Stopping Speed and Distance

Date: 08/28/2001 at 20:43:01
From: Adam Seaberg
Subject: an 18 wheeler vs. a motorcycle in stopping

t school today I was asked the question "out of a 18-wheel tractor- 
trailer and a 2-wheel motorcycle, which would be able to stop faster 
or in a shorter distance? I responded with the 18-wheeler because it 
has more cubic sq. in. of rubber on the ground compared to weight 
or mass than a motorcycle does. I was told that I was wrong but I 
don't believe it, so I decided to go to someone who would help me 
prove it. 

I feel that this is a very very interesting question and I would 
really appreciate your help on this 

Thank you,
Adam Seaberg

Date: 08/29/2001 at 14:26:46
From: Doctor Rick
Subject: Re: an 18 wheeler vs. a motorcycle in stopping

Hi, Adam.

Yes, it's an interesting question. My first guess is the opposite of 
yours. My brother has driven fairly large trucks, and he complains 
about people who pull in front of him while approaching a red light. 
He says people don't realize that a truck has a longer stopping 
distance.

Let's consider the physics of stopping. A moving vehicle has kinetic 
energy equal to half the mass times the velocity squared. If the truck 
and motorcycle are moving at the same speed, their kinetic energies 
are thus proportional to their masses.

If a constant stopping force is applied to a vehicle, the energy 
dissipated is equal to the force times the distance. The kinetic 
energy goes to zero (the vehicle stops) in a distance such that the 
force times the distance equals the kinetic energy. The stopping 
distance is thus the kinetic energy divided by the stopping force.

The maximum stopping force that can be applied without skidding is 
equal to the normal force (the force with which the vehicle is pulled 
against the road) times the coefficient of static friction for the 
tires on the road. If we assume that the truck tires and motorcycle 
tires have the same coefficient of static friction, then the maximum 
stopping force is proportional to the normal force, which is 
proportional to the mass.

Why doesn't the stopping force depend on the number of tires? The 
weight of the vehicle is distributed among the tires. If you double 
the number of tires, you halve the average weight supported by each 
tire. The maximum stopping force is proportional to the weight 
supported by the tire; thus the stopping force from each tire is 
halved. Half the force from twice as many tires means the same total 
stopping force, independent of the number of tires.

The stopping distance is proportional to the kinetic energy divided by 
the stopping force, and each of these is proportional to the mass, so 
this proportionality cancels out. The stopping distance, according to 
my reasoning, is independent of the mass of the vehicle; it should be 
about the same for a motorcycle or a tractor-trailer.

Does this mean we were both wrong? I do find it surprising - it goes 
against what I had always assumed - so I checked on the Web and I 
found this site that supports my reasoning:

   Stopping Distance for Auto - Dr. Rod Nave, HyperPhysics, 
     Georgia State University
   http://hyperphysics.phy-astr.gsu.edu/hbase/crstp.html   

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/