Question

Two bumper cars move toward each other. The cars hit and bounce off of each other. The momentum before the collision will be equal to the
after.

Answer 1

Yes — the total momentum of the two-car system is conserved (provided no net external impulse acts during the short collision). Vectorially,
m1 v1 + m2 v2 = m1 v1' + m2 v2',
where v1, v2 are the velocities before and v1', v2' after.

Note: "Conserved" refers to the total momentum of the system, not necessarily each car's momentum. Kinetic energy need not be conserved — some energy can convert to sound, deformation, heat, etc., unless the collision is perfectly elastic.