kinetic energy is a relative quantity. As you might guess, this means that when you're using energy conservation, you have to stay within a single frame of reference; all that energy conservation tells you is that the amount of energy as measured in any one frame stays the same over time. You can't meaningfully compare the amount of energy measured in ... 16 Of course, it does, since:$$\frac{\partial E}{\partial t} = \frac{\partial }{\partial t} \left(m \cdot c^2 \right) $$Very little, though 16 Can photons push the source which is emitting them? Yes, photons have momentum and momentum must be conserved. The source is pushed in the opposite direction of the photons. If yes, will a more intense flashlight accelerate me more? Yes, more photons means greater momentum. Does the wavelength of the light matter? Yes, shorter wavelength ... 16 Work is calculated as force times distance.$$W = Fd$$The purpose of a simple machine like a screw jack is to lessen the force required. However, the work needed is still the same, so the distance over which you exert the force has to increase. Halving the force requires doubling the distance. In this problem, you want to lift 2000 lbs a distance of 1 ... 15 1. If the universe is expanding, what is it expanding into? The universe isn't expanding into anything. Space-time isn't curving into a higher-dimensional space. So what do we mean by "curved" and "expanding", words usually having a meaning only for objects in space? The answer is it is just an analogy. Mathematicians have found properties of space an ant ... 15 If you write down the formulas for kinetic energy,$$E_k = \frac{1}{2} m v^2$$and momentum$$p = mv$$you see that you can write the energy in terms of momentum via$$E_k = \frac{p^2}{2m} So, if two objects have the same energy $E_k$, they only have the same momentum if they also have the same mass. Since the bull has a much larger mass than the bullet, ...

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Waves always travel. Even standing waves can always be interpreted as two traveling waves that are moving in opposite directions (more on that below). Keeping the idea that waves must travel in mind, here's what happens whenever you figure out a way to build a region in which the energy of such a moving wave cancels out fully: If you look closely, you will ...

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I think that the other two answers miss an interesting point of your question. Lets put a light bulb inside 100% reflecting sphere. All emitted photons will stay inside the sphere and intensity of light will continuously increase to infinity. Real mirrors are never 100% reflective but there are mirrors with 99.98 reflectivity (these are dichroic mirrors ...

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Magnetic field in this case (a set of magnets in space, no relativity involved) is conservative, which means it has a potential -- each positional configuration of charges (or dipoles in this case) has its fixed energy which does not depend on history or momenta of charges. So, the work you put or get from displacing them is just exchanged with the potential ...

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The energy conservation becomes vacuous or invalid in the general theory of relativity and especially in cosmology. See http://motls.blogspot.com/2010/08/why-and-how-energy-is-not-conserved-in.html Why and what does it imply? First of all, Noether's theorem makes the energy conservation law equivalent to the time-translational symmetry. In general ...

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In non-relativistic and non-gravitational physics (both conditions have to be satisfied simultaneously for the following proposition to hold), energy is only defined up to an arbitrary additive shift. In this restricted context, the choice of the additive shift is an unphysical, unobservable convention. Special relativity However, in special relativity, ...

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