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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 ... 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 ... 25 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 ... 21 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 (there is no ... 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 ... 14 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 ... 14 Repeat lifting books to height h several times per minute for an hour or two, and observe who gets tired. You do work lifting the book, burning up biological energy stored in ATP molecules in your cells, to activate muscles, which apply some force upward on the book (as well as suffer some internal friction). The biological energy used up is converted ... 13 Note: This question was cross posted by the OP on the Mathematics Stack Exchange. Here is a copy of my answer for it there. This is it. The perfectly centered billiards break. Behold. Setup This break was computed in Mathematica using a numerical differential equations model. Here are a few details of the model: All balls are assumed ... 10 As the magnet approaches the solenoid, a current is induced. The current generates a magnetic field. The field repels the magnet, slowing it's approach. The amplitude of the oscillations diminish. If there was no resistance, this would work in reverse as the magnet receded from the solenoid. The magnetic field would accelerate the magnet. The magnet would ... 9 Yes, when you fire a pistol the hammer hits the bullet with a relatively small initial kinetic energy but the kinetic energy of the hammer and bullet after the collision is considerably higher. This may seem a silly example, but I think it actually highlights the important principle involved. In general when two bodies undergo an inelastic collision part of ... 8 "The end state in A and B is the same." This is the fallacy. The total state has to include the potential field of the gravitational field. If you do things via A, the potential field disturbance has not yet propagated to the right-most body, so the potential energy field has not yet settled down. After a few more seconds, the disturbance will reach the ... 8 A simple counterexample: Imagine two particles with opposite direction and equal speed. The center of mass does not move, yet the kinetic energy of the system is non-zero. Now let both particles come to rest (by friction, hitting a wall, whatever). The kinetic energy is now zero, and total momentum has been conserved, while energy is not. The crucial ... 8 You say: Imagine a book that we lift it with a force that is exactly equal to the force of gravity so the forces cancel out and the book moves with a constant velocity. so I'm guessing your reasoning is that the net force on the book is zero so the amount of work done on the book is zero. And you are absolutely correct - no work is done on the book and ... 7 Gravity assists don't change speed in the two body problem. An object approaching a lone gravitating body will enter and leave the vicinity of that body with exactly the same speed. All that a lone gravitating body can do is change the direction in which the object is heading. The body that provides the assist needs to be moving with respect to the target ... 7 The answer is yes. The de Broglie wavelengths of freely propagating particles (i.e. forget the influence of interactions and gravity perturbations, just consider the Universe as a whole) are redshifted by the expansion of the universe. Another way of saying this is that their peculiar momenta with respect to a co-moving local volume decrease as the inverse ... 7 Gravitational potential energy is usually measured as a negative value. We do this because an object that is so far away from a gravity well that it practically is unaware of it shouldn't be considered as having any potential energy. So as r\to\infty, PE\to0. As an object falls into a gravity well, it loses potential energy, so gravitational PE is a ... 7 Noether's theorem states that to every continuous symmetry of a physical system there is an associated, conserved quantity. The conserved quantity associated with time translation invariance (i.e. it doesn't matter if you perform an experiment now or tomorrow, provided you set it up the same way) is what we call energy. Therefore, somewhat tautologically, ... 7 This is an experimental physicist's answer: The linked article is careful to state: That means that conservation of energy can appear to be violated, but only for small values of t (time) Italics mine. Conservation of energy is an experimental fact that has been validated in innumerable experiments. This means, as far as the correspondence with ... 6 Suppose the ramp wasn't there, then the trajectory of the object would the same as if it fell off a cliff: To get the equation of motion you simply note that the horizontal and vertical coordinates are given by (neglecting air resistance):$$ x = ut  y = \tfrac{1}{2} g t^2 $$So you can get the trajectory by substituting for t to get:$$ y = ...

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Courtesy of the book Carl found we have an answer! Consider the element of the liquid helium at a height $h$ above the fluid surface and distance $y$ from the wall. To raise that element above the fluid surface costs an energy $mgh$, but because there is a Van der Waals attraction between the helium atoms and the wall you get back an energy $E_{VdW}$. ...

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Does it get converted to mass and added into the black hole mass? Yes, exactly. But asking this question in the way you have exposes a subtlety in the definition of mass: the amount of energy that counts as a system's mass doesn't all have to be "intrinsic mass." Different kinds of energy can contribute to mass as well. The mass of any system is really ...

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You should think of the formula the other way around, i.e. $$\mathrm{d}W = F\mathrm{d}x$$ which means that the infinitesimal work done along an infinitesimal path is just the force $F$ times the length $\mathrm{d}x$ of the path along which the force was exerted. If we are now given a real path $\gamma : [a,b] \to \mathbb{R}^3$, the total work done along ...

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Good question! Many astronomers think that the moons of Mars, Phobos and Deimos, are captured asteroids. Others object precisely because of the issues that you raised. Capture is not easy. Sans a collision, capture is impossible in the Newtonian two body problem. A hyperbolic trajectory stays hyperbolic. On the other hand capture in the multi body problem ...

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I will try to give a short introduction into the ideas of scientific truth as I understand them. In mathematics, the world is beautifully simple. We have axioms that the set to be true, and from these we can deduce a plethora of statements to be undoubtedly true - given that the axioms are true. There may be undecidable statements about which we cannot say ...

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Temperature means energy. The heat energy is still here. It's just that the "object" (the Universe) grown bigger so this energy had to spread through it. The more energy in a single point, the hotter it is. That's why they say it got cooler. It's like the expanding gas from your spray deodorant is cold when it leaves the can, but it was at room temperature ...

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From the official website: Honestly, I think this basically says all you need to know.

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If the universe is flat does it mean it doesn't exist? What kind of incoherent question is this? "If the universe is flat" presumes the universe exists and is spatially flat. Your question amounts to "Does the existence of a spatially flat universe mean the non-existence of a spatially flat universe?". Isn't the answer analytically no?

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I was surprised to see an effect that's actually real, and not a hidden motor or something like that. I believe this experiment can feasibly be repeated. The principles behind it make sense. The driving force comes from the density difference in the cup versus in the pipes. The cup has very few bubbles in it compared to the pipe. Why? Because: The ...

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