Imagine we have something very heavy (i.e supermassive black hole) and some object that we can throw with 0.999999 speed of light (i.e proton). We are throwing our particle in the direction of hole. The black hole is so heavy that we can assume that in some moment acceleration of gravity would be say 0.0001 speed of light/s^2. So the question is what will be the speed of proton in few a seconds later, assuming we have such distances that it will not hit the black hole before.
General Relativity – Can Gravity Accelerate an Object Past the Speed of Light?
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To answer your question you need to be clear what coordinates you're using. If you use coordinates that are co-moving with the rock falling into the black hole then the rock will see the event horizon pass at the speed of light.
External observers, using Schwarzchild coordinates, will see the rock slow down as it approaches the horizon, and if you wait an infinite time you'll see it stop.
External observers obviously can't comment on the speed of the rock after it has passed the event horizon because it takes longer than an infinite time to get there. If you use the rock co-moving coordinates then you can ask what speed you hit the singularity and ... actually I'm not sure what the answer is. I'll have to go away and think about it.
Incidentally http://jila.colorado.edu/~ajsh/insidebh/schw.html is a fun site describing what happens when you fall into a black hole.
A black hole of mass $10^{36}$ kg would have a Schwarzschild radius (the distance from the center to the event horizon) of about $1.5\times 10^{9}$ m.
So your choice of "orbital" distance is inside the black hole, and it won't orbit, it will just fall inward. Outside the event horizon there is a minimum distance for a stable circular orbit which is three times the Schwarzschild radius.
You cannot use Newtonian physics for these calculations; you need full blown general relativity. If you try mixing these two systems together you get nonsense. It is only at distances where the gravitational field is weak that you can apply Newtonian physics with good accuracy.
Regarding why we are so confident that we know what happens inside a black hole (like everything falling inward) when we can never test that directly it comes down to the way the theory is designed. The theory for what happens outside the event horizon and inside the event horizon are the same theory and as we have verified that theory for outside the event horizon to a very, very high level of confidence so there is a very high level of confidence in it's predictions for inside the event horizon. The basic core the theory of general relativity is that physics must work the same way everywhere. The event horizon doesn't make any difference in this sense.
Best Answer
This the classic "hurling a stone into a black hole" problem. It's described in detail in sample problem 3 in chapter 3 of Exploring Black Holes by Edwin F.Taylor and John Archibald Wheeler. Incidentally I strongly recommend this book if you're interested in learning about black holes. It does require some maths, so it's not a book for the general public, but the maths is fairly basic compared to the usual GR textbooks.
The answer to your question is that no-one observes the stone (proton in your example) to move faster than light, no matter how fast you throw it towards the black hole.
I've phrased this carefully because in GR it doesn't make sense to ask questions like "how fast is the stone" moving unless you specify what observer you're talking about. Generally we consider two different types of observer. The Schwarzschild observer sits at infinity (or far enough away to be effectively at infinity) and the shell observer sits at a fixed distance from the event horizon (firing the rockets of his spaceship to stay in place).
These two observers see very different things. For the Schwarzschild observer the stone initially accelerates, but then slows to a stop as it meets the horizon. The Schwarzschild observer will never see the stone cross the event horizon, or not unless they're prepared to wait an infinite time.
The shell observer sees the stone fly past at a velocity less than the speed of light, and the nearer the shell observer gets to the event horizon the faster they see the stone pass. If the shell observer could sit at the event horizon (they can't without an infinitely powerful rocket) they'd see the stone pass at the speed of light.
To calculate the trajectory of a hurled stone you start by calculating the trajectory of a stone falling from rest at infinity. I'm not going to repeat all the details from the Taylor and Wheeler book since they're a bit involved and you can check the book. Instead I'll simply quote the result:
For the Schwarzschild observer:
$$ \frac{dr}{dt} = - \left( 1 - \frac{2M}{r} \right) \left( \frac{2M}{r} \right)^{1/2} $$
For the shell observer:
$$ \frac{dr_{shell}}{dt_{shell}} = - \left( \frac{2M}{r} \right)^{1/2} $$
These equations use geometric units so the speed of light is 1. If you put $r = 2M$ to find the velocities at the event horizon you'll find the Schwarzschild observer gets $v = 0$ and the (hypothetical) shell observer gets $v = 1$ (i.e. $c$).
But this was for a stone that started at rest from infinity. Suppose we give the stone some extra energy by throwing it. This means it corresponds to an object that starts from infinity with a finite velocity $v_\infty$. We'll define $\gamma_\infty$ as the corresponding value of the Lorentz factor. Again I'm only going to give the result, which is:
For the Schwarzschild observer:
$$ \frac{dr}{dt} = - \left( 1 - \frac{2M}{r} \right) \left[ 1 - \frac{1}{\gamma_\infty^2}\left( 1 - \frac{2M}{r} \right) \right]^{1/2} $$
For the shell observer:
$$ \frac{dr_{shell}}{dt_{shell}} = - \left[ 1 - \frac{1}{\gamma_\infty^2}\left( 1 - \frac{2M}{r} \right) \right] ^{1/2} $$
Maybe it's not obvious from a quick glance at the equations that neither $dr/dt$ nor $dr_{shell}/dt_{shell}$ exceeds infinity, but if you increase your stone's initial velocity to near $c$ the value of $\gamma_\infty$ goes to $\infty$ and hence 1/$\gamma^2$ goes to zero. In this limit it's easy to see that the velocity never exceeds $c$.
In his comments Jerry says several times that the velocity exceeds $c$ only after crossing the event horizon. While Jerry knows vaaaaastly more than me about GR I would take him to task for this. It certainly isn't true for the Schwarzschild observer, and you can't even in principle have a shell observer within the event horizon.