Why didn't the debris collapse back into the Earth at the time of Moon's formation? The most accurate theory for the formation of our moon is the Giant impact hypothesis, which says that a Mars sized body collided with our early Earth and after this collision all the debris got spreaded and that debris moved in an orbit (somehow) and then recombined to form our moon.
But why didn't that debris fall back to the earth? They didn't have any perpendicular component of velocity to follow an orbit as suggested in this answer.
They were projected radially from the earth's and the collider's crust (Right ?)
So why did that debris followed a curved path to form the moon  ?
 A: Hypersonic collisions of solid matter results in vapor. Much of the "debris" must have been in the gas state. Gas doesn't generally follow Keplerian orbits: instead it collapses into a disk. The subsequent evolution of the disk, condensation and accretion, formed the Moon in a process similar to the formation of planets. At least that's the current understanding...
A: The Giant Impact might have liquified and even vaporized a large part of both the impactor and the Earth.  If this was the case, gas expansion may have added velocity to ejecta post-ejection, to alter the orbits of the ejecta.
Lock et. Al  propose the possibility that the impact that created the Moon imparted so much energy that the resultant system exceeded the corotation limit (CoRoL), which is the hottest, highest AM system possible.  They theorize the result would be a rapidly spinning vaporized torus of gas called a synestia instead of a planet. Here is a figure from their paper:

They talk about the synestia contracting and shedding angular momentum similar to how the accretion disk of our solar system is thought to have shed momentum as the Sun and planets were formed.  The formation of the Moon would then be analogous to the formation of the planets in the Solar System.
Note: I took some of the material for this answer from another answer of mine here: https://astronomy.stackexchange.com/questions/26253/how-fast-was-the-earth-spinning-directly-after-the-moon-formed/40781#40781
A: A lot of debris has probably fallen back to earth. To stay in orbit you need enough angular momentum to overcome attraction. But if the collision happened at an angle a portion of the debris could have enough angular momentum to sustain orbit. Here is a nice video of how the collision could have happened.
Here are some snapshots from the video in case the link breaks



Over time any debris that had enough angular momentum to stay in orbit will eventually collect into the moon (or fall back to earth).

Edit based on the comments.
If you have some experience with orbital mechanics you might expect the debris to follow an ellipse. Since an ellipse forms a closed orbit which started at the surface you might expect that after one period all the debris would have fallen exactly where it came from. This isn't the case though: ellipsoidal curves only occur in two body systems. The blob of mass that is ejected is large enough that it has a gravitational field of its own and this complicates things a lot. Combine this with the fact that the blobs can collide with other blobs and stick to each other. This give some pretty complicated interactions and to say anything meaningful you would have to run a simulation at some point. In this case the interactions make some of the matter go into orbit instead of falling back down.
A: Newton's cannonball seemingly applies here, which I suspect is the basis for your question.

An unpowered orbiting body will revisit its position, so anything that got shot off from the surface of the Earth must revisit the Earth. That is correct.
However, unpowered is the issue here. If something interferes with this object mid-orbit, it can alter its trajectory.
As a simple thought example: if I throw a ball up, it will come down again. It will not "bounce" in the air. Launch, upwards, downwards, landing. That is the sequence of events.
But if I throw a second ball and manage to hit the first ball with it, I can alter its trajectory, potentially changing it from moving downward to moving upward again.
Given the right force and the right timing (and non-atmospheric altitude), the first ball could change trajectory into a sustainable orbit.
It's important to realize that all these bits of debris were bumping into each other, constantly altering the trajectory. The bits that came up hit the bits that came down. While the odds of hitting the other bit with the exact force for one (or both) to achieve orbits is rather low, there was a lot of debris.
Given the law of big numbers, it's statistically likely for a subset of the debris to achieve a stable enough orbit. Everything else fell back down or escaped Earth's gravity well.
Note also that the dust didn't need to achieve an orbit that's close to circular. What is more correct is to say that the Moon's current orbit is basically an averaging out of all of the dust particle's trajectories, which very basically averaged out as they impacted each other and became a single object.
These original dust trajectories could have all been wildly different, and it is a fair presumption that they indeed were, before forming the Moon.
A: I can see where you're getting this question from: If you fire a cannon from the surface of an airless planet with enough velocity to achieve some sort of orbit, then that cannonball will inevitably come back around and hit the cannon in the back.
But the situation gets a lot more complicated when you have vast amounts of debris flying into space, because the debris field isn't just a single cannonball in a nice neat gravitational field.  Each speck of debris is gravitationally bound by the planet it just left, but also to all the other debris around it.  A low-flying grain of sand might not have launched with enough velocity to make more than a long arc through the sky, but it's being pulled upward by a trillion grains that nearly acquired escape velocity, and meanwhile the sand grains that got close to escape velocity are being pulled back down by the grains that are below them. It's almost a fluid in the vacuum, drawn together by gravity rather than surface tension.  Much of the mass would fall back down or be launched into an independent solar orbit, but as the smear of mess interacts with itself and congeals, a lot of it is going to find a more or less stable orbit, which will lead to coalescence into as single body over many million years.
Or to put it another way, the impact threw up so much debris that the planet's barycenter moved during the impact, so that orbits that previously would have intersected the surface of the planet were now pointing at empty space all the way around.
But even that's not the whole story. I think you're vastly underestimating the effect on the Earth itself. The formation of the Moon is less "blasting a lot of debris off the surface" and more that the planet itself got completely re-melted and stretched into a blobby oblong by the impact. Don't even think of it like bits of rock flying off the surface into space; think of it like stretching a ball of Play-Doh. So much of the planet went 'up there' that concepts like stable orbits didn't even make a lot of sense for a while. It's like the whole planet got spread out into a protoplanetary disc and reformed into a different shape. (Technically it wasn't really a disc, just a long blob, but still.)
A: Theories of the moon's formation are hypothetical and a subject of active research. While the impact theory is the most popular at the moment, it's not a done deal. So the true answer is that we don't know.
But proposals for how it might have happened generally suggest a glancing collision, where the second body (Theia) ploughs through the mantle of the Earth, missing the iron core. This explains why the moon has no iron core and has much the same mineral content as the Earth's mantle. This means the debris would start off in a range of elliptical orbits grazing the surface, major axes of the ellipses spread out across the hemisphere that suffered the collision (and thus crossing over in high orbit). There are then a couple of mechanisms commonly proposed for circularising the orbits - tidal circularisation, and collision.
When a body in elliptical orbit is subject to tidal forces, the force is different when it is close to the planet than when it is further away. This applies a repetitive torque on the body that tends to make the orbit more circular, and also drags it towards the rotation of the planet. Tidal forces are stronger and more non-linear if the planet has a non-spherical mass distribution, as might be expected after a planetary collision.
The other mechanism is collisions between bodies in orbit. Initially, the particles have a range of energies and angular momenta, are of different sizes and masses, ranging from huge lumps to vaporised rock (lots of heat, so rock will be boiling off, which will push the bits around like tiny rockets). When chunks collide in orbit, they exchange a part of their energy and angular momentum depending on the parameters of the collision, but given the random properties of the bits, may be considered to be at random. It will spread out in all directions from the point of impact. Some of it will be dropped into even more elliptical orbits and collide with the Earth. Some of it will be knocked into a more circular orbit. A certain proportion enters circular-ish orbit outside the Roche limit and stays there, the rest falls back to Earth.

A: I agree you raise a valid point.
It is of course the case that material that is thrown up will not enter circular orbit automatically. Compare launching of satellites. The rocket sets its payload up such that the payload will reach an orbital apogee near the intended altitude, and then some more propulsion is needed to circularize the orbit.
So the question becomes: in the case of the proto-Moon, what effect caused circularization of the orbital motion of the material that went on to form the Moon?
My guess is that the material that was thrown up extended over such a large angle (relative to the Earth), that some debris that was starting to fall down again encounted debris that was still moving upwards.
I imagine that statistically this can have a circularizing effect: the more eccentric the orbit of a particular piece of debris, the larger the probability of a collision with some other piece of debris.  Conversely, the more circular the orbits of a population of pieces of debris the lower the probability of a high speed collision within that population.
For comparison: the rings of Saturn. In the case of Saturn's rings the supposition is that tidal effects from other Saturn moons tend to randomize the motion of the particles of the rings, preventing those particles from assembling to another moon. But the rings are very close to circular because any piece not in circular orbit has an elevated probability to collide with other pieces.
In the case of the proto-moon of the Earth there were no other moons to prevent aggregation of the proto-moon to the eventual Moon.

[Later edit]
I recommend the answer by SoupDragon
More details than in my answer, and the picture makes precicely the point I was trying to make in my answer.
