Creative Education
2012. Vol.3, Supplement, 114-116
Published Online December 2012 in SciRes (http://www.SciRP.org/journal/ce) DOI:10.4236/ce.2012.37B030
114
The Physical Foundation of Car-following Model
Xiaolin Shu, Shuo Jin, Ying Zhang, Hongbo Zhou, Guanghong Lu
College of Physical Science and Nuclear Engineer, Beihang University, Beijing, 100191, China
Received 2012
Based on the Newton’s law of motion, the motion equation of the microscopic car-following model is in-
troduced. The moving vehicles comply with the idea of Newton's equations of motion. But the interaction
between vehicles is different from their traction. The inertia of the car is not its mass. We discuss the dif-
ferences and present how to treat them. This article can deepen and widen the students' understanding of
Newton's laws of motion. It can also provide the materials for university physical teaching. It is suitable
for the students to exploratory study.
Keywords: Component; Newton’s Law of Motion; the Car Following Model; Inertia
Introduction
Mechanics study the interactions of objects as well as their
motion under the interactions. The Newton's three laws of
motion are the basis of mechanics. The motion states of a
particle can be determined by the Newton motion equation. In
middle school, mechanics is the main part of the physics
teaching. The students have known very well about how to
solve linear motion and uniform circular motion with Newton's
law. In college physics teaching, the basic laws of mechanics,
including Newton's three laws of motion, work-energy theorem
and impulse-momentum theorem, have been teaching in
classroom systematically. The particle dynamics problems can
be divided into two types. (1) Knowing particles equation of
motion, to find the action on it force. This kind of problem can
be adopted by the differential method. (2) Knowing force on a
particle, to find the equation of motion. The kind of problem
can be worked out by integral method. A variety of cases of
these two kinds of problems are teaching in c lassroom, such as
the external force on particle is a function of time, or position
or velocity, etc. It is also given out the solving steps for every
cases. Through the accumulated effect of exerting a force on a
body that undergoes a displacement, work-energy theorem is
introduced. By the accumulation effect of the net force acting
on the object during the interval, the impulse-momentum
theorem is introduced. On the classroom teaching
systematically and large exercise training after class, the
students master the fixed steps and thinking mode on solving
problems by Newton's law. In principle, students should be able
to solve all the particle dynamics’ problems.
But some factors becomes the obstacles to develop the ability
of applying knowledge to solve actual problems. Such as the
teaching materials are out of line with the actual living cases,
and the teaching goal is for the entrance examination, etc. Even
though students know well about the teaching physical content
and have a large number of similar repeated training exercises,
it is restrained the students’ minds of students. This paper
analyzes that the Newton’s motion equation applied in traffic
problem to deduce the microscopic car-follow model equation.
On the one hand, to promote the students' interest in university
physics, on the other hand, to display application of physical
basic principles in the solution actual problem.
High efficient transport system is the development founda-
tion of modern industrial society. With the rapid development
of industry, the traffic volume increasing rapidly in many coun-
tries is over the capacity of the highway. It has seriously hin-
dered the development of society. Understanding transportation
process and the dynamics law is the basis of solving the traffic
problems. As in 1935, Greenshields [1] started to study the
vehicle traffic problems. Transportation process shows so many
physical phenomena, such as dynamic jam, phase change, crit-
ical phenomenon, metastability, self-organized critical state,
nonlinear wave (solitary wave), etc. These phenomena attracted
the physicists’ wide interests. They study the traffic problems
from physics thoughts. It is said that the development of the
modern traffic theory is the organic combination of the com-
puter technology and modern physics technology. Helbing [2,3]
carried on the detailed conclusion on traffic physical model.
Microscopic car-following model is a good example for apply-
ing the classical Newton’s motion equation to solve the traffic
problem.
Microscopic Car Following Model
Newton's second law of motion describes the reason for the
variation of the motion state of the particle . It is the relationship
between the net force and the changes of motion state. It states
as that the net force vector is equal to the mass mα of the object
times the acceleration of the object.
( )
( )
2
α
ααβ 2
β(α)
dr t
m= Ft
dt ≠
∑

 (1)
where αβ F

is the force between particles α and β, which is
related with position or velocity of two particles.
Consider the traffic flow as a particle system and the simple
case of linear movement as the friction Ffr(t)=- μαvα(t) with
friction coefficient μα, so the Newton motion equation of parti-
cle (1) can be expressed as:
( )( )
0ααααααβ
β(α)
mxt= F (x(t),t)μv (t)+Ft
≠
−∑

(2)
where 0α
F (x,t)
is the vehicle driving force, which is the

X. L. SHU ET AL.
115
interaction between the particles. From this equation, it is
deduced systematically to many physica l quantities, such as the
density of particles, momentum or average rate, etc.
Highway traffic can be regarded as the self-driving system.
The moving car can be treated as particles in the system (as
shown in Figure 1 below). Besides the external force, there are
the internal forces from other particles in the system. The driv-
ing force of between cars can be used 0
α
F(t) to instead
0α
F (x,t)
.
The inertia is the property of the tendency of an object to keep
moving once it is set in motion. Usually, the mass is a quantita-
tive measure of inertia. In the traffic flow, the vehicle mass is
not the inertia because it is the self-driving car. The inertia is
related to the vehicle mass. But it is also related to the other
factors, such as the driving force and braking force etc. So it is
hard to de cide. In the model, the quality mα can be moved t o t he
right side of the equation and combined with the force together
as the force, which is also undetermined. This method is also
applied to the calculation of density functional theory.
Let
00
ααα
F (t)=γv (t),
/
ααα
γ=mτ
,
αβααβ
F(t)= mf(t)
, the equa-
tion (2) is as
( )
00
αααα
αβ
β(α)
α
dv (t)v (t)e (t)v (t)
=+ ft
dtτ
≠
−
∑
(3)
The meaning of the first term in the right of the equation is
that in the relaxation time τα, the particle speed tends to the
wanted velocity 0
α
v (t) as the exponential function form on the
effects of the driving force
0
/
αα
v (t)τ
and friction force
/
αα
v (t)τ−
. The
αβ
f (t)
is the action from other particles. Ob-
viously if the action is an attractive force with particle α, it will
lead to gather effect to form a blockage. So, it is usually use to
reducing or exclusive action.
In 1950s, physicists Pipes put forward the microscopic traffic
car-following model [4]. He assumed that the acceleration of
vehicle α just determined by its front neighboring. Therefore,
equation (3) reduces to:
( )
0
1
ααα
α,( α)
α
dv (t)vv (t)
=+f t
dtτ−
− (4)
It should pay attention that the Newton’s third law of motion
is not suiting to in the case. The leading vehicle has an effect on
its following vehicle, but the following vehicle has no action on
its leading car.
The force 10
α,α
f (t)<
−, shows the repulsive action of vehicle
α and its leading vehicle (α-1). It is influenced by three factors:
(1). the relative speed
α
Δv (t)
, (2). the safe distance
α
s (t)
, (3).
the speed α v(t) of the vehicle α. Forces 1α,α f (t)
− is the func-
tion of three amounts,
( )
1α,( α)ααα
ft=f(s (t),v (t),Δv (t))
− (5)
when the vehicle is in the motion of constant acceleration, it
can introduce the traffic correlation function of speed.
0
e
αααααα
v (s,v,Δv)=v +τf(s ,v ,Δv) (6)
lα
lα-1
l1
lα+1
lN
1
α
α-1
α+1
Figure 1.
Schematic plot for car flow [4].
Substitute (6) into (4), it can simplify model equation as
e
ααααα
dvv (s,v,Δv) v
=
dtτ
−
(7)
Equation (7) is the basic equation microscopic car-following
model.
Bando [5] supposes a specific form
0/ 2tanhtanh
'
e cc
v(d)= (v)[(dd)+d]− (8)
where V0 the maximum speed of the vehicle, dc the safe dis-
tance. Using the optimal speed of traffic instead of the correla-
tion function, the car-following model can be represent as
'
aeαa
dv(t)v (d(t))v(t)
=
dtτ
− (9)
In Bando’s simulation, the traffic appeared three different
areas: stable area, metastability area and unstable region. When
it is in stable condition 1
2τ
'
eαeα
aα
dv (d)dv(s)
=>
dd ds, which means
the relaxa tion time is lar ger or the velocity is larger, only a tiny
disturbance will cause traffic jams. Figure 2 is a system evol u-
tion plot of time and location structure by Bando [5]. In the
picture, the black areas represent a higher density of vehicle
which is easy to form traffic jams areas, gray area stand for the
low density areas of vehicle. The high density areas is called
jam phase, low density areas is called the movement phase.
Conclusion
From the above derivation process of microscopic traffic
car-following model, the interaction between the vehicles is not
the traction of cars. It is related to the safe distance of the car,
the speed of the car and the speed difference from the leading
car. The inertial is not the mass of vehicle because of the
self-driving system. It can be treated as combining in the inte-
raction. The Newton’s third law is not suitable in that case. But
the idea of Newton’s motion law can still be used in the traffic
problem because the motion and its state of vehicle are deter-
mined by the interaction.
In the microscopic car-following model established process,
it is a very good example for how to treat and simplify the
physical concept and the basic principles of physics. It is pre-
sented the practical application of the physical ideas and me-
thods. This is the missing part of the classroom teaching. It is
also
Figure 2.

X. L. SHU ET AL.
116
Plots of the positions of all vehicles [5].
important ability cultivation for students using learning know-
ledge to solve actual problem. This paper hope to provide ma-
terials for college physics teacher to teaching in class and for
students to make creative learning
REFERENCES
B. D. Greenshields, “A Stud y of Traffic Capacity ,“ Proceedings of the
highway research board (Highway Research Board, Washington, D.
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Dirk Helbing, “Traffic and related self-driven many-particle systems,”
Review of Moder n Physics, vol . 73, pp. 1067-1141, 2001.
T. Nagatani, “The physics of traf f ic jams,” Rep. Prog . P hys., vo l. 65 pp .
1331-1386, 2002.
Louis A. Pipes, “An operational analysis of traffic dynamics,” Journal
of Applied Physics, vol. 24, pp. 274-281, 1953.
M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama,
“Dynamical model of traffic congestion and numerical simulation,”
Phys. Rev. E., vol. 51 pp. 1035-1042, 1995.