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Back Pain / Low Speed Rear-Impact Collisions - Physical Laws Related to Injuries Incurred
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The subject of low-speed rear-impact collisions, as related to injuries resulting from them, is an ongoing topic for debate. Particularly useful in this regard is an understanding of the underlying physical laws and engineering principles related to motion and force. These principles help explain how injury can be a byproduct of a rear-end impact, despite the fact that the vehicle involved incurs minimal to no damage.
This article describes principles of conservation of momentum, elasticity, coefficient of restitution, magnification of acceleration, effect of crush distances, braking effect, and G force as they relate to the potential for injury.
At the core of all physics and engineering principles are underlying laws, which define force and energy. One of these is the law of conservation of momentum, better known as Newton's Third Law. Momentum is inertia or energy in motion and is the product of the mass of an object and its velocity. The law of conservation of momentum states that energy and momentum are conserved in a collision:
Total Momentum (Before Collision) = Total Momentum (After Collision)
In other words, the total amount of energy that enters into a system is equal to the amount released from that system. Consequently, the momentum of the first vehicle, plus the momentum of the second vehicle, must be the same after the collision as it was before the collision. This can be expressed mathematically as:
M1V1 + M2V2 = M1V11 + M2V21
Where M1 and M2 are the masses of the two vehicles, V1 and V2 are their initial velocities, and V11 and V21 are their velocities after impact.
When the vehicle being struck (M2) is at rest, it has no velocity and, hence, no momentum. The vehicle that is striking (M1) has a momentum based on its velocity and mass.
The critical elements of this law are (1) the velocity (V1) and mass (M1) of the striking vehicle and (2) how that relates proportionately to the vehicle being struck (M2). A vehicle (MI) that weighs 10,000 pounds striking a vehicle (M2) that weighs 2,000 pounds will be imposing a momentum five times that of two cars colliding of equal mass or weight. Therefore, a school bus rear-ending with an automobile at a low velocity can impose significant momentum at a very low speed.
To further define conservation of momentum, physicists classify a collision as either an elastic collision or an inelastic (plastic) collision. An elastic collision occurs when objects collide without lasting deformation or the generation of heat. An inelastic collision takes place when objects colliding become deformed. In an absolute sense, an elastic collision is not possible in the everyday world, because of the force of gravity and resulting friction. However, in the first milliseconds directly after a rear-end collision, the net momentum of colliding vehicles is essentially the same.
When a moving billiard ball hits another billiard ball that is at rest, head-on, the first ball comes to rest, while the second ball begins to move at the initial velocity of the first ball. The momentum of the striking ball is transformed to the second ball. This is an example of elastic behavior. If the billiard balls were made of a breakable material, such as glass, the striking impact would cause breakage; this is an example of plastic deformation.
Another mathematical formula used to determine acceleration, a, involves the relationship between velocity and crush distance. Mathematically, this is expressed as:
a = V2/2S
where V is velocity and S is crush distance. Crush distance is the measurement of actual plastic deformation sustained by a vehicle.
The following example illustrates the significance of crush distance. A vehicle striking a solid brick wall at 10 mph (4.46 m/sec), resulting in a crush of 5 inches (0.127 in), is compared with a vehicle also traveling at 10 mph (4.46 m/sec), with a crush of 2 inches (0.0254 in). Entering these data into the a = V2/2S equation (using the metric equivalents), we find that 10 mph at 5 inches of crush results in a force of 8G, while 10 mph at 2 inches of crush results in 20G. Thus, as vehicle crush distance decreases, the acceleration produced increases -to both the vehicle and occupant-provided velocity is constant.
Based on this principle, Robbins noted, "Motor vehicle bodies or bumper-to-bumper chassis offer little or no crushing effect on arresting obstacles when impacted; thus, relatively high G forces can be experienced by occupants when rear-ended, resulting in whiplash injury."
The formula used in engineering to determine the degree of elasticity, e, mathematically is referred to as a coefficient of restitution. This can be calculated by solving:
E = (V21 - V11) divided by (V1 - V2)
In this equation, V1 and V2 equal the velocities of the struck and striking vehicles, respectively, before impact, and V11 and V21 equal the velocities of the struck and striking vehicles, respectively, after impact. For a perfectly elastic impact, e = 1; whereas, for a totally plastic impact, e = 0.
The underlying importance of elastic and plastic deformation, as it relates to low-speed rear-end impacts on occupants, was described by Navin and Romilly:
"It is known that during a collision, the vehicle structure deforms, converting the system's kinetic energy into sound, thermal and strain energies. The rate of deformation defines the vehicle stiffness characteristics, while the amount of recoverable deformation is a function of its elastic properties. At high impact speeds, very little elastic recovery occurs and the vehicle generally behaves as a plastic body. At low impact speeds, however, plastic behavior may be absent, allowing more of the total impact energy available to be recovered in elastic rebound. For the occupant, the best ride down profile occurs when the vehicle behaves as a plastic body where large deformations reduce the overall acceleration".
Based on their research at the University of British Columbia (UBC), Navin and Romilly note that:
"To illustrate this concept, one particular vehicle tested at the Capital UBC Accident Research Facility showed no structural damage after a 15 km/6 barrier impact and was indicative of the fully elastic case. The overall change in velocity, Delta V, experienced by an occupant in this vehicle would have been approximately 30 km/b, due to the nearly equal and opposite rebound velocity which resulted after the barrier impact. If this vehicle had behaved in a fully plastic manner upon impact, the same occupant would have experienced a Delta V of only 15 km/h. Thus, based on an assumption of equal impact times during both collisions, the average acceleration experienced by the occupant in the elastic vehicle would be approximately twice that of the plastic vehicle. This theory implies that vehicles which do not sustain damage in low-speed impacts can produce correspondingly higher dynamic loadings on their occupants than those which plastically deform under the same or possibly more severe impact conditions."
Research shows that the relationship between elasticity and injury is not linear. Testing done by Thompson et al. revealed that at speeds above 15 km/h, the G forces incurred by occupants actually begin leveling off. This is the speed at which plastic deformation, in the form of buckling, begins to occur. The important point is this: As plastic deformation begins to take place, the increase in occupant loading does not increase dramatically. Thompson et al. demonstrated loadings as high as 10G to the occupants of a vehicle traveling at speeds of 15 km/h.
As numerous studies have documented, a vehicle at impact moves forward before the occupant's head begins to move. Further, in whiplash accidents the shoulder moves forward before the head, resulting, initially, in an extension.
The principles of magnification of acceleration come into play, as the head attempts to catch up to the shoulders, resulting in flexion. This forward flexion results in the head attaining 2 to 2-1/2 times the acceleration or G force incurred by the vehicle itself. Hence, a force of 3G sustained by a vehicle can result in a force of 6G to 7G to the occupant's head. The whole sequence of events, from the striking of vehicles to the rapid deceleration of the head, takes place in approximately 300 milliseconds.
Magnification of acceleration was documented by West, Gough, and Harper. They utilized human test subjects, wearing helmets with accelerometers attached in the X, Y, and Z directions. At 8.7 km/h (5.4 mph), the vehicle's peak G acceleration was 2.8G, while the occupant's peak head acceleration in the X direction was 7.5G. Consequently, the head sustained a G force of 2.67 times that of the vehicle.
Another important principle in physics that is involved in automobile collision is the relationship of acceleration to velocity and time. Acceleration, a, is equal to the change in velocity (delta V) over time (t), and is expressed in the formula:
a = delta V/t
The important criteria influencing acceleration are, consequently, the maximum velocity that occurred and the time that elapsed in attaining delta V.
Some argue that applying the brake during a lowspeed rear-end collision results in no significant change in acceleration. The relationship between braking and time and velocity was studied by Emori and Horiguchi of the engineering department at Seki University. These researchers found that the maximum speed of the struck vehicle decreases with increasing braking force. However, because duration of acceleration becomes shorter with increasing braking force, maximum acceleration of the struck vehicle is nearly the same with and without braking.
"G force" is defined as the acceleration imposed by earth's gravity, or 32.2 feet per second, squared. A force of 5G, for example, means that a body is accelerating at five times the force of gravity. One researcher, I. Macnab, has described the effect of acceleration to injuries of the neck:
"If, as a result of an accident, the head accelerates in relationship to the trunk -backward, forward, or sideways -injury to the neck may result. Because lesions produced in this way differ from those resulting from forced passive movements of the head, it seems worthwhile to differentiate them by the term acceleration injuries of the neck. In acceleration injuries, the force applied to the neck is roughly equivalent to the weight of the head multiplied by the speed that the head is moving".
The human head has an average weight of 10 pounds. Consequently, a 5G force results in a potential loading of approximately 50 pounds to the head.
West, Gough, and Harper found that at 11.6 km/h (7.25 mph), vehicle peak acceleration was 3. 1 G, while the occupant's head accelerated in the X direction at a force of 8.3G. This is equivalent to an 83-pound force acting on the head. Extension or X direction acceleration of the cervical spine will result in multiple-plane loading components, creating a shearing force, tensile force, and a compressive or axial force.
Working on cadavers, Przybylski et al. performed ligament uniaxial tension testing on C2-C7. They found that the anterior longitudinal ligament had a mean ultimate load of 107 ± 63 Newtons. Furthermore, according to the Society of Automotive Engineers, the limit on shearing force in the cervical spine is 231 Newtons, or approximately 52 pounds, and the limit on axial force for the upper cervical spine is 249 Newtons, or approximately 56 pounds. One Newton is equal to 0.225 pounds of force.
Watts, Atkinson, and Hennessy performed theoretical mathematical calculations, assuming a delta V of the torso at I I mph. Their calculations revealed that a 60-degree extension would occur in 0.086 seconds, which is not enough time for muscles to be enervated and thereby act as a protective mechanism for the head and neck. Based on prior studies that had detected rupture of the neck ligaments at 178 pounds and cervical disc maximum loading of 230 pounds, their calculations revealed possible disc and/or ligament damage for a delta V of I I mph. Their calculations also demonstrated that cervical injury could be incurred, reaching loadings of 190 pounds of torque, at speeds as low as 7.5 mph in an untensed neck of a normal, healthy individual.
Similarly, Barzelay and Lacy observed that the risk of injury appears to be greatest in collisions with impact velocities between eight and 20 mph. They also concluded that the force to the head can reach 100 pounds in collisions where speed at impact is no more than 15 mph.
Watts and his colleagues observed that tolerance levels may be even more restricted in individuals with prior health problems:
"Consider a person with a prior neck problem such that his or her natural neck extension is limited to only 20 degrees and damage level is only 70 lbs, due to scar tissue. With only a small initial tensing of the neck muscles, this damage level will be reached at a push speed of 2.5-3.0 mph! Note that the lower push speed range is similar to or even lower than the speeds typical for causing vehicle damage. Thus, human body damage can occur with either 0 or very small amounts of vehicle damage and it is not necessary to have copious vehicle damage in order to hurt the human body".
The injuries sustained in accidents can have long-term sequelae. Hohl, a physician researcher, studied 146 patients, with no pre-existing cervical degenerative changes, who had sustained soft-tissue injuries resulting from automobile accidents. He concluded, after a five-year study period, that 39 percent of the patients showed degenerative changes.
An understanding of the basic physics and engineering principles of conservation of momentum, elastic and inelastic collisions, effect of vehicle crush distance, coefficient of restitution, magnification of acceleration, braking effect on acceleration, and effect of G force is a prerequisite to engaging in a meaningful discourse on the relationship of bodily injury and low-speed rear-impact collisions. Applying these principles demonstrates clearly why it is possible to see injuries following low-speed rear-impact collisions.
Engineering studies and/or accident reconstruction data can imply potential injury. However, whether an occupant of a specific low-speed rear-end impact collision actually sustains injury can only be determined after a thorough clinical evaluation. Utilizing traditional orthopedic, neurologic, and spinal examination procedures, in addition to any applicable radiography and computerized diagnostics, the examining doctor trained in the specialty of spinal/body mechanics is able to objectively determine if injury has actually occurred.
In addition, the inverse relationship between vehicular damage and occupant injury imposes a challenging dilemma for manufacturers of vehicles and government regulatory agencies. Decreasing the costs of body shop expenses for an accident damaged vehicle, while at the same time increasing the probability that the occupants of the vehicle will suffer some injury, is neither cost-effective nor sound public policy.
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