Conservation of Momentum
The law of conservation of momentum states that the total momentum of an isolated system with no external forces will be conserved. The momentum can be transferred from one object to another, but the total momentum can neither increase nor decrease.
Deciding whether momentum is conserved in a collision is easy. Momentum is conserved in all collisions. When doing a physics homework problem involving a collision, the total momentum is always the same before and after the collision. Always use the conservation of momentum equation.
Remember that momentum is a vector. In a two or three dimensional collision problem it is absolutely essential to add the momenta of the different objects according to the rules of vector addition. Divide all momenta in the problem into x and y components (and z for a three dimensional problem). Leaving out this step will virtually guarantee a wrong answer.
Conservation of Kinetic Energy
Energy is one of the fundamental quantities that is always conserved. The total amount of energy in an isolated system can neither increase nor decrease. Energy can however change form. That means that the total amount of kinetic energy in a system can change. Kinetic energy can decrease if it is converted to some other form of energy. If another form of energy is converted to kinetic energy, the total kinetic energy of a system can increase.
Working with kinetic energy equations can in some ways be easier than with momentum equations, but it can also in some ways be more difficult. Energy is a scalar rather than a vector quantity, so there is no need to divide energy into components. However velocity is squared in the kinetic energy formula, so solving kinetic energy equations often requires solving a quadratic equation.
In some collisions the initial kinetic energy can change form. For example if the collision produces a noise, kinetic energy transformed into sound energy. If the collision deforms the objects, some of the kinetic energy goes into deformation. Hence Kinetic energy may not be conserved in a collision.
Types of Collisions
Kinetic energy is conserved in some but not all collisions. Whether the kinetic energy is conserved depends on the type of collision. Physicists classify four types of collisions.
Elastic collisions: Kinetic energy is conserved in elastic, which are also called completely elastic, collisions. To solve these problems, use both momentum and kinetic energy conservation.
Inelastic collisions: Kinetic energy is not conserved in inelastic collisions. To solve these problems use momentum conservation but not kinetic energy conservation.
Completely inelastic collisions: In completely inelastic collisions, the objects stick together after the collision. That means they have the same velocity after the collision. To solve these problems, use momentum conservation and use the same velocity after the collision for the objects. Do not use kinetic energy conservation.
Explosive collisions: In explosive collisions kinetic energy increases. The extra kinetic energy usually comes from stored chemical potential energy. To solve these problems. use momentum conservation only.
Conservation of momentum applies to all collision homework problems. Understanding the different types of collisions helps students know when to use conservation of kinetic energy to solve physics collision problems.
Further Reading
Wilson, J.D., Buffa, A.J., and Lou, B., College Physics 6th ed., Pearson, 2007.