# 9: Linear Momentum and Collisions

In this section, we develop and define another conserved quantity, called *linear momentum*, and another relationship (the *impulse-momentum theorem*), which will put an additional constraint on how a system evolves in time. Conservation of momentum is useful for understanding collisions, such as that shown in the above image. It is just as powerful, just as important, and just as useful as conservation of energy and the work-energy theorem.

- 9.0: Prelude to Linear Momentum and Collisions
- The concepts of work, energy, and the work-energy theorem are valuable for two primary reasons: First, they are powerful computational tools, making it much easier to analyze complex physical systems than is possible using Newton’s laws directly (for example, systems with nonconstant forces); and second, the observation that the total energy of a closed system is conserved means that the system can only evolve in ways that are consistent with energy conservation.

- 9.1: Linear Momentum
- Momentum is a concept that describes how the motion of an object depends not only on its mass, but also its velocity. Momentum is a vector quantity that depends equally on an object's mass and velocity. The SI unit for momentum is kg • m/s.

- 9.2: Impulse and Collisions (Part 1)
- When a force is applied on an object for some amount of time, the object experiences an impulse. This impulse is equal to the object’s change of momentum. Newton’s second law in terms of momentum states that the net force applied to a system equals the rate of change of the momentum that the force causes.

- 9.2: Impulse and Collisions (Part 2)
- Since an impulse is a force acting for some amount of time, it causes an object’s motion to change.

- 9.3: Conservation of Linear Momentum (Part 1)
- The law of conservation of momentum says that the momentum of a closed system is constant in time (conserved). A closed (or isolated) system is defined to be one for which the mass remains constant, and the net external force is zero. The total momentum of a system is conserved only when the system is closed.

- 9.4: Types of Collisions
- An elastic collision is one that conserves kinetic energy. An inelastic collision does not conserve kinetic energy. Momentum is conserved regardless of whether or not kinetic energy is conserved. Analysis of kinetic energy changes and conservation of momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one-dimensional, two-body collisions.

- 9.5: Collisions in Multiple Dimensions
- The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Momentum is conserved in both directions simultaneously and independently. The Pythagorean theorem gives the magnitude of the momentum vector using the x- and y-components, calculated using conservation of momentum in each direction.

- 9.6: Center of Mass (Part 1)
- An extended object (made up of many objects) has a defined position vector called the center of mass. The center of mass can be thought of, loosely, as the average location of the total mass of the object. The center of mass of an object traces out the trajectory dictated by Newton’s second law, due to the net external force. The internal forces within an extended object cannot alter the momentum of the extended object as a whole.

- 9.7: Rocket Propulsion
- A rocket is an example of conservation of momentum where the mass of the system is not constant, since the rocket ejects fuel to provide thrust. The rocket equation gives us the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass.

*Thumbnail: The concepts of impulse, momentum, and center of mass are crucial for a major-league baseball player to successfully get a hit. If he misjudges these quantities, he might break his bat instead. (credit: modification of work by “Cathy T”/Flickr).*

### Contributors

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).