Question

Estimate the skidding distance of a car travelling at 30 mph if the wheels are locked and sliding assume u = 0.34 in dry conditions

Answer 1

Hey K,

u is your coefficient of kinetic friction, which determines the friction force on the car when it is moving. If we assume a car weighs about 1000 kg, then we have enough info to solve the problem.

The force of friction on the car is going to be u * the normal force acting on the car. The normal force is the force of the earth holing up the car, keeping it from falling further due to gravity. The normal force is equal in magnitude to the weight of the car. The weight of the car is (negative meaning gravity is pulling down):
\[W=-mg\]
Our normal force acts pushes up on the car(therefore positive):
\[N=mg\]
And therefore the force of friction on the car on the car is:
\[F = -{\mu}N = -{\mu}mg\]

Th easiest way to solve this is by conservation of energy, and the work-energy theorem. If our car is going 30 mph, then it has a kinetic energy:

\[KE = \frac{ 1 }{ 2 }m v^{2}\]

In order for the car to stop, some force must do work on it so that its kinetic energy is 0. Work is equivalent to force x distance:

\[Work = Fd\]

By conservation of energy, the initial energy must equal the final energy. In this case the final energy is 0. Our initial energy is the kinetic energy of the car, and the heat energy produced by friction.
\[KE - Work = 0\]

Solving this equation we get:
\[KE = Work\]
\[\frac{ 1 }{ 2 } mv^2 = {\mu}mgd\]
\[d=\frac{ 1 }{ 2 }\frac{ mv ^{2} }{ {\mu }mg}\]

d is our skidding distance.

Answer 2

simplifying a little...

\[d=\frac{ 1 }{ 2 }\frac{ v^{2} }{ {\mu}g }\]