Skidmark Forensics

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. (Photo from The Traffic Accident Reconstruction Origin.)

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 objects 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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Sam Hokin / samh@fusion.kth.se