Firstly, my apologies as I forgot this website does not allow Check My Work type of discussions. Thank you to those who answered the reasonings by the way,
Basically, my question was as follows:
In a 2D collision, is there a reason why you cannot find the resultant velocities of an object before and after a collision and use that to find impulse? Why do you have to find each impulse component first using a change in velocity for x and y) before you solve using Pythagorean theorem?
Recall that impulse is defined as $$\vec{J} = \vec{P}_f - \vec{P}_i$$ If you just want the magnitude, you can consider the magnitude of both sides of the equation above $$|\vec{J}| = |\vec{P}_f - \vec{P}_i|$$ For any two vectors in general $$| \vec{P}_f - \vec{P}_i| \neq | \vec{P}_f| - | \vec{P}_i |$$ A good example to consider is two opposite pointing vectors, say $\vec{A} = 4 \hat{i}$ and $\vec{B} = -4\hat{i}$. Here, $\vec{A} - \vec{B} = 8\hat{i}$. So, $| \vec{A} - \vec{B}| =8$ while $| \vec{A}| - | \vec{B}|=0$. Since you want the magnitude of the impulse, you must find the vector $\vec{P}_f - \vec{P}_i$ first and then find its magnitude with Pythagoras and not the other way around.
