Not only can momentum be preserved; it will be preserved, no matter what. For more information, I suggest the Wikipedia article on "Conservation of momentum".
because when it is bouncing, has more momentum than when it doesn't
More or less. Actually, a moving object has momentum - defined as mass times velocity. The word "impulse" is used for transfer of momentum, for example, in a collision. It has the same units as momentum, but the use of the word "impulse" seems inappropriate in this context.
Newton's Laws of motion. Specifically, his Second (F = ma, or more generally, F = dp/dt) and Third (F1 = -F2) Laws.
While energy is ALWAYS conserved, this isn't always useful for calculations, since MECHANICAL ENERGY - the energy that can be easily calculated - is NOT always conserved. On the other hand, momentum is always conserved, whether a collision is elastic or inelastic. (In an elastic collision, energy is also conserved.) Thus, conservation of momentum is often more useful for calculations involving collisions.
That means that there is a quantity, called momentum, that is conserved. Momentum is defined as the product of mass times velocity. If you add (product x velocity) for several items, or particles, before and after a collission for example, the total you get will not change. Note that, since velocity is a vector quantity, so is momentum, so if your items move in more than one dimension, you have to use vector addition.
In this context "conserved" means the total kinetic energy of all the objects is the same after the collision as before the collision. Note, the TOTAL is the same but the individual kinetic energies of each object may be different before and after. When two or more objects are about to collide they have a certain total kinetic energy. It is common that during the collision some of the kinetic energy is transformed into heat. So after the collision the total kinetic energy is less then before the collision. This is a non-elastic collision. There are some collisions, however, in which none of the kinetic energy is changed to heat. These are called ELASTIC collisions. So the total kinetic energy doesn't change, or is "conserved". There is another possible non-elastic collision. If during the collision there is an explosion, then its possible for the objects to have a larger total kinetic energy after the collision as they aquire some of the explosive energy. Finally note, that in all collisions the TOTAL vector momentum is the same just before and just after the collision. So in a collision momentum is always conserved.
If you're suggesting something like an auto accident, the energy of the collision is used to deform materials in the structural elements of the vehicle(s). It also heats them. The primary design features of cars includes a lot of thought to where the energy of a collision can go. Bumpers collapse, body panels and their strengthening members fold and become compressed, and a top or roof can collapse down. All this sinks ("sucks up") energy. And if it all works in an optimal way, you can climb out and walk away.
In elastic collisions, momentum is a completely conserved quantity, meaning that the total momentum of the system before the collision should equal to the total momentum of the system after the collision. In this case, the p initial was equal to 0, that means p final should have also been 0, the only way that could be achieved is if the momentum of both carts had the same magnitude but in the opposite direction. p = m*v so if p is the same, the cart with the heavier mass would necessarily have a slower speed than the light cart.
In a closed system, the TOTAL initial momentum before an "event" is the same as the TOTAL final momentum (at the end).
In an elastic collision, all initial kinetic energy is fully restored as final kinetic energy. where nothing is converted into noise, heat or any other form of energy. In an inelastic collision, kinetic energy is "lost" to thermal or sound energy.
A high speed collision between two cars would cause more damage than a low speed collision between the same two cars because they have more kinetic energy as their velocity increases. The greater the kinetic energy upon impact, the greater the resultant damage.