Your teacher's definition is the correct one. Your math is unnecessarily complicated
unless there is a possibility of each object gaining or losing mass during the collision.
Quote:
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((initial velocity of object A * mass of object A) – (final velocity of object A * mass of object A)) + ((initial velocity of object B * mass of object B) – (final velocity of object B * mass of object B))
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I would put it this way:
Change in momentum (Impulse) of object A = mass of object A (final velocity A - initial velocity A)
Change in momentum (Impulse) of object B = mass of object B (final velocity B - initial velocity B)
Your addition of the two will simply give you the "total impulse" for the system.
Does that make sense?
Nate