LINEAR MOMENTUM COMPUTATIONS

SECONDARY SLAP - TWO COLLISIONS OR ONE?

One of the nightmares of anyone investigating vehicular collisions is the case in which one of the vehicles rotates after the collision and there is secondary contact between the two vehicles. The term generally applied to this contact is "secondary slap."

This secondary slap is generally relatively close to the separation of the two vehicles after their initial contact in both distance and time. It usually involves much less force than the initial collision, but creates a much greater problem for the investigator than the initial collision. The problem is that, in the past, we have always had to treat them as separate collisions. After all, we have four different departure angles from two different collisions, and that creates havoc in the linear momentum computations.

In dealing with the four different departure angles, we must also deal with the fact that we have four different departure speeds. For this reason, the safest way to handle the problem was to make two separate problems out of this type of collision.

At this point, it is necessary to say a word about vectors and vector addition. A momentum vector is the result of multiplying the velocity of an object by the mass (frequently weight is substituted for mass) of the object. This vector has both quantity (momentum) and direction. A resultant vector is the result of adding two (or more) vectors together. A resultant momentum vector is the result of adding two momentum vectors together. Vectors can be added together graphically by placing the tail of the second vector on the head of the first vector and drawing a line from the tail of the first to the head of the second.

In a linear momentum computation, we use the Law of Conservation of Linear Momentum. This law states that the total momentum into collision must be equal to the total momentum out of the collision (P_R₌P_R'). This does not mean that the speed, direction, and momentum for each of the vehicles will remain the same. Far from it, they have probably changed significantly. It means that the resultant momentum vector before collision (P_R) is the same as the resultant momentum vector after the collision (P_R').

First, we must handle the secondary slap as a linear momentum problem in order to obtain the necessary information to deal with the primary collision. This involves a determination of the departure angles out of the secondary slap and a determination of the separation speeds from the secondary slap. We can use this information along with the approach angles of the centers of mass of the vehicles into the secondary slap in order to determine the speeds of the two vehicles at the impact of the secondary slap.

The impact speeds of the centers of mass at the secondary slap should be very close to the post impact speeds of the centers of mass out of the primary collision, certainly within the margins of error mandated by the uncertainty of the measurements. Due to the close proximity in both distance and time of the primary collision to the secondary slap, the effects of the coefficient of friction between the vehicles and the surface should be minimal.

Now we are able to tackle the linear momentum problem involved in the primary collision. The post impact speeds are known, as they are the pre-impact speeds determined in the problem involving the secondary slap. The departure angles of the centers of mass of the two vehicles out of the primary collision will usually be very close to the approach angles of the two vehicles into the secondary slap.

This leads to an apparent contradiction. How can the departure angles out of the primary collision and the approach angles into a secondary slap be the same? This is possible because of the same attribute that causes the secondary slap. It is the shape of each of the two vehicles and the changing orientation of these individual shapes about the center of mass of each vehicle that makes this apparent contradiction possible. The centers of mass are departing a collision based on the orientations of the shapes of the vehicles and at the same time approaching a second collision based on the orientation of the vehicles for the secondary collision.

With known departure angles and speeds for the primary collision, the only unknowns are the approach angles for the primary collision. Once these have been measured, the impact speeds for the primary collision can be computed.

The method described above will certainly give us impact speeds, and if all of our measurements have been accurate, should give us accurate speeds, at least within the margin of error resulting from the uncertainty of our measurements. However, is all of this necessary?

Let us examine the above collision using a slightly different approach. The law of Conservation of Linear Momentum will still be our primary tool in this re-examination of the collision, and in fact, it is this very law that must be invoked in order to re-examine the collision.

In the secondary slap, which is truly a separate collision, what really happened was an exchange of momentum between the two vehicles. In fact, this is what happens in all collisions. The force that results in the transfer of momentum from one vehicle to the other is called impulse. This impulse is equal in magnitude and opposite in direction for each vehicle. It is an equal exchange between the two vehicles and results in their individual momentum vectors being altered. Because the impulse is equal, the change in the momentum vectors must also be equal. Consequently, there is no change in the resultant vector. This, remember, is the Law of Conservation of Linear Momentum - the momentum before impact and the momentum after impact must be the same (P_R₌P_R').

What effect does this have on our approach to solving the problem? It has the effect of allowing us to use the post impact information, the angles and speeds, from the secondary slap as the post impact information for the primary collision. If the post impact resultant vector of the secondary slap is identical to the pre-impact resultant vector for the secondary slap, and the post impact resultant vector from the primary collision is the pre-impact resultant vector for the secondary slap, then the post impact resultant vector from the primary collision must be identical to the post impact resultant vector for the secondary slap, and therefore, interchangeable.

Law of Conservation of Momentum:

P1 + P2 = P1' + P2'

P11 + P22 = P11' + P22'

P1' = P11

and

P2' = P22

then

P1' + P2' = P11' + P22'

THEREFORE:

P1 + P2 = P11' + P22'

What does not follow is that the post-impact momentum vector for vehicle 1 out of the primary collision (P1') is the same as the post-impact momentum vector of vehicle 1 out of the secondary slap (P11'). Nor would it be correct to say that the post-impact momentum vector for vehicle 2 out of the primary collision (P2') is the same as the post-impact momentum vector of vehicle 2 out of the secondary slap (P22'). Indeed, because the departure angles of the vehicles out of the primary collision will probably be different from the departure angles of the vehicles out of the secondary slap, the departure speeds of the vehicles out of the collisions should likewise be different.

It would be correct therefore to say that in most circumstances:

P1' <> P11'

and

P2' <> P22'

where <> indicates INEQUALITY.

Therefore, it would be improper as well as inaccurate to use the departure angle of one of the vehicles out of the primary collision in combination with the speed of this vehicle out of the secondary slap.

In other words, we may correctly use the post impact data out of the primary collision or the post impact data out of the secondary slap in order to determine the impact speeds of the vehicles at impact in the primary collision. We cannot interchange the individual components of the post impact data between the primary and secondary collisions.

This is a case of being able to act as if we are ordering from a menu. We are allowed take two vectors from either column A (P1' and P2') or column B (P11' and P22'), but we do not have the option of taking any two of the four vectors to inject into the equation in order to reach a solution. We are limited to picking two vectors from the same collision, but it may be from either the primary collision or the secondary collision.

By studying the underlying fundamental principles involved, we can see that the solution is not necessarily as complicated as it first appears. While the rules involving the application of the Law of Conservation of Momentum must be strictly adhered to, they may also be used to eliminate computations that merely become redundant.