### Unit 8: Momentum and Collisions

We use the term momentum in various ways in everyday language. For example, we often speak of sports teams gaining and maintaining the momentum to win. Generally, momentum implies a tendency to continue on course (to move in the same direction) and is associated with mass and velocity. Momentum has its most important application when analyzing collision problems. Like energy, it is important because it is conserved. Only a few physical quantities are conserved in nature, and studying them yields fundamental insight into how nature works, as we shall see during our study of momentum.

**Completing this unit should take you approximately 4 hours.**

Upon successful completion of this unit, the student will be able to:

- define linear momentum;
- explain how momentum and force are related;
- use Newton's Second Law of Motion in terms of momentum;
- describe the relationship between impulse and momentum;
- define elastic, inelastic, and totally inelastic collisions;
- use conservation of momentum to solve linear collision problems;
- use conservation of momentum to solve situations involving rotation; and
- give several everyday examples of conservation of angular momentum.

### 8.1: Linear Momentum

We define linear momentum as the product of an object's mass and velocity. It can be written as , where is linear momentum, is mass, and is velocity.

Linear momentum is a vector quantity because velocity is a vector quantity, and the linear momentum will have the same direction as the velocity. The units for linear momentum are kgm/s.

This text goes into the concepts of linear momentum, impulse, and how force is used to change momentum over time.

This video discusses some of the concepts we will explore later in this unit.

This video gives another way of thinking about momentum.

### 8.2: Momentum and Newton's Second Law

In Unit 4 we learned to write Newton's Second Law of Motion as . While this is the most common way to write and use this law, it was not how Newton originally wrote it. Newton wrote this law in terms of momentum rather than force and acceleration:

This shows that the net force equals the change in momentum divided by the change in time. This equation certainly appears different from the familiar we used in Unit 4.

As you read, pay attention to how we can derive from the Second Law in terms of momentum in equations 8.9, 8.10, 8.11, and 8.12. Furthermore, we define impulse as change in momentum. Using Newton's Second Law of Motion, we can write this as .

See how to use Newton's Second Law in terms of momentum in Example 8.2. This problem calculates the force applied to a tennis ball: there is a change of velocity of the ball but no change in mass, so pay special attention to how change in momentum is calculated in equation 8.14.

When we calculate impulse, we assume the net force is constant during the time we are interested in. In reality, force is rarely constant. For example, in Example 8.2, we assumed the force on the tennis ball was constant over time. In reality, the force on the tennis ball probably changed from the beginning to the end of the swing of the tennis racquet. Nevertheless, the change in force was probably not significant, and we assume it is constant to make our calculations easier.

Read this text as it expands on linear momentum and Newton's Second Law to define a new quantity, impulse.

Also This video introduces the concept of impulse.

This video describes how to use the impulse-force equation to solve problems.

This video presents a graphical analysis of force and time and how they relate to impulse.

### 8.3: Elastic, Inelastic, and Totally Inelastic Collisions

When two or more objects interact physically, we say the objects collide or experience a collision. Here, we consider three types of collisions for solving physics problems. They are all based on the energy transfer in the collisions. By definition, an elastic collision is a collision where the internal kinetic energy is conserved in the interaction. So, in an elastic collision, all the kinetic energy remains kinetic energy. That is, no kinetic energy is converted to heat, friction, or other types of energy.

As you read, pay attention to the diagram of two metal boxes interacting in an elastic collision on an ice surface in Figure 8.6.

Watch this video, which accompanies what you just read.

Unlike an elastic collision, an inelastic collision is a collision where the internal kinetic energy is not conserved. In inelastic collisions, some kinetic energy of the colliding objects is lost to friction, heat, or even work. Inelastic collisions are what we mostly observe in the real world. Watch this video for an overview of inelastic and elastic collisions.

Read this text. As we learned in the previous video, in reality, no collisions are perfectly elastic because some kinetic energy is always "lost" by being converted to other forms of energy. Another example of an elastic collision is if two balls collide on a smooth icy surface. Because the ice has almost no friction, little kinetic energy would be lost to friction.

See an example of two blocks experiencing a totally inelastic collision in Figure 8.8. See a good example of an inelastic collision in Figure 8.9. In this example, a hockey goalie stops a puck in the net. Although the ice surface is essentially frictionless, some kinetic energy of the puck is converted to heat and sound as the goalie stops it. A totally inelastic collision (also called a perfectly inelastic collision) is an inelastic collision where the objects "stick together" upon colliding.

Watch this video, which accompanies what you just read.

This video compares elastic and inelastic collisions.

This video looks more closely at the case for perfectly inelastic collisions. What makes something perfectly inelastic is that the objects stick together after the collision. This means they have the same final velocity.

Watch this video for another explanation of elastic and inelastic collisions in one dimension.

### 8.4: Solving Problems Involving Conservation of Linear Momentum in Collisions

When solving problems for elastic collisions, it is important to remember that the kinetic energy is conserved. Therefore, the total kinetic energy at the start of the collision must equal the total kinetic energy at the end of the collision. We can write this as

Moreover, we know that momentum must be conserved in the collision. Therefore, the total momentum at the start of the collision must equal the total momentum at the end of the collision. That is, for two objects (object one and two) colliding, we can write . Using conservation of momentum, we can usually set up these problems so we only have to solve for one unknown.

As you read, pay attention to Example 8.4. In this example, one of the objects is initially at rest (its velocity equals zero), so it does not have an initial momentum. This lets us simplify the conservation of energy momentum equations. Then, by using the equations for conservation of energy and momentum, we can solve for final velocity after collision.

Read this text, which shows that we can also solve for final velocities after inelastic collisions. In these problems, it is important to remember that kinetic energy is not conserved but momentum is conserved.

In Example 8.5(a), the conservation of momentum equation is used to determine the final velocity of the object (the hockey goalie) in an inelastic collision. In inelastic collisions, some kinetic energy is converted to other forms of energy. The energy difference before and after collision can be calculated to determine how much kinetic energy was lost.

In Example 8.5(b), the amount of energy lost is calculated. The total kinetic energy in the system is calculated before and after collision based on the mass and velocities of the objects. The difference in kinetic energy shows how much kinetic energy was converted to other forms of energy during the collision. Example 8.6 is similar.

This video gives an overall overview of linear momentum, impulse, and the conservation of momentum law.

This video gives an example of using the conservation of momentum law to solve collision problems (problems that deal with two objects combining).

This video gives an example of using the conservation of momentum to solve explosion problems (problems that deal with two objects separating from each other).

These videos expand on the conservation of momentum law for two-dimensional collision problems. Note that the conservation of momentum simply needs to be applied twice: once for each of the two dimensions of motion between objects.

### 8.5: Conservation of Angular Momentum

We define angular momentum as . It is similar to the momentum defined for linear motion. As such, angular momentum in a system is conserved in the same way that linear momentum is conserved. Therefore, we can say that , where is the initial angular momentum in a system and is the final angular momentum in the system. We can also write this as:

We see conservation of angular momentum in many everyday examples. As you read, pay attention to the example of the spinning figure skater in Figure 10.23. In the first picture, the figure skater is spinning with her arms out on a frictionless ice surface. In the second picture, she pulls her arms in, and her rotational velocity increases. When the figure skater pulls in her arms, she lowers her moment of inertia. Because angular momentum is conserved, because her moment of inertia decreases, her angular velocity must therefore increase.

Read this to understand angular momentum, how torque plays a role, and how angular momentum is conserved without torque.

This video gives a demonstration and a brief explanation behind the conservation of angular momentum.

This video goes into solving for the conservation of angular momentum for a particular situation.

### Unit 8 Assessment

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