Momentum Is a Vector

Momentum is p = mv. Mass is always positive, but velocity points somewhere, so momentum inherits that direction. That single fact changes how you combine momenta. Two objects moving toward each other don't have momenta that add up to something bigger — their momenta can partly or fully cancel, because one points one way and the other points the opposite way.

To add momentum vectors, pick a positive direction and stick with it. Anything moving that way gets a positive momentum value; anything moving the opposite way gets a negative one. Then you just add the signed numbers. A 4 kg cart at 3 m/s east has momentum +12 kg m/s. A 2 kg cart at 5 m/s west has momentum -10 kg m/s. Total: +2 kg m/s, meaning 2 kg m/s east. Treat it as scalar addition of magnitudes and you'd wrongly get 22.

This vector nature is exactly why momentum conservation splits cleanly into x and y pieces. Forces and velocities along x never affect what's happening along y, so you can track east-west momentum and north-south momentum as two separate conservation stories running at the same time. That's the bridge into collision problems: when carts collide at an angle, you conserve p_x and p_y independently, each with its own positive direction and its own bookkeeping of signs.