# Linear Momentum and Collisions

## Collisions of Point Masses in Two Dimensions


### Learning Objectives

By the end of this section, you will be able to:
1. Discuss two dimensional collisions as an extension of one dimensional analysis.
2. Define point masses.
3. Derive an expression for conservation of momentum along x-axis and y-axis.
4. Describe elastic collisions of two objects with equal mass.
5. Determine the magnitude and direction of the final velocity given initial velocity, and scattering angle.
In the previous two sections, we considered only one-dimensional collisions; during such 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 we shall see that their study is an extension of the one-dimensional analysis already presented. The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along perpendicular axes. Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously.
One complication arising in 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 by one another, 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


, so that momentum 

 is conserved. The simplest collision is one in which one of the particles is initially at rest. (See .) 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


- and -axes 

will also be conserved, but with the chosen coordinate system,




 is initially zero and





is the momentum of the incoming particle. Both facts simplify the analysis. (Even with the simplifying assumptions of point masses, one particle initially at rest, and a convenient coordinate system, we still gain new insights into nature from the analysis of two-dimensional collisions.)
Along the -axis, the equation for conservation of momentum is 
Where the subscripts denote the particles and axes and the primes denote the situation after the collision. 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 -axis have the form . Because particle 1 initially moves along the -axis, we find . 
Conservation of momentum along the -axis gives the following equation:
where  and  are as shown in .
Along the -axis, the equation for conservation of momentum is
or
But  is zero, because particle 1 initially moves along the -axis. Because particle 2 is initially at rest,  is also zero. The equation for conservation of momentum along the -axis becomes
The components of the velocities along the -axis have the form . 
Thus, conservation of momentum along the -axis gives the following equation:
The equations of conservation of momentum along the -axis and -axis are very useful in analyzing two-dimensional collisions of particles, where one is originally stationary (a common laboratory situation). But two equations can only be used to find two unknowns, and so other data may be necessary when collision experiments are used to explore nature at the subatomic level.

### Elastic Collisions of Two Objects with Equal Mass

Some interesting situations arise when the two colliding objects have equal mass and the collision is elastic. This situation is nearly the case with colliding billiard balls, and precisely the case with some subatomic particle collisions. We can thus get a mental image of a collision of subatomic particles by thinking about billiards (or pool). (Refer to  for masses and angles.) First, an elastic collision conserves internal kinetic energy. Again, let us assume object 2  is initially at rest. Then, the internal kinetic energy before and after the collision of two objects that have equal masses is
Because the masses are equal, . Algebraic manipulation (left to the reader) of conservation of momentum in the - and -directions can show that
(Remember that  is negative here.) The two preceding equations can both be true only if
There are three ways that this term can be zero. They are
1. : head-on collision; incoming ball stops
2. : no collision; incoming ball continues unaffected
3. : angle of separation  is  after the collision 
All three of these ways are familiar occurrences in billiards and pool, although most of us try to avoid the second. If you play enough pool, you will notice that the angle between the balls is very close to  after the collision, although it will vary from this value if a great deal of spin is placed on the ball. (Large spin carries in extra energy and a quantity called angular momentum, which must also be conserved.) The assumption that the scattering of billiard balls is elastic is reasonable based on the correctness of the three results it produces. This assumption also implies that, to a good approximation, momentum is conserved for the two-ball system in billiards and pool. The problems below explore these and other characteristics of two-dimensional collisions.

### Test Prep for AP Courses


### Section Summary

1. 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 -axis parallel to the velocity of the incoming particle.
2. 
Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the -axis), stated by
 and along the direction perpendicular to the initial direction (the
-axis) stated by
.

3. The internal kinetic before and after the collision of two objects that have equal masses is 
4. Point masses are structureless particles that cannot spin.

### Conceptual Questions


### Problems & Exercises
