I. INTRODUCTION:

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
dynamic friction.

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

= 30000

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

back to the TECHNICAL SERVICES Home Page...