# Momentum

## Elastic and Inelastic Collisions


### Section Key Terms


### Elastic and Inelastic Collisions

When objects collide, they can either stick together or bounce off one another, remaining separate. In this section, we’ll cover these two different types of collisions, first in one dimension and then in two dimensions. 
In an elastic collision, the objects separate after impact and don’t lose any of their kinetic energy. Kinetic energy is the energy of motion and is covered in detail elsewhere. The law of conservation of momentum is very useful here, and it can be used whenever the net external force on a system is zero.  shows an elastic collision where momentum is conserved.
An animation of an elastic collision between balls can be seen by watching this video. It replicates the elastic collisions between balls of varying masses.
Perfectly elastic collisions can happen only with subatomic particles. Everyday observable examples of perfectly elastic collisions don’t exist—some kinetic energy is always lost, as it is converted into heat transfer due to friction. However, collisions between everyday objects are almost perfectly elastic when they occur with objects and surfaces that are nearly frictionless, such as with two steel blocks on ice.
Now, to solve problems involving one-dimensional elastic collisions between two objects, we can use the equation for conservation of momentum. First, the equation for conservation of momentum for two objects in a one-dimensional collision is
Substituting the definition of momentum p = mv for each initial and final momentum, we get
where the primes (') indicate values after the collision; In some texts, you may see i for initial (before collision) and f for final (after collision). The equation assumes that the mass of each object does not change during the collision.
Now, let us turn to the second type of collision. An inelastic collision is one in which kinetic energy is not conserved. A perfectly inelastic collision (also sometimes called completely or maximally inelastic) is one in which objects stick together after impact, and the maximum amount of kinetic energy is lost. This lack of conservation means that the forces between colliding objects may convert kinetic energy to other forms of energy, such as potential energy or thermal energy. The concepts of energy are discussed more thoroughly elsewhere. For inelastic collisions, kinetic energy may be lost in the form of heat.  shows an example of an inelastic collision. Two objects that have equal masses head toward each other at equal speeds and then stick together. The two objects come to rest after sticking together, conserving momentum but not kinetic energy after they collide. Some of the energy of motion gets converted to thermal energy, or heat.
Since the two objects stick together after colliding, they move together at the same speed. This lets us simplify the conservation of momentum equation from
to
for inelastic collisions, where v′ is the final velocity for both objects as they are stuck together, either in motion or at rest.

### Solving Collision Problems

The Khan Academy videos referenced in this section show examples of elastic and inelastic collisions in one dimension. In one-dimensional collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and just as we did with two-dimensional forces, we will solve these problems by first choosing a coordinate system and separating the motion into its x and y components.
One complication with two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass each other, they will spin in circles. We will not consider such rotation until later, and so for now, we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point masses—that is, structureless particles that cannot rotate or spin.
We start by assuming that Fnet = 0, so that momentum p is conserved. The simplest collision is one in which one of the particles is initially at rest. The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in . Because momentum is conserved, the components of momentum along the x- and y-axes, displayed as pand p, will also be conserved. With the chosen coordinate system, pis initially zero and pis the momentum of the incoming particle.
Now, we will take the conservation of momentum equation, p1 + p2 = p′1 + p′2  and break it into its x and y components.
Along the x-axis, the equation for conservation of momentum is
In terms of masses and velocities, this equation is
But because particle 2 is initially at rest, this equation becomes
The components of the velocities along the x-axis have the form v cos θ . Because particle 1 initially moves along the x-axis, we find v1= v1. Conservation of momentum along the x-axis gives the equation
where 
 and 
 are as shown in .
Along the y-axis, the equation for conservation of momentum is
or
But v1y is zero, because particle 1 initially moves along the x-axis. Because particle 2 is initially at rest, v2y is also zero. The equation for conservation of momentum along the y-axis becomes
The components of the velocities along the y-axis have the form v sin  
 . Therefore, conservation of momentum along the y-axis gives the following equation:

### Practice Problems


### Check Your Understanding


### Section Summary

1. If kinetic energy is conserved, the collision is elastic. If they stick together, the collision is perfectly inelastic.
2. Kinetic energy is conserved in an elastic collision, but not in an inelastic collision.
3. The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the x-axis parallel to the velocity of the incoming particle.
4. Two-dimensional collisions of point masses, where mass 2 is initially at rest, conserve momentum along the initial direction of mass 1, or the x-axis, and along the direction perpendicular to the initial direction, or the y-axis.
5. Point masses are structureless particles that cannot spin.

### Key Equations


### Concept Items


### Critical Thinking


### Problems


### Performance Task


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
