Just because "It'll be out into space and past the Moon's orbit in seconds", why assume that one must measure anything at the height of the Moon's orbit? The whole experiment could be done over the height of a meter.
You arte the one assuming things. You are claiming that the gravitational pull of the Earth is approximately constant over distances of hundreds of thousands of miles. You're wrong. I'm not assuming anything, you are.
The results of Newton's equations for a uniform gravitational field can be correct to a certain number of significant digits and for a given experiment, despite that gravity is not uniform
Yes and you obviously have no clue about how to work out the accuracy of your assumption.
Let's compare them shall we?
1. Assuming gravity is constant, so to raise an object of mass m through a height of h you need E(h) = mgh Joules.
2. Gravity goes like $$F = \frac{GMm}{r^{2}}$$ and to raise an object from the Earth's surface to a height h above the surface is $$E(h) = U(R+h) - U(R) = -\frac{GMm}{R+h} + \frac{GMm}{R}$$
We have that $$g = \frac{GM}{R}$$. We also want to consider 2 different cases, one whee h<<R (you don't raise the object much) and another where h>>R, you raise the object a lot. For the first case
$$\frac{GM}{R+h} = \frac{GM}{R}\left(1 + \frac{h}{R} \right)^{-1} = \frac{GM}{R}(1 - \frac{h}{R} + \ldots \right) = g(1 - \frac{h}{R})$$. Therefore the change in energy is $$E(h) = mgh + \ldots$$
For the second case
$$\frac{GM}{R+h} = \frac{GM}{h}\left(1 + \frac{R}{h} \right)^{-1} = \frac{GM}{h}(1 - \frac{R}{h} + \ldots \right) $$
Now let's compare the approximation you claim is valid with what Newtonian physics actually says. For h<<R you have E(h) = mgh and Newton has E(h) = mgh + ..., so they are approximately correct. This is why you are taught such an approximation in high school, its not too far from the truth
provided h/R is small. But what if h/R is big? Then you have the following Newtonian result :
$$E(h) = -\frac{GM}{h}(1 - \frac{R}{h} + \ldots \right) + \frac{GM}{R}$$
In the limit of h->infinity, ie it never coming back and forever going up and up, you have to give it $$\frac{GM}{R}$$ energy. Compare this wiith your claim, E(h) = mgh. If h is infinite you need to provide infinite amounts of energy. This is NOT what Newton says. This is not some minor correction in the 10th significant figure, you're claiming HUGE changes. For instance, escape velocity is the velocity at which an object has sufficient kinetic energy to go up and up and up and never fall back to Earth, it can keep going. If gravity always behaved like mgh escape velocities would not exist, since you could never overcome the gravitational forces of a planet. Escape velocity for a rocket from the Earth is about 11km/s. This would sling the rocket out past the Moon, the Sun, into empty space. By your equations/claims that rocket wouldn't get more than about 60 million kilometres from Earth. We've sent probes out past that distance, we know how gtavity behaves. You're claiming its
impossible to do what we've done. This isn't a small correction, you're claiming energies 2, 10, 100, a million times more should be needed.
You are not agreeing with Newton or Einstein or experiment, what you have said is
false. You mention Newton's equations for constant gravity but there are no such equations!! Newton's equation for gravity is $$F = G\frac{Mm}{r^{2}}$$, which is dependent on the distance r. Only when you work out how all the particles in a planet or star affect something on the surface and that something doesn't move very much can you even get close to a constant gravitational field. Newtonian gravity is not constant, just as you can only approximate relativistic gravity to be constant when you consider very very little amounts of motion.
Seriously, you're trying to argue physics, which you admit you haven't studied, with someone who has. You're trying to claim things about Newtonian physics and relativity which are
false and unlike you I've actually learnt something about them. You're saying "a few sig figs" when you actually have no clue as to the accuracy of your claims or the level of detail already done by physicists.
Likewise, SR's equations for uniform acceleration (which replace Newton's for a uniform gravitational field, given the principle of equivalence) can return correct results to a certain number of significant digits for a given experiment. To accurately find whether or not objects thrown upward at close to the speed of light accelerate away from us as we measure (not to be confused with the value for g input into the equation), only a certain number of significant digits are needed and only a small height is needed (even a meter will do; if not, make it a centimeter), so SR's equations work fine.
No they fucking dont. Unlike you I've actually calculated quantities in GR and I can tell you absolutely categorically that SR
doesn't cut it when you consider gravity. Ever use a GPS route finder? It uses GR, it is
proof you're wrong.
First, answer me this: Do you agree that SR tells us that, in principle, a spaceship could travel from our Solar system to the nearest other star system, say 4 light years away as we on Earth measure, in just one month on the ship's clock?
That's time dilation. It doesn't tell you the path the rocket should be set on in order to get there, you need to account for gravity to do that and SR cannot do it.
You might not realise this but when you turn on a rocket's engines you generally have to know where to point it in order to get to its destination. I'm sure you're experience it in a car, you have to know which way to drive in order to get to where you want to go. Yes, SR says if you drive quickly your clocks is measure less time but it won't tell you how to get there, which direction to go in. GR does because it tells you how gravity alters the motion of the rocket.
which the equations should agree with.
I know the equations,
you are wrong.
You have no clue what physics involves and yet you're stupid enough to think that because you had a dream, a fucking
dream, then suddenly you can tell physicists how to do physics. Obviously you weren't the sharpest tool in the shed before your dream if you think that a dream can make you knowledgeable.
