COLLISION TIMES FOUR

Copyright Ó George M. Bonnett, JD 1997 All rights reserved.

1. Vehicle 1 strikes an identical vehicle (Vehicle 2) head-on (collinear). The overlap is 100%. Both vehicles are travelling at exactly 30 miles per hour, for a closing velocity of 60 miles per hour.
2. Vehicle 3, travelling at 60 miles per hour, strikes Vehicle 4, which is stopped. The closing velocity is 60 miles per hour. The vehicles impact front to front. The overlap is 100%. Both vehicles are identical, and are identical to Vehicles 1 and 2.
3. Vehicle 5, travelling at 45 miles per hour, strikes Vehicle 6, which is travelling 15 miles per hour in the opposite direction. The closing velocity is 60 miles per hour. The vehicles impact front to front. The overlap is 100%. Both vehicles are identical, and are identical to Vehicles 1 and 2.
4. Vehicle 7, identical to all of the other vehicles, strikes an immovable, non-deforming, barrier at 30 miles per hour. The closing velocity is 30 miles per hour.

How are these collisions similar? How do they differ? How does the damage to the vehicles compare in the collisions? Are any the collisions basically identical? Should any of the post-collision vehicles have similar or identical damage? If the collisions produce different results, what is the difference and what causes it?

If you feel uneasy about the answers you are probably in the majority. Even those reconstructionists with years of experience may be unsure of the answers. Many may come up with answers that are incorrect.

Let's examine each accident in detail and find out what they have in common and what they don't. First, a few simplifying assumptions:

• There is no restitution;
• There is no grade;
• The vehicles are absolutely identical before impact;
• The weight of each vehicle is 1 unit.

Collision 1

• Both vehicles are travelling at the same speed.
• Both vehicles have a weight of 1 unit.
• Both vehicles have the same kinetic energy at impact, 30.0621 units based on the formula Kinetic Energy = .5 * Mass * Velocity^2.
• Total collision kinetic energy at impact is 60.1242 units.

Collision 2

• Vehicle 3 has 120.2484 units of kinetic energy at impact.
• Vehicle 4 has no kinetic energy at impact.
• Total collision kinetic energy at impact is 120.2484 units.

Collision 3

• Vehicle 5 has 67.6397 units of kinetic energy at impact.
• Vehicle 6 has 7.5155 units of kinetic energy at impact.
• Total collision kinetic energy at impact is 75.1552 units.

Collision 4

• Vehicle 7 has 30.0621 units of kinetic energy at impact.
• Total collision kinetic energy at impact is 30.0621 units.

In Collision 1, the kinetic energy on both sides of the equation is identical. The net kinetic energy after the collision is zero, therefore all of the energy must be absorbed in the form of heat, sound, light, and damage to the vehicles. Since the vehicles have the same resistance to deformation, the damage to each vehicle must be the same.

In Collision 2, all of the kinetic energy is on one side of the equation. The post-collision energy results in vehicular motion. This motion is in the same direction (determined through momentum) as that of Vehicle 3 prior to the collision. This motion is dependent upon the inertia (resistance to change of velocity) of Vehicle 4. Energy not converted into motion, light, heat, or sound, must be absorbed in the form of damage by the vehicles. Any damage must be absorbed equally by the vehicles since they both have an identical resistance to deformation.

In Collision 3, there is an imbalance of energy on the opposite sides of the equation. The post-collision energy results in vehicular motion. The kinetic energy of Vehicle 5 is greater than the kinetic energy of Vehicle 6, and results in motion. Momentum dictates that the direction will be in the same direction as that of Vehicle 5 prior to impact. Energy not converted into motion, light, heat, or sound, must be absorbed in the form of damage by the vehicles. Any damage must be absorbed equally by the vehicles since they both have an identical resistance to deformation.

In Collision 4, all of the energy is on one side of the equation. By definition, the barrier is immovable and non-deforming. All of the energy that is not given off in the form of light, heat, or sound, must be absorbed by Vehicle 7. The energy absorbed by Vehicle 7 is the same amount of energy absorbed by Vehicle 1 and by Vehicle 2. Since the resistance to deformation of all three vehicles is identical, the damage must be identical.

While all of the proceeding arguments used the Law of Conservation of Energy as their foundation, almost identical arguments can be made using the Law of Conservation of Momentum. The difference between the approaches is that the Law of Conservation of Momentum does not involve damage. Energy can result in damage or motion. Momentum can only result in a change in the velocity vector, if the mass is not altered. The velocity vector has both scalar velocity (speed) and direction. A change in the velocity vector (delta V) can be a change in the scalar velocity (speed), or a change of direction. Since all of the above collisions are collinear, inline collisions, there is no force that could cause a change of direction except the opposite direction. Therefore, the delta V resulting from the four collisions can result in a change of direction or speed or both.

Do the facts given for Collisions 2 and 3 contain sufficient information to determine the post impact speeds of the vehicles? Will the vehicles separate after impact?

The collisions are collinear, with no restitution. There is no force to separate the vehicles in these collisions, and therefore they must have the same velocity. Because the total momentum before impact must equal the total momentum after impact, and the velocities and masses of the vehicles are all identical, the post impact velocities for Collisions 2 and 3 can be computed. The post impact velocity, based on momentum, must be 30 miles per hour for Collision 2, and 15 miles per hour for Collision 3.

If the post impact velocity for each vehicle in Collision 2 is 30 miles per hour and the weights are the same, doesn’t this mean that all of the energy went into motion? The speed of each vehicle is 30 miles per hour, and the mass is the same for each vehicle. It would seem that the energy is the same before and after collision and therefore no energy remains that will convert to damage. We all know that a vehicle travelling at 60 miles per hour that strikes a vehicle that is stopped, should result in at least some damage. In fact we might guess that the damage would be extensive.

Is something wrong with the Law of Conservation of Energy, or the Law of Conservation of Momentum? Maybe, just maybe, there is something wrong with our math?

The post-impact velocity of each vehicle in Collision 2 is 30 miles per hour. Using the formula for kinetic energy, the energy for each post-impact vehicle is 30.0621 units. The total kinetic energy is 60.1242 units for both vehicles. The total kinetic energy at impact was computed to be 120.2484 for the collision. This is a difference of 60.1242 units, or 30.0621 units of energy that is absorbed in the form of damage by each vehicle (Vehicle 3 and Vehicle 4). This is exactly the same as the damage absorbed by Vehicles 1 and 2 in Collision 1, and by Vehicle 7 in Collision 4.

In Collision 3, the total post-impact energy is 15.031 units. The total kinetic energy at impact was computed to be 75.1552 units. Subtracting 15.031 from 75.1552 leaves 60.1242 units to be absorbed equally by both vehicles (Vehicle 5 and Vehicle 6). Each vehicle must absorb 30.0621 units of energy.

Vehicles 1, 2, 3, 4, 5, 6, and 7 all absorb the identical amount of energy as damage. If the energy is identical, and the vehicles are identical, and the resistance to damage of each vehicle is identical, the damage must be identical for each vehicle in each of the collisions.

The "trick" is that in the energy computations, the velocity, in feet per second, is squared. It is the squaring of this term that accounts for what appears to be a contradiction.

While the post impact motion of the vehicles may be different, the damage is identical. Understanding how to properly use of the Law of Conservation of Energy, and the Law of the Conservation of Momentum, along with the addition and subtraction of vectors can be the greatest weapons of the reconstructionist. They are some of most basic tools of our profession.

Copyright Ó George M. Bonnett, JD 1997 All rights reserved.