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Whoa, there! You made another very basic mistake in the original post. I assume you are trying to find the fourier transform of [math]\exp(-at^2/\Delta t^2)[/math]. The correct way to do this is via the definite integral [math]\int_{-\infty}^{\infty} \exp(-at^2/\Delta t^2)\,\exp(-i \omega t)\,dt[/math] Note very well: This is a definite integral, not an indefinite integral.

The denominator is a constant. It is just a number. You can take it outside the integral. BTW, what is [math]i^2[/math] ?

Firstly, you did not specify the variable of integration, which I assume is [math]t[/math]. Secondly, you used the fact that [math]\exp(a)\exp(b) = \exp(a+b)[/math] for the very first step. That's fine. What's wrong is that was use that this relation works both ways. Use this relation again, but in reverse.

If you mean gravitational effects, the answer is no. At the atomic level gravity itself is but a minor perturbative effect. Tidal gravity is essentially a non-effect. If on the other hand you mean are there any residual forces analogous to tidal forces being a kind of residual effect of gravitation, the answer is yes. Van der Waal forces are residual effects of the electrostatic force, and the nuclear force is a residual effect of the strong interaction.

Whoever gave you that explanation did you wrong. That is nonsense. The force is perpendicular to the motion for a circular orbit only. While Earth and Venus have nearly circular orbits, Mercury's orbit is somewhat eccentric (e=0.206), and comets have highly eccentric orbits (e.g., 0.967 for Halley's comet). A good place to start is Kepler's laws. The orbit of a planet around the Sun is an ellipse, with the Sun at a focus of the ellipse. For any given planet, the line between the Sun and the planet sweeps a constant area per unit time. The square of the period of a planet's orbit is proportional to the cube of the semi-major axis of the orbit. Kepler's laws answer the question raised in the original post in the sense that they say that planets do not fall into the Sun. They follow elliptical paths instead. The answer provided by Kepler's laws is not particularly satisfying. Why do planets follow ellipses? The answer is because Newton's law of gravitation says that is what the must do. Newton's law of gravitation combined with Newton's laws of motion form a second-order differential equation. The solutions to this differential equation are in the form of conic sections: Circles, ellipses, parabolas, and hyperbolas.

There you go!

The hint ed84c gave is a bit cryptic (and appropriately so). Several hours have passed. A bit less cryptically, the idea is the solve for [math]e^x[/math]. Set [math]u=e^x[/math]. The equation in the original post becomes [math]u+1/u=3[/math]. Now multiply both sides by u. What kind of equation results from this?

Yep. That will do, also, and that is probably what is wanted for this (what appears to be homework) problem. Of course saying acosh(1.5) is much easier.

There is no teasing that x out of there. Bignose suggestion to look into the hyperbolic functions is exactly right.

Your numbers are incorrect, sjbouscher. Read Bignose's post. Read the correct answer in post #6. Read the problem. It says right up front that the cow eats the same amount as the duck and goat combined and that the duck and goat eat all the grass in 90 days. You set the problem up incorrect in post #10 and arrived at a nonsense answer. What is the right way to set up the problem? The underlying assumption is that the grass grows at some unspecified linear rate, call this rate r. If the initial height of the grass is h, the height h_0 (the subscript 0 denotes no critters are eating the grass) at time t is [math]h_0(t)=h+rt[/math]. That is assuming no critter is munching on it, of course. This is not a particularly realistic assumption (grass stops growing at some height), but it becomes a lot closer to reality when critters keep the grass trimmed. Suppose some critters are munching on the grass -- the cow and goat, for instance. These critters are going to eat fixed amounts of grass per day. Denote the amount eaten by the cow as c and the goat as g. Now the height as a function of time is [math]h_{c,g}(t) = h+(r-(c+g))t[/math]. The cow and goat eat all the grass in 45 days. In other words, the height is zero at t=45. In math, [math]h+45(r-(c+g)) = 0[/math] or [math]45(c+g-r)=h[/math]. Continuing in this regard, [math]45(c+g-r)=h[/math] [math]60(c+d-r)=h[/math] [math]90(d+g-r)=h[/math] [math]c=d+g[/math] That final equation reflects the given information that the cow eats as much as the duck and goat combined. One thing that makes this problem interesting is that the problem is uniquely solvable even though there are only four equations and five unknowns (c, d, g, h, and r).

The gravitational potential energy between two objects rises toward some horizontal asymptote as the distance between these objects approaches infinity. Calling this horizontal asymptote zero is convenient, but not necessary. In gravitational potential energy, as in any other form of potential energy, it is only the change in potential energy that matters. Gravitational potential energy is not "negative energy" in the sense that it represents an over-unity source of energy. The sole reason gravity can be represented in the form of a potential is because gravity is a conservative force. You can draw energy out of a gravitational system, but that means the total energy of the system, potential+kinetic, has decreased. There is no free lunch with gravity, or any other conservative force. In Newtonian mechanics, a pair of isolated gravitating objects will maintain a constant total energy. However, Newtonian gravity is not quite right. In general relativity, two gravitating objects will emit gravity waves because gravity makes the objects accelerate. The total energy of the two objects will decrease over time.

Deceleration is only useful in the context of one dimensional motion, and only with respect to some predefined reference frame. Deceleration is a concept of very limited applicability, and is fully covered by the concept of acceleration in general. Acceleration is any change in velocity.

Just to add to what Bignose has already said, look at the problem statement. The answer to the question of how long it will take the cow by itself to eat the grass is right there. "The duck and the goat eat the grass in 90 days. The cow eats the same amount of grass as the duck and the goat together." In other words, the cow will eat the grass in 90 days, not 7.5.

I predict that in 2012 The US will elect a Democrat or Republican as President. A political scandal will take place somewhere in the world. A famous celebrity will die. News agencies will cover this death to death. The year will end on December 31. I will of course accept cash contributions in recognition of these amazing prognostic capabilities.

"Realistic only" and you guys immediately launch into discussions of a bunch of things that are pipe dreams only? You might want to reword the question.

Any erection that lasts longer than four hours, regardless of cause, is called priapism. Among other things, it can lead to gangrene, and the only solution for that is amputation. The TV ads for all of these products mention this problem because even if it occurs only rarely it is apparently perceived as a huge selling point.

Ask your girlfriend to become a bit more handsy. And eat a slice or two of watermelon.

Davis, T.M. and Lineweaver, C.H. (2001), "Superluminal recession velocities", AIP Conf. Proc. 555, 348, DOI:10.1063/1.1363540 http://arxiv.org/abs/astro-ph/0011070 What stage we're currently at: The state of the art is chemical propulsion. Mankind has a grand total of five tiny vehicles on escape trajectories from the solar system: Pioneer 10 and 11, Voyager 1 and 2, and New Horizons. The first four are well beyond any of the planets. New Horizons is en route to Pluto. That is the best we have done with, and is close to the best we can possibly do with chemical propulsion. Beyond the state of the art: There's ion drive propulsion. Not much oomph, yet. Ion propulsion has moderately high specific impulse but incredibly low thrust. It took SMART-1 over a year to attain go from mid Earth orbit to lunar orbit. VASIMR has promise to bridge the gap between the high thrust / low impulse world of chemical rockets and the extremely low thrust / high impulse world of ion propulsion. Ion propulsion will not get us to the stars. Any kind of propulsion that requires the vehicle to carry the fuel with it will not get us to the stars. The non-relativistic Tsiolkovsky rocket equation is a very brutal equation. The relativistic rocket equation makes the Tsiolkovsky rocket equation look tame. Propulsion that does not require carrying fuel with the vehicle is the realm science fiction. Solar sails have been tested on a very small and very limited scale. Bussard ram jets are science fiction, but plausible science fiction. Beyond that is science fantasy.

Not quite. The distant galaxies are a lot further further away than can be explained without considering the expansion of space. The recession rate of those very distant galaxies is greater than the speed of light, thanks to the expansion of space. So, to answer the question raised by the title of the thread: We are traveling at light speed, and then some -- with respect to those distant galaxies, that is. That of course does not answer the intent of the question. The answer to the intent of the question is to stop reading so much science fiction and get a bit more down to Earth, or at least get a bit closer to the Milky Way.

Several posts moved to this thread. Note: The subject of this new thread lies more in the domain of quantum mechanics (i.e., Quantum Theory) than relativity.

You weigh slightly (and very, very slightly) less when the Moon is directly overhead / underfoot compared to when the Moon is on the horizon. However, that does not mean you weight less when the tides go up / more when they come down. The ocean tides don't necessarily rise and fall with the Moon. It's a lot more complex than that. The image below displays the amplitude and timing of principal lunar semidiurnal (or M2) component of the ocean tides. Source: http://svs.gsfc.nasa.gov/stories/topex/tides.html Note that there are several dark blue patches at the center of which several white lines coalesce. The points at which those lines come together are called nodes: Points where the M2 component of the tide is zero. The white lines coming out of those nodes are called cotidal lines. All points on a given cotidal line will have high and low tides occur at exactly the same time. Move toward the next cotidal line and the tides will be earlier or later, depending on which direction you are heading. You will also see several spots on the Earth such as the North Sea, Hudson's Bay, southeast South America, and the east coast of Maylaysia where there are a lot of white cotidal lines. The timing of the tides is very complex in those areas. For example, at any given time of day, there is always a high tide somewhere in the North Sea.

Did you read it? Another post by the same author: http://www.physicsforums.com/showpost.php?p=675711 Yet another: http:// http://www.physicsforums.com/showpost.php?p=865368 An interpretation of the post in question: http://www.physicsforums.com/showpost.php?p=1212576 More from physics forums: http://www.physicsforums.com/showpost.php?p=856187 Light never travels with a speed that is anything less than c. http://www.physicsforums.com/showpost.php?p=3992497 In fact, photons never travel at any speed other than c. We have working physicists at this forum, too: http://www.scienceforums.net/forum/showpost.php?p=508561 photons always travel at c http://www.scienceforums.net/forum/showpost.php?p=179222 So saying that photons always travel at c is correct.

Phase velocity can much less than c. Of course, it can also exceed c. You are evading the issue. Are you of the opinion that photons do not always travel at c?

You apparently need to go back to school, then. insane_alien and iNow are correct. Photons always travel at c. Period. You are conflating the group velocity of a stream of photons with velocity of the photons that comprise the stream.

What measurements? The question is about Maxwell's equations.