Collision of two satellites In a theoretical situation in which two identical satellites travelling around the Earth in opposite directions collided head on so that there velocities relative to the earth both became 0, would they both suddenly accelerate down towards the Earth? I think the answer to this question is yes, so if so, how far away from the Earth would they have to be for this to happen? And how would you calculate this distance?
 A: Your question highlights a common misconception.  A satellite in orbit around the Earth is accelerating towards the Earth right now.  Any object moving in a circular path has an acceleration towards the center of the circle because the direction, and therefore the velocity, of the object is constantly changing.  This acceleration, called centripetal acceleration is given by $$a_c=\frac{v^2}{r}$$ where $v$ is the orbital velocity of the satellite, and $r$ is the distance of the satellite from the center of the Earth.
Now you might find something unexpected.  The International Space Station (ISS) is in a low Earth orbit, orbiting about 400 km above the surface of the Earth.  When you consider that the radius of the Earth is 6,371km (on average) you get an orbital radius of about 6,771km.  The ISS also has an orbital velocity of about 7.7 or 7.8 km/s.  When you use that information to find the centripetal acceleration of the ISS, you find that $a=8.7 m/s^2$  Since gravity is the force that is providing this acceleration to the satellite, this is the same as saying that the acceleration due to gravity at this height is $8.7 m/s^2$.  Note that the acceleration due to gravity at the Earth's surface is about $9.8 m/s^2$, so this is only a decrease of about 11%.
So here's the answer to your real question.  If two satellites were to collide they would continue to accelerate towards the Earth at $8.7 m/s^2$.  The difference is that neither satellite will have an orbital velocity tangent to the path of the orbit anymore.*  As a result, the motion of the satellites will now be towards Earth.  As the satellites get closer to the surface of the Earth, the acceleration due to gravity will increase, but the satellites will also encounter increasing air resistance as they get closer to the surface of the Earth.
*I am assuming the satellites collide completely inelastically.  I consider this to be a reasonable assumption to make, at least for the sake of this thought experiment.
A: IMHO, closest analogy would be the 2009 Iridium / Kosmos hyper-velocity impact...
https://en.wikipedia.org/wiki/2009_satellite_collision
To put it politely, there was a strew of debris, sent every which way. Tracking suggested that a subsequent meteor shower was just that rather than collision debris, which gradually decayed from orbit over many months.
