9: Linear Momentum and Collisions

*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)*

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 non-constant 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. In other words, a system cannot evolve randomly; it can only change in ways that conserve energy.

In this chapter, 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.

So far, we’ve shown that a force causes a change in velocity. In most examples up until now, this Δv happened over a relatively long period of time and resulted in a small, constant acceleration. Now imagine throwing a ball against the wall. What forces act on the ball while it is in contact with the wall? How long does the wall exert a force on the ball? This is an example of a collision. A collision refers to two objects hitting one another, interacting with (probably very large) forces for some (probably very short) amount of time, and then continuing along (probably in radically altered paths, and maybe pretty squooshed!). Some examples can be a baseball bat striking a ball, a superball bouncing off the floor (the two objects are earth and ball there!), two cars running into one another head on. There are different kinds of collisions, as we will see, and at first glance they all seem very hard for us to make sense of – forces that are HUGE and very short lived – we probably can’t assume constant acceleration, so how can we make predictions about what happens before and after? The answer comes from a big idea: conservation of momentum. It turns out that we can often make EXACT predictions about what’s going to happen, even if we know almost NOTHING about the details of the crash, or the materials involved!

#### 9.1 Linear Momentum

- Explain what momentum is, physically
- Calculate the momentum of a moving object

#### 9.2 Impulse and Collisions

- Explain what an impulse is, physically
- Describe what an impulse does
- Relate impulses to collisions
- Apply the impulse-momentum theorem to solve problems

#### 9.3 Conservation of Linear Momentum

- Explain the meaning of “conservation of momentum”
- Correctly identify if a system is, or is not, closed
- Define a system whose momentum is conserved
- Mathematically express conservation of momentum for a given system
- Calculate an unknown quantity using conservation of momentum

#### 9.4 Types of Collisions

- Identify the type of collision
- Correctly label a collision as elastic or inelastic
- Use kinetic energy along with momentum and impulse to analyze a collision

#### 9.5 Collisions in Multiple Dimensions

- Express momentum as a two-dimensional vector
- Write equations for momentum conservation in component form
- Calculate momentum in two dimensions, as a vector quantity

#### 9.6 Center of Mass

- Explain the meaning and usefulness of the concept of center of mass
- Calculate the center of mass of a given system
- Apply the center of mass concept in two and three dimensions
- Calculate the velocity and acceleration of the center of mass