CATAGORY: Mechanics
 QUESTION: We have the length of a car tire's skid mark. We are
guessing that the co-efficient of friction for car tires is about .5 or
so. What is the proper formula for the calculating of the velocity of this
vehicle?

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We want to dissipate the kinetic energy of the car into work done on the
car
by friction.  Thus

(1)       Change in kinetic energy = work done by friction

             final KE - initial KE = work done by friction
             
                  0  - (1/2)mv^2   = -Uk mg L
                  
Where m is the mass of the car v is the initial speed of the car, Uk is
the
coefficient of friction, g = magnitude of the acceleration due to gravity
= 9.8
m/s^2 and L is the length of the skid.  The work is negative because the
force 
is
acting against the displacement and thus drains energy away from the car.
The
mass cancells in (1). Also divide by -1 to give

(2)                   (1/2)v^2   =  Uk g L

solving for v in terms of L gives

(3)                      v = sqrt(2 Uk g L)


So your "proper" formula is

                 **************************
                 *                        *
                 *   v = sqrt(2 Uk g L)   *
                 *                        *
                 ************************** 
                 
It of course assumes that ALL the kinetic energy is dissipated in the skid
and not part in the skid and part in colliding with something else.