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Skidmark Forensics
Photo from The Traffic Accident
Reconstruction Origin.
An important task in auto accident recostruction is the analysis of
skidmarks, which I call "skidmark forensics". Armed with data on the
car's tires and the road surface, a accident reconstruction engineer
can make a good estimate of a car's speed just before
the driver hit the brakes. We can get the basic idea using a very
simple model of friction.
Sliding Friction
A common model of friction comes from the observation that the heavier
an object is, the harder it sticks to a surface. If you call the
force you have to use to push a sliding object f, then the
simplest model of friction that makes f increase with
object weight is one that just says f is proportional to
weight mg (where m is the mass of the car and
g=9.8 m/s² is the acceleration of gravity):
f = µmg
The constant µ (the Greek letter 'mu') is
known as the coefficient of kinetic
friction, and accident investigators have tables for all sorts of
tires and road surfaces.
So how long is a skidmark for a given initial car speed? There are a
couple ways of figuring this out. I'll use an energy technique here.
The main idea is this: the car is hurtling along at speed
v, which means it has a lot of kinetic energy of
motion associated with it. If the car has mass m, then the
kinetic energy K is given by
K = ½mv²
All of this energy gets converted to heat in the tires and road and
air as the car skids to a stop. The conversion of kinetic energy to
heat is done by the work of friction, W, which is
just the friction force times the distance d the car skids:
W = fd = µmgd
So, if we equate the initial kinetic energy K to the work W
done by friction in slowing the car down, we get an expression for the
skidmark distance d:
W = K
µmgd = ½mv²
=> d = v²/2µg
Since the distance d increases as the square of the
speed v, it is quite sensitive: a doubling of speed
quadruples d. Also note that the car's mass
does not matter: a heavier car has just as much more kinetic energy as work done
by friction per meter skid, so the skid length is the same. The key
thing is the coefficient of friction µ for the particular
tire/road combination.
Example
Of course, what the police measure is d, from which they
deduce v =  2µgd. For
example, if they measure a 30 m
skidmark with a car whose tire/road combination gives
µ=0.7, they would deduce that the car was traveling with
initial speed v = 20 m/s, which is 73 km/hr.
Equations
- friction force: f = µmg
- kinetic energy: K = ½mv²
- friction work: W = fd = µmgd
- skidmark distance: d = v²/2µg
Summary
- A simple model of friction says the friction force is
proportional to an object's weight.
- The work done by friction, given by friction force times
distance, goes into heat.
- The skid distance can be found by equationg the kinetic energy of
motion to the work done by friction.
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