one-dimensional collision
A one-dimensional collision is a head-on hit, where everything moves along a single straight line — before and after. Two train cars on a track, two carts on an air track, two balls rolling toward each other down the same groove: the action lives on one axis, so we can drop the arrows and just use plus and minus signs for direction.
On a line, momentum conservation is one tidy equation: m1 v1 + m2 v2 = m1 v1' + m2 v2', where the primes mark the velocities after the collision and a leftward velocity simply carries a minus sign. For an elastic 1D collision you add the kinetic-energy equation, and the two together give clean closed formulas for both final velocities. For a perfectly inelastic 1D collision the two objects share one final velocity, and momentum conservation alone solves it.
The one-dimensional case is where beginners should start, because the vector bookkeeping shrinks to careful sign-keeping. Master it and the two-dimensional collision is just the same rules applied to the x-part and the y-part separately.
Cart A (2 kg, +3 m/s) meets cart B (1 kg, -3 m/s) head-on and they stick. Momentum: 2(+3) + 1(-3) = +3 kg m/s = (2+1) v', so v' = +1 m/s — the joined carts drift slowly to the right, the direction A's larger momentum was pointing.
On a line, direction is just a sign; here the plus side wins by 3 kg m/s.
Keep signs consistent: pick one direction as positive and stick with it for every velocity, before and after. A velocity aimed the other way must carry a minus sign, or the momentum will not add up.