Much of the work of accident analysis and reconstruction involves the application of general physical laws to the particular situation at hand. These physical laws are expressed in terms of Newtonian mechanics, which was developed by the great English scientist Sir Isaac Newton in the seventeenth century. Newtonian mechanics is the base science of physics and engineering. Today the development and application of basic Newtonian mechanics is almost exclusively the province of engineers in that physicists, for the most part, conduct basic research only in "new physics" areas such as quantum mechanics, relativistic physics, etc.
II. KINEMATICS: THE ANALYSIS OF MOTION
1. Velocity. Velocity is the rate of change of the position of a body over time (velocity = distance/time). Velocity is a vector, which means it has both a magnitude and direction. Thus, 30 miles per hour north, or 20 meters per second along the X axis, are both velocities. Note that the more common term, Speed, is a velocity without a direction, so that 30 miles per hour or 20 meters per second are speeds, and not velocities, since no direction is specified.
2. Acceleration. Acceleration is the rate of change in velocity (acceleration = velocity/time). Thus, any time a body changes its rate of travel or its direction of travel, it is said to be accelerating. Acceleration is a vector requiring both magnitude and direction. Thus, for example, 32.2 ft/sec/sec. , downward toward the center of the earth, is the acceleration vector of all bodies on the surface of the earth. If the support collapses, for example, if a person falls off a ladder, they will increase their velocity at a rate of 32.2 ft/sec. per second. At the end of one second of free fall, they will be traveling at a rate of 32.2 ft/sec. which is approximately 22 miles per hour. They will have fallen approximately 16.1 feet in this one-second interval. (D = 1/2 a t x t = 1/2 x 32.2 x 1 x 1 = 16.1).
III. NEWTON'S LAWS
Newton wrote three laws of motion which relate kinematic phenomena to something else-- forces. These are:
1. First Law: "A body does not alter its state of motion without the influence of an external force." That is, there is no change in the velocity of a body (neither in magnitude nor in direction) unless some force acts on that body.
2. Second Law: "The net resultant force applied to the body is equal to the first time-derivative of the momentum function." Or roughly:
Force = Mass x Acceleration.
This relationship is not so much a natural law as a rule for assigning a magnitude to forces.
3. Third Law: "For every applied force there is an equal and oppositely directed reactive force." You push on the wall- the wall pushes back. The pusher and the pushed, the striker and the struck both experience forces of the same magnitude but of opposite direction.
IV. DYNAMICS: THE ANALYSIS OF FORCES AND THEIR EFFECTS
1. Force. A force in physics or engineering is something close to what we call "forces" in everyday life. Any push, pull, twist, etc., involves a force or forces. Forces are measured in physics according to their effects-- according to Newton's Second Law, Force = Mass x Acceleration, or F= MA. Note that "mass" is that property of physical objects that causes them to resist changes in motion. "Inertia" is another term for mass.
2. Weight. Weight is a special force which results from the fact that a mass is being acted on by a gravitational field. A body on the surface of the earth has a certain weight because it has mass which, when acted on by the earth's gravitational field, would cause the body to be accelerated if it lost its support. If it could fall, its "weight" would result in its being accelerated toward the center of the earth.
3. Torque. A torque is a twisting force. It is a
force that tends to induce rotary motion rather than straight
line motion. Torques are typically measured in lb-ft. Thus,
if we pull on a one-foot long wrench with a force of 100
lbs., we exert 100 ft-lb of torque on the nut. The same
torque is generated by 50 lbs. on a two-foot wrench, etc.
4. Friction. Friction is a special kind of force
produced by two bodies that are in contact. If a book is at
rest on a desk and we try to push it, our efforts are
resisted by what is known as static friction. Once we get it
moving, if we stop pushing it, it comes to rest almost
immediately. The retarding or stopping force is known as
Rolling Friction is the relatively low retarding force associated with the free rolling of objects, e.g. tires. Sliding friction is much greater than rolling friction and occurs whenever two objects in contact move with respect to one another without the benefit of any revolving elements. When a car is driven down the road without braking, it is being retarded by rolling friction (also by air drag, another type of force). When the wheels are locked, the vehicle is retarded by the sliding friction between the tires and on the road. In between these two cases, in situations with braking without wheel lock-up, the retarding forces are a complicated manifestation of the operation of the brakes, tires, and suspension system of the car. One measure of the sliding friction of cars is the coefficient of drag (Cd). This gives an indication of how hard it is to push the car along the road with its wheels locked. A typical Cd is 0.7. In this case, a force equal to 0.7 x the weight of the car is required to keep it sliding along the road. With a Cd and skid mark of known lengths, it is possible to estimate a vehicle's velocity before the start of the slide.
S(mph) = 5.5 x sqrt (Cd x length of skid)
5. Momentum. Momentum is the product of
mass and velocity (momentum = mass x velocity). Momentum is a
vector quantity as are velocity, acceleration and force.
Momentum is conserved in impacts. That is, the sums of the
various momentums of the bodies before the collision are the
same as momentums after the collision. Thus, we can
frequently compute the velocities of vehicles before a
collision by knowing their speeds and directions of travel
after the impact. Momentum is not the same as, and should not
be confused with, energy.
6. Work. When a force acts on a body, work is said to be done on that body. The quantity of work, W, is given by the formula: W = F x D, where "F" is the force that acts on the body and "D" is the distance through which it acts. Thus, a 100-lb. force acting for a distance of 10' (e.g., pushing an object against a resistance of 100# for 10') results in work = 1000 lb-ft being done on the object.
7. Energy. Energy is the capacity to do work. In Newtonian physics, the energy of a body is computed in two ways: either by computing its kinetic energy: KE = 1/2 M x V x V where "M" = mass and "V" = velocity; or by computing its potential energy with respect to a system of forces capable of doing work on the body. For example, the potential energy of a body in the earth's gravitational field is PE = W x H, where "W" is the weight of the body and "H" is its height above some reference point.
Computations involving kinetic energy are tricky but informative. For example, if a moving body crashes into a solid, non-yielding wall at 20 miles per hour, the kinetic energy dissipated in the crash = 1/2 M x 20 x 20 = 1/2 M x 400 = M x 200; but if it crashes into the wall at 40 miles per hour, the energy dissipated in the collision is 1/2 M x 40 x 40 = 1/2 M x 1600 = M x 800. Thus, four times as much energy is involved in the second crash as in the first (M x 800/M x 200 = 4). The speed has doubled but the energy has quadrupled!
8. Power. Power is the rate at which work is done. When work is done at the rate of 33,000 ft-lbs. per minute, one horsepower is produced. A "horsepower" is merely a conveniently sized unit for measuring power. Its connection with equine work capacity is tenuous. A human being can apparently work at a rate of about .35 horsepower for short periods of time, if all their skeletal muscles are being effectively used.
V. SPECIAL TERMS AND EXAMPLES OF NEWTONIAN ANALYSIS
1. G's. The term "g" forces or "g" loads is a convenient descriptive tool for technical discussions involving accelerations due to impact forces. One "g" is an acceleration equal to that generated by a free fall in the earth's gravitational field, i.e., 32.2 feet per second per second. Thus, a body acted on by a 0.5 g acceleration experiences a force equal to half its weight. This force acts in the direction of the acceleration.
Frequently, during high speed collisions, accelerations in the range of 25 to 50 g's are generated. If a 150-lb. person is acted on by a 50-g retarding force during an accident, then a force of 7,500 lbs. acts on his body. Note that while a "g" value is really an acceleration, it is sometimes discussed as though it were a force. This is technically inaccurate but not harmful if it is clear what body the force acts on and if the mass of that body does not change during the application of the "g load."
2. Moment. The word "moment" in physics is really another word for torque. Generally, when we have two or more torques acting on a body, the result is said to produce a "moment" or net torque, which acts to rotate the body in a direction determined by the combined torque vectors.
3. Pressure. Pressure is force per unit area. Thus, if a 10-lb. weight has a contact area with another body of 1 square in., the pressure is 10 lb per square in. Normal atmospheric pressure at sea level is about 14.7 lb./sq.-in. (psi) This is the weight of a column of air 1 inch square extending up from sea level to the outer limit of the earth's atmosphere--roughly 80 miles high. A barometer reading of about 30 inches of mercury (30 in hg) is about one atmosphere pressure or 14.7 psi. Thus, a 30-inch column of mercury weighs about as much as 80 miles of air since they both exert the same weight force per unit area.
4. Stress. Stress, like pressure, is also a force per unit area. However, while pressures acts on bodies, stresses act in a body. If I pull on both ends of a steel bar of 1 in. cross-sectional area with a force of 100 lbs., I generate a tensile stress inside the bar of 100 psi . If I pull across the bar so as to try to split it in half, I generate a sheer stress. Note that a mild steel bar would be able to absorb tensile stresses in excess of 50,000 psi without breaking so that my efforts to pull it apart result in only trivial stresses in the bar.
VI. ACCIDENT RECONSTRUCTION PROBLEMS
1. Suppose a 3000 lb. car were to crash into the rear of a truck that weighs 30,000#. The road is asphalt and the coefficient of drag is 0.7, the car is going 10 miles per hour, the truck is at rest.
The momentum of the system before the collision is:
Momentum = Mass (car) x Velocity (car) + Mass (truck) x Velocity (truck)
= 3000 x 10 + 0 x 30,000
Since the external forces acting on the system during the impact are minimal (the brakes of the truck are off) momentum is conserved. Thus, after the accident the following relation holds true:
Momentum before =Momentum after
30,000 = 3000 x V + 30,000 x V
V = 30,000/33,000 = 0.9 mph
where "V" is now the post impact velocity of the vehicles. (We assume here that the truck and the car are moving with the same velocity after impact)
Thus, the car loses 9.1 mph = 13.3 ft/sec due to the action of the retarding impact force.
If the car is shortened about 8" in this impact, then the distance through which the retarding force acts is about 12" (the truck starts moving during the impact; assume it moves about 4") so that the car travels about 1' during the impact with an average velocity of about 8 ft/sec. Thus, the duration of the impact can be estimated as follows:
Distance = Velocity x Time
Time = Distance/Velocity
T = 1 ft/(8 ft/sec) = 0.125 sec
So the car decelerates from 14.7 ft/sec to 1.3 ft/sec in a time of 0.125 sec. Thus, its average deceleration is:
13.4 ft/sec / 0.125 sec = 107 ft/sec/sec = 3.33 g
The average force acting on the car then is 3.33 x 3,000# = 9990#
This is also the force that acts on the truck (Newton's Third Law: Every force has an equal and opposite reaction force): So that the average acceleration of the truck is: 30,000/9990 = 3 ft/sec/sec = 0.1 g
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