## One Dimensional Inelastic Collisions

In this type of collisions only the linear momentum is conserved and not so the mechanical energy. In general, assuming that two objects are moving toward a collision with each other with the linear momentum and before the collision, the two object will stick together during the collision and, after, they will move as a single object with the final linear momentum given by , .

 In the above animation, m1 correspond to the loaded train wagon (left wagon) and m2 is the wagon with only the passenger inside (right wagon). The magnitude of the velocity before the collision for the right wagon is four time the magnitude of the velocity of both wagons moving together after the collision. At the same time, the magnitude of the velocity of the left wagon is twice the magnitude of the velocity of the two wagons moving together after the collision. With the previous information, it is possible to calculate the relation between the mass of the left wagon and the mass of the right wagon (see below).

Before the collision, the total linear momentum of the system constituted of the two cars is . An equation can be set using the conservation of linear momentum for all kinds of collisions.

In this case, the conservation of the linear momentum is . From this equation the following relation is established:

The presiding equation can be solved for one of the quantities. In most cases, the unknown is one of the velocities.

In summary, the following steps were used in order to derive the previous equation:

1. Calculation of the individual linear momentum for all the parts of the system, and .

2. Calculation of the total linear momentum of the system. This result is obtained by developing the vector addition of the linear momentum of the parts that form the system. This total linear momentum corresponds to the initial linear momentum of the system, .

3. Understanding that the linear momentum is conserved in this situation because it is a collision, .

4. State the final linear momentum of the system in terms of the masses and velocity. Inelastic collision between two object results in a single object after the collision. Thus, there is only one velocity after the collision and one object made up of the entire mass of the system, .

5. Finally, establish the equation that combines the conservation of the linear momentum with the partial results of points 2 and 4, .

Accordingly with the conditions established above, in the case of the left train wagons, the velocity is  ; where the negative sign accounts for the fact that, before the collision, the right wagon moves toward the left (negative) while after the collision both wagons move toward the right (positive). Also, based on the description above, the left wagon velocity is . Substituting these relation on the conservation of linear momentum relation which was presented in step 5 of the previous enumeration,

Thus, the mass of the wagon originally moving from the left toward the right is five times the mass of the wagon in the beginning moving from the right to the left.

Given the different values of the velocity, the kinetic energy before the collision can be compared to the linear momentum after the collision.

 Two or More Dimensions Inelastic Collisions

In this case, the vector nature of the linear momentum is fundamental in order to understand the results of two dimensional (or more dimensions) collisions. The linear momentum conservation mathematical equation takes the following form,

where the vector nature is explicitly shown. In addition, it can be said that

Each component of the linear momentum is conserved independently of the others.

 Example

The following example shows how to operate exercises where the linear momentum is conserved in more than one dimension.

 1) Question N a) b) c) d) e) None of the above.

Try again!

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com
 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com