Janus

Thrust is not the issue, exhaust velocity is. The relativistic rocket equation is [math]V_f = c \tanh \left( \frac{Ve}{c} \ln(MR) \right )[/math] With Ve being the exhaust velocity and MR is the mass ratio (initial mass/final) of the craft. Rearranged to solve for MR, and given that a Nuclear pulse rocket can deliver an exhaust velocity of 30 km/s, we get a mass ratio of 2.1e43 just to get to 1/100 of the speed of light. To put this in perspective, the upper estimated mass of the entire galaxy(dark matter included), is only 2e42 kg. So it would take the order of a mass equal to ten galaxies in fuel per kilogram of payload mass to reach even 1% of c with a nuclear pulse drive. The only foreseeable way that interstellar travel within a lifetime will be possible is to develop propulsion systems that are capable of much higher exhaust velocities.

By the fact that the light form the front flash reaches him first and the sources of the flashes are equal distant from him NO! . The only frame in which the flashes occur when M and M' are co-located is the embankment frame. In both frames, the Flashes occur when each end of the train is co-located with a spot on the embankment where the lightning strikes. These points are at the same spots of the embankment in both frames. However, The relative distance between these points and the two ends of the Train is not equal in both frames. In the embankment frame the train has relative motion and is length contracted, thus it is the contracted length of the train that fits exactly between the simultaneous flashes. However, in train frame, the train is its non-contracted proper length and it is the embankment that has relative motion and is contracted. Thus the distance between the spots on the embankment will be shorter than the length of the train. The front of the train will reach its point of the embankment where the lightning strikes before the back of the train does. Since the lightning strikes and the co-location of the ends of the train and the strike points on the embankment are all co-located, both frames have to agree to to this fact. The train frame has to conclude that the lightning strikes occured at separate times All explained above and in the animations if you bother to study them.

M &M' are co-located when the flashes occur in the embankment frame, they are not so in the train frame. Events according to the Embankment frame: Same events according to the Train frame: Both frames agree that the flashes occur when the Ends of the Trains are co-located with the red dots. Both frames agree that the embankment observer sees the flashes simultaneously, and what point of the train is co-located with the observer when he sees the flashes. Both frame agree that the train observer sees the flashes at separate times, And what points of the embankment are co-located with the observer when he sees each flash. The frames will not agree as to whether the flashes occurred simultaneously, or how far each observer was from the flash when they occurred.

Sun is moving in an orbital path around the galactic center, which is static in space. Static in space according to what? the Milky way in turns orbits the center of mass of the local group, which in turn is being drawn towards the Great Attractor. And it is the barycenter of the Solar system that orbits the galactic center. There is no such thing as a "real" orbital path. There is no absolute reference frame from which to judge such a path. You can only deal with paths relative to some reference point, and the center of mass of the Solar system is just such a reference point and is just as "real" as any other. It also is the reference point that is the least complicated to work from when dealing with the relative motions of the elements of the Solar system. The subsequent movement of Solar systems CoG with respect to the Galactic center is of no consequence in this respect.

Every body, including the Sun orbits the Center of Mass of the combined Solar System. In other words, if you look at the whole of the Solar system as a closed system, then the center of Mass of this system is a fixed point around which everything orbits. Since the various bodies of the Solar system changed position with respect to each other, the paths the planet take around this fixed point vary.

Sorry, but your understanding is wrong. Again, wrong. In SR, as long as A and B have a constant relative motion with respect to each other it does not matter how that motion came to be. As far as time dilation, length contraction, etc are concerned all that counts is the relative motion. A Will see B as time dilated, and B will see A as time dilated. It is only during the actual acceleration that this symmetry is broken (how the symmetry is broken depends on both the direction of the acceleration and the distance between A and B with respect to the acceleration direction). You are violating the Principle of Relativity. If there is no absolute rest, then there is no way that you can say that it the ladder that is moving. Let's put it this way, just because you applied a force to the ladder does not always mean that it is moving. A force can also bring a moving object to a halt. The only way that could say that the force accelerated or decelerated the ladder would be to assume that there is an state of absolute rest by which to measure motion. For instance, once the ladder finishes accelerating, I can establish a new inertial reference frame in which the ladder is at rest and the garage is moving. Now according to you, in this frame the Garage will appear longer. I apply a force to the ladder so that matches the speed of the garage. By your argument above, the ladder should length contract due to its acceleration, making it even shorter than the garage. However, it is now in the same rest frame as the garage, and so as seen from the garage and ladder should be the same length. This is a contradiction. So the only way for this to work as you say is for there to be a absolute rest frame by which to judge the various motions and accelerations. The principle of Relativity forbids this however.

I was wondering the same thing. At the most, one might be able to say that granpa's answer was incomplete because it only gave the answer for one frame, but by the same token, my answer could be considered incomplete because I didn't mention that there was a frame in which A and B never differ in age.

Actually, who is "younger" depends upon which frame you are considering it from. Here's the space-time diagrams for the situation as seen from both the staring inertial rest frame of A and B and from their final inertial rest frame. From the original frame, A (the blue world-line) ends up younger than B,(the red World-lines) However, when you switch to the final rest frame, B is younger than A.

You seem to be mixing up signs. On one hand you are saying that the correct answer is negative, but then say that negative means going upward. You can't drop a something and have have it move upward, unless the acceleration itself is upward, and if upward is negative, then the acceleration is negative. You have to be consistent throughout the problem when applying signs. So assuming that the correct answer is negative, then that means that Up is positive and Down is negative. Therefore: v(initial) =0 x=0 x(initial) = 4 A = -9.8m/s² and v² = 0²+2(-9.8)(0-4) v² = 2(-9.8)(-4) = 78.4 v=sqrt(78.4) = (+/-)8.85 You have to decide on whether the answer is positive or negative based on the context of the problem.

Actually, Doppler effect can be quite useful in analyzing the twin paradox. To wit: After Brian accelerates off, both He and Andy will receive pips from each other at the rate of: [math] \sqrt {\frac{1-0.5}{1+0.5}} = 0.577[/math] per day. (assuming the pips were sent at the speed of light.) When Brian reaches 100 days according to his clock, he has received 57.77 pips from Andy. He heads back to Earth. On the return, he receives pips at a rate of [math] \sqrt {\frac{1+0.5}{1-0.5}} = 1.732[/math] per day So after and additional 100 days (by his clock) he has received an additional 173.2 pips from Andy for a total of ~231 days Thus he says the Andy's clock ticked off 31 more days than his did. Andy also receives pips at a rate of 0.577 per day from Brian as Brian sped off. After 100 days, he has received 57.7 pips from Brian. However, Brain kept going until his clock read 100 ticks, so Andy will not start receiving the return trip pips until he finishes receiving the away trip pips. At 0.577 pips per day, this takes an additional 73.31 days. He then starts receiving pips at the rate of 1.732 pips per day. Again, he has to receive all the pips that Brian sent on the return trip, so this takes 57.7 days. Thus he receives 200 total pips from Brian while counting 100 + 73.31 + 57.7 = ~231 days on his own clock. Exactly what Brian determined. The difference here comes form the fact, that Brian, in turning around and heading back toward Andy started receiving pips at a fast rate immediately, while Andy had to wait until the information the Brian had turned around to reach him across the distance separating them. I should also point out that the time difference between the two is equal to 200/231 = 0.866, the same result you get by using the time dilation formula.

Your question is meaningless. Here's why: Look at this animation from the embankment frame of the standard train example for the Relativity of Simultaneity: Call the event of the the front of the train being even with the right red dot "Event A" Call the event of the the rear of the train being even with the left red dot "Event B" As shown in the animation, these events are simultaneous in the embankment frame. Now look at the animation for the same events from the train's frame. Note that events A and B are not simultaneous and happen at different instants. Now you question is: "What two points in the moving frame when translated to the rest frame coordinates are simultaneous to the moving frame when R1 and R2 are simultaneous in the rest frame?" But that assumes that there is some moment in the Train frame that corresponds to the moment A and B are simultaneous in the embankment frame. But there isn't. There are single events that correspond to single events in the embankment frame. Event A and Event B both happen. and even though they both happen at the the "moment of simultaneity" in the embankment frame, you cannot transform them into a single "moment" in the train frame. You could pick one or the other, but which one is more representative of the "moment of the simultaneity"? And whichever one you pick will have different events that are simultaneous to it in the train frame. To make things worse, there aren't just two events that are simultaneous in the embankment frame when A an B are simultaneous, there are an infinite number of them. Every point along the train is even with some point of the embankment when A and B occur in the embankment frame, and each of those co-locations are an "event". Events that will not be simultaneous to each other in the train frame. And each one of those events could make an equal claim to being representative of the "moment of simultaneity" of the embankment. And each of them have different events that are simultaneous to them in the train frame. So to put it simply, your question has no answer because what you are asking for is disallowed by the very nature of the scenario.

It's known as an analemma: http://en.wikipedia.org/wiki/Analemma It is caused by both the eccentricity of the Earth's orbit and the axial tilt of the Earth.

This how events occur according to the embankment observer: The same events according to the train observer:

Just as an aside: "Troll" comes from "trolling''(trawling), which is a type of fishing where baited lines are slowly pulled through the water in hopes that the fish will take the hook.

A few numbers: To produce a 1g acceleration, your centripetal acceleration would have to be 9.07 m/s, this equals a bank of just a little under 68° from the horizontal. For a track with a radius of 100m, you get a speed of 30 m/s or 21sec per revolution. If the track is 10m wide, then the inner edge will have to be angled at 67.32° to keep the force normal to the floor, and that normal force will be reduce to .977g. The outer edge will have a steeper bank and a higher normal force.

The rate of the Earth's slowing is 2 milliseconds per century. IOW, it would take 50,000 yrs before the day lengthens by 1 sec, and 540,000,000 years before the day reaches 27 hrs in length. This would be a little shorter than the time between now and when the first fish evolved. There is no doubt that life could adapt fast enough to adjust to this change.

Its the more "natural" unit. For example, angular momentum can be simply expressed as L= Iw, where 'w' is the angular velocity measured in radians/sec. If you were to use degrees/sec, you would have to add a factor of pi/180 to get the right answer.

this was just for fun, as I realize that an animated logo is a bit much (and hard to put on a t-shirt1).

Here's just two reasons why this doesn't work. 1. When we look at far distant galaxies, we see them as they were billions of years ago. If dark matter were the remnants left over from earlier stars, the further the galaxy is away from us, the more of these earlier stars we would still see. In other words, as we look at further and further galaxies, we should detect more and more baryonic matter and less and less dark matter. We don't. 2. If all these earlier stars produced heavy elements, we would see a higher percentage of heavier elements than we do. It doesn't matter how much of this remnant formed latter stars, the mere fact of larger concentrations of heavy elements would change the relative abundance of heavy to light elements in the spectrum we get from second generation stars. The ratio of elements that we measure in our observations precludes the existence of much more baryonic matter than we see.

Unfortunately, a recent paper has thrown a wet blanket on the Alcubierre warp. It is purely a classical theory which does not take into account quantum mechanics. It turns out that when you add QM into the picture, the warp in not stable and even if you could maintain it, everyone in a craft propelled by it would be fried by Hawking radiation. There is no way for us to presently create the exotic matter needed even in theory. There isn't any evidence that it is even possible for such exotic matter to exist.

The ability to visibly see it is not the point. Any electromagnetic energy emitted by the brain can still be detected. And then there are the numbers involved. Given the total power usage of a human brain multiplied by the population of the Earth you get about 2e11 watts. This is less than 1/16 of the total energy used just by the US, and some 870,000 times less than what the Earth receives from the Sun. Considering that the Earth intercepts just a small fraction of the Sun's output, and the Sun is just one of hundreds of billions in our galaxy alone, and that dark energy is estimated to make up over 70% of the Universe, one can see that any energy emitted by the human brain is insignificant compared to what is needed to explain dark energy.

Dark matter and dark energy are two different things. Dark matter is matter that does not react electromagnetically, and thus does not absorb, reflect or emit light or any other part of the electromagnetic spectrum. We have some ideas for candidates for dark matter, but but as of yet haven't discovered the actual culprit. Dark energy is a term used for whatever is causing the apparent acceleration of the Universe. In this case, the label "dark" just means "unknown". The "dark thoughts" statement is just too silly for serious comment.

The Alan Parsons Project even did an album entitled Pyramid, which included the song Pyramania:

Velocities do not add linearly, but by the equation [math]w= \frac{u+v}{1+\frac{uv}{c^2}}[/math] So for example, if the craft were moving at .99c relative to the external observer, and you where moving at .99c relative to the ship, the external observer would measure your velocity as 0.9999494975c For the example of 50 mph and 20 mph, the answer comes out to 69.99989546 mph, Which differs from 70 mph by only 6.6236 inches per hour. So you can see why before Relativity we thought velocities added linearly.

But we can detect antimatter. The positron is the antimatter version of the electron, and is produced by some radioactive isotopes. Such isotopes are used in PET scans. We have also been able to create and store small amounts of antimatter.