In a two-body collision in the center-of-mass reference frame the momenta of the particles are equal and opposite to one an- other both before and after the collision. Is the line of relative motion necessarily the same after collision as before? Under what conditions would the magnitudes of the velocities of the bodies increase? decrease? remain the same as a result of the collision?

							
Yes, the line of relative motions necessarily the same after collision as before.
In accordance to the law of conservation of momentum, the total momentum of the particle having mass m1 and the particle having mass m2 before the collision equals their total momentum after the collision. The changes in momentum of the two objects have equal magnitudes and opposite signs, a necessary consequence of the law of conservation of momentum. Therefore the line of relative motions necessarily the same after collision as before.
In an elastic collision, the final velocity v1f of the body having mass m1 is,
v1f = (m1 – m2/ m1 + m2) v1i + (2m2/ m1 + m2) v2i
and the final velocity v2f of the body having mass m2 is,
v2f = (2m1/ m1 + m2) v1i + (m2 – m1/ m1 + m2) v2i
Here, v1i is the initial velocity of the body having mass m1 and v2i is the initial velocity of the body having mass m2.
As the target particle (m2) is at rest, thus, v2i = 0.
Thus, v1f = (m1 – m2/ m1 + m2) v1i and v2f = (2m1/ m1 + m2) v1i
Therefore the magnitude of the velocities of the bodies will depend upon the mass of the two bodies and initial velocities of the two bodies by the above two equations.
As, m2 » m1, thus,
v1f ≈ - v1i + 2 v2i and v2f = v2i
When the massive particle is moving slowly or at rest, then
v1f ≈ - v1i and v2f = 0
As, m1 >> m2, thus,
v1f = v1i and v2f ≈ 2v1i - v2i
Therefore, as, m2 >> m1 or m1 >> m2, then the final velocity of the body will be equal to the initial velocity of the body.