yes. If the asteroid had equal and opposite momentum to the moon, then when they 'stick together' due to collision, then the new object would not be moving, so it would fall into the earth. (This type of collision is called inelastic, because the mechanical energy is clearly not conserved). In reality, energy is conserved, but it is simply turned into other forms, e.g. heat.
And for your other question: We assume the moon is initially in a circular orbit, and then we want to imagine an asteroid (moving in the same direction as the moon), collides with it, and we want to know the momentum of the asteroid which we require to cause the new 'asteroid+moon' object to completely fly out of orbit of the earth.
Ok, so first let's name the variables. m_m \ , \ v_m to be the initial mass and speed of the moon and m_a \ , \ v_a to be the initial mass and speed of the asteroid. And finally, m_e to be the mass of the earth, and r to be the initial distance between the moon and earth.
First, we assumed the moon was initially in circular orbit around the earth, so we have:
m_m \frac{{v_m}^2}{r} = \frac{m_e m_m G}{r^2}
(Where G is the gravitational constant). And rearranging:
v_m = \sqrt{\frac{m_e G}{r}}
So that's fine, for the initial speed of the moon. Now we can also define m_c \ , \ v_c are the mass and speed of the 'combined asteroid and moon' object, then we want the kinetic energy plus gravitational energy of the new object to equal zero, since this is the least amount of energy required for the object to escape Earth's gravity. So:
-\frac{m_e m_c G}{r} +\frac{1}{2} m_c {v_c}^2 = 0
This last equation is the total energy of the new object, which we want to equal (at least) zero. Also, notice that the gravitational energy of the new object must be negative, this is because I am defining the gravitational energy to equal zero at infinity. Now, rearranging the last equation:
v_c = \sqrt{\frac{2 m_e G}{r}}
Right. So, to make it clear, this is the speed of the 'combined moon and asteroid object', just after the collision has happened. Also, there is another equation we can use: conservation of momentum:
m_m v_m + m_a v_a = m_c v_c
And now, if we use our equations for v_m and v_c, we get:
m_m \sqrt{\frac{m_e G}{r}} + m_a v_a = m_c \sqrt{\frac{2 m_e G}{r}}
We can use one more equation, which is conservation of mass:
m_a + m_m = m_c
And we can use this for the value of m_c, so then we have:
m_m \sqrt{\frac{m_e G}{r}} + m_a v_a = (m_m + m_a) \sqrt{2} \sqrt{\frac{m_e G}{r}}
And from here, doing some rearranging, we get:
v_a = (\frac{m_m}{m_a} (\sqrt{2} - 1) + \sqrt{2} ) \sqrt{\frac{m_e G}{r}}
So this equation tells us what the minimum speed of the asteroid needs to be, in terms of the other variables. Does my working look good? Hopefully I haven't made any mistakes, It is 4am where I am :)
