Marisa Michelini, Gian Luigi Michelutti, Department of Physics, University of Udine, Italy
1. Introduction
Researchers and teachers often find themselves faced by the difficulties which students have with
the mathematical instruments which physics uses both on the descriptive level and also on the
interpretative level; almost as a logical, unquestionable consequence, they look for ways to avoid or
reduce involvement on the formal level [1,2,3]. For example, they try to give greater weight to
experimental activities (the educational role of which is beyond discussion for an experimental
subject) [4,5,6], or they try to entrust to operativity and/or informal education the connection
between the perception and observation of phenomenology and the physical description [7,8,9].
Didactic projects done in the 60s and 70s throughout the Western world [10] are examples of a
translation of the various pedagogical theories into operational strategies for effective teaching.
These projects mainly relied on experimental activity to construct a gradual awareness of the formal
relationships between the significant variables in selected experiments (PSSC, IPS, PS2), even
when the formulation was of a historic type (PPC). Such experiments were conducted over a wide
scale, and established that it is not sufficient to optimize teaching in order to achieve good learning,
and that other types of difficulties occur [11], such as those linked to the lack of connection between
common-sense interpretation and physical interpretation [12,13], or those linked to the ability to use
ways of representing things which physics uses, for example graphs [14].
The difficulties in mechanics are particularly well-known [15-21]. If we examine them we notice
that the difficulties are of a conceptual type, both with regard to the significance of the elements of
formulas which physics introduces to describe and interpret, and also with regard to the styles of
formalization which physics assumes.
Studies on learning processes [22,23], and theories on conceptual change [24] have given useful
indications for involving students, for the processes of building knowledge, for ways to encourage
the contextualization of concepts, and for the ways in which to help students look at the world from
a physical point of view. Research into the use of the computer in physics teaching has given an
important contribution to the ability of looking at processes from a physical point of view, to
reading and using graphs [25-29]. Such research has contributed decisively to putting into students'
hands the process of constructing formalized physical models starting from qualitative hypotheses
With regard on how to make students aware of the ways physics uses mathematics to deal with
descriptive and interpretative problems in various circumstances: this problem is still open. It seems
to us that this cannot be considered a secondary problem for a subject like ours, which has assumed,
as a work style, a predictive capacity based on the description of phenomena by means of
mathematical tools. It is a style which is a part of the epistemic roots of physics and we do not think
it is possible to give this up, if we want to give young people the opportunity to develop a passion
for this discipline [35]. Therefore, we need contributions which show the ways physics uses
mathematics, which familiarize students with these ways, and which give young people the
opportunity to operate on this level without inhibitions, overcoming the prejudice that the symbolic
language is impossible for them to manage.
This work wants to give a contribution to this end, and offer a new way of recognizing mechanical
energy as a constant of motion.
2. Definition of the proposal
The point of view which directs this proposal favors the principles of energy conservation in the
knowledge of physics and associates them with the existence of one or more constants of motion. In
this conference we shall consider the conservation of mechanical energy, which allows to identify
interesting characteristics of the motion of a material point, establishing a relationship between
position and velocity.
In the teaching of physics, the principle of conservation of mechanical energy is traditionally
introduced by using the concept of work. In reality, in many basic physics problems, it is this very
same concept which takes on significance and usefulness from the formulation of the principle of
energy conservation.
We therefore propose a didactic definition which sees the principle of energy conservation as the
pivot of a mechanism which holds the concept of work as marginal and reserved only for dissipative
This approach develops from the second law of dynamics to a search for quantities which are
constants of motion, that is, which re-write the law in terms of temporal derivatives of quantity,
which express it in the direction of the motion.
In order to do this, we consider the scalar products with the velocity v of both terms of Newton's
second law.
On the didactic level, in carrying out this first step, we point out to students that the scalar product
of two vectorial sizes selects the contribution of one in the direction of the other. Moreover, we get
the students used to exploring the significance of the temporal variation of one size with respect to
its value: this usually helps us to know new properties, as happens in the description of phases in
space. By representing the quantity of motion as a function of the position for a mass-spring
oscillator, we see immediately if the motion of the system is dissipative or not.
The expression of the kinetic energy emerges from the search for a quantity whose derivative gives
the scalar product of the quantity of motion with the velocity. The expression of the potential
energy is defined as that quantity whose temporal derivative gives the scalar product of the force
with the velocity and it is recognized that this is possible only if the force has constant components
in all directions. Since the two scalar products are equal, students recognize the conservation of
mechanical energy both in classical dynamics and also in relativistic dynamics. The student finds
himself face-to-face with a new way of looking at the characteristics of force. It must be subjected
to verification and the general characteristics must be examined. From the expressions already
found it is easy to obtain the known expressions of mechanical energy in the case of weight force,
elastic force, gravitational force, Coulomb force and Lorenz force. The definition of conservative
force emerges as the consequence of the fact that the scalar product of force times the velocity is
equal to the temporal derivative of the potential energy. Work is obtained by examining the
variation of mechanical energy when forces of friction are in play.
This approach uses elementary mathematical tools, without losing the formal elegance and
generality of the ordinary treatment based on infinitesimal calculus. It can therefore be given to first
year students on no-calculus courses or as more detailed work for secondary school students
specializing in science.
3. The formal itinerary proposed
Newton's second law
= p& ,
compares the resulting force F acting on the material point with mass m and velocity v with the
derivative of its quantity of motion p = m v.
To recognize the contribution in the direction of motion, we consider the scalar products with the
velocity v of the two members of the law.
F ⋅ v = p& ⋅ v
& ⋅v
3.1 The classic case of the scalar product p
It is known that the second member of (1) in the classic case of constant mass can be written as
dp d
p& =
= (mv) = m
= ma
dt dt
and hence the scalar product p& ⋅ v which is the second member of (2) can be written as
dv y
ma ⋅ v = m(v x x + v y
+ vz z ) .
Each term of (3) contains the component of the velocity vector in that direction and its temporal
derivative, we shall therefore look at it as the temporal derivative of a single quantity. It is
recognized that:
dv x
d 1
=  v x2 + C x  , v y y = d  1 v 2y + Cy  , v z dvz = d  1 v 2z + Cz  ,
dt  2
dt dt  2
dt dt  2
where Cx , Cy and Cz are arbitrary constants, from the second member of (3) we get
d 1
+ vy y + v z z =  v 2 + C ,
dt dt 2
Due to the principle of composition of motions we can put
v 2x + v 2y + vz2 = v 2 and C = Cx + Cy + Cz .
Thus we obtain
d 1
⋅ v = ma ⋅ v =  mv 2 + C '  .
dt  2
putting C'=0 so that the kinetic energy Ec (v ) =
1 2
mv , is cancelled when velocity is nil, we obtain
⋅v = c
& ⋅v
3.2 The relativistic case of the scalar product p
In relativistic dynamics the expression of the quantity of motion vector is
 v2 
p = mγ v = m1 − 2 
 c 
−1 2
where m is constant and γ (v ) = (1 − v 2 c 2 )
−1 2
 v2 
= m1 − 2 
 c 
By definition, therefore,
−3 2
. Deriving (5), therefore, we find
 v2 
v dv
v + m1 − 2 
c 2 dt
 c 
−1 2
 v2
F = m1 − 2
 c
−3 2
 v2
v + m1 − 2
 c
−1 2
4. The law of force and mechanical energy in free motion
We want to show that there are four basic cases where the law of force, acting on a free material
point, allows us to find the constant of motion mechanical energy, using elementary rules of
If the force acting on the material point is constant and can be written as
F = F0 = F0,x xˆ + F0, yyˆ + F0,z ˆz .
The scalar product of the force times the velocity gives
F ⋅ v = F0, x
dz d
+ F0, y
+ F0, z
= (F0, x x + F0, y y + F0, z z + C ) (4.2)
dt dt
Thus we can introduce the function of the coordinates of potential energy
− E p (x,y,z) = F0, x x + F0,y y + F0,z z + C ,
where C usually represents an arbitrary constant
By choosing C = 0 , so that Ep (0,0,0 ) = 0 we can write
F⋅v =
(− E p ) .
F ⋅ v = p& ⋅ v .
Replacing in (4), we find
−E p )= (Ec ) ,
that is to say,
(E + E p ) = 0 .
dt c
We can interpret (5) saying that mechanical energy
Em = Ec + E p
remains constant during the motion of the material point. We point out that the expression of kinetic
energy is always given by (2.4) or by (2.9), whereas the expression of potential energy changes,
case by case, and is deduced from the equation F ⋅ v = E& p .
Replacing (4.3) in (3.1) we find, in classic dynamics,
Em =
1 2
mv − F0, x x − F0, y y − F0,z z
or, in relativistic dynamics,
Em = mc 2 (γ − 1) − F0,x x − F0, y y − F0,z z .
Let us examine the case of weight force and take, to fix our ideas, F0 = −mgzˆ . (4.4) becomes
Em =
1 2
mv + m gz
that is
1 2 1 2 1 2
1 2
1 2
1 2
mvx + mvy + mvz + mgz = mv0, x + mv0, y + mv0,z + mgz0 .
From (4.6), since the Cartesian components of acceleration a x e a y are nil, we deduce the relations
between the components of the position vector and the velocity vector in the motion of free fall of
v x = v0, x ,
v y = v0,y ,
v 2z − v 20, z = g(z0 − z ).
5. Elastic force
Let us consider a material point subject to a force, which obeys Hooke’s law,
F = -k x xˆ .
By calculating the scalar product of force times the velocity, we find
F ⋅ v = -k x
dx d  1 2
= - k x + C .
dt dt  2
By comparing (5.2) with (3.1), we can put
Ep =
1 2
kx + C.
If we take C = 0 , so that Ep (0 ) = 0 , we find
Em =
1 2 1 2
mv + k x
or, distinguishing the relativistic case from the classic case,
1 2
Em = mc (γ − 1) + k x .
6. Gravitational force and Coulomb force
On the level of macroscopic objects, motion is governed by two basic forces only, gravitational
force and Coulomb or electromagnetic force. Mathematically, these two forces can be represented
by the same formula,
F = k r r = k (x + y + z ) (xxˆ + yyˆ + zzˆ ).
If we put k = −G Mm , then (6.1) identifies the gravitational force which the material mass point M ,
located at the origin of the reference system, exerts on the material mass point m , located in the
point identified by the position vector r . On the contrary, if we put k = Qq 4πε r ε 0 then (6.1)
identifies the Coulomb force which the punctiform load Q exerts on the punctiform load q .
If we calculate the scalar product of the force times the velocity, from (6.1) we directly obtain
−3 2
F ⋅ v = k x2 + y2 + z2
) (xv
−3 2
+ yv y + zv z ) .
On the other hand, if we take u = r 2 = x 2 + y 2 + z 2 e u −1 2 = r −1 = (x 2 + y 2 + z2 )
d −1 2
1 −3 2
1 2
2 −3 2
u = − u = − (x + y + z ) ,
−1 2
d 2
u = (x + y + z )= 2(xvx + yvy + zvz ).
, we find
Consequently, by combining (6.3) and (6.4), we have
d −1 2 d −1 2 d
2 −3 2
u =
u = − (x + y + z ) (xvx + yvy + zvz ).
Therefore, by replacing (6.5) in (6.2), we can write
F⋅v = −
d k
k u −1 2 + C ) = −  + C  .
dt  r
By comparing (6.6) with (3.1), which provides the definition of potential energy, we can put
Ep = + C .
If we choose C = 0 , so that the potential energy is annulled at an infinite distance from the origin,
the formula for the constant of motion mechanical energy is
Em = Ec + E p =
1 2 k
mv +
in classical dynamics, while in relativistic dynamics it becomes
Em = Ec + E p = mc (γ − 1) + .
7. Lorentz force
Lorentz force acts on a punctiform electric load moving in a magnetic field
FL = qv × B ,
where q is the algebraic value of the movable load, v is the velocity of the load, B is the magnetic
induction of the field where the load moves. The scalar product of the Lorentz force for the velocity
is nil,
FL ⋅ v = (q v × B ) ⋅ v = 0 ,
because in the double mixed product the same velocity vector appears twice. In this case, therefore,
the motion constant of mechanical energy coincides with the kinetic energy,
Em = Ec .
8. Conservative forces
We underline the fact that the mechanical energy of a material point on which any force F acts, is a
motion constant if F ⋅ v = − E& p . This consideration also suggests the way to define the concept of
conservative force.
9. The exact differential F ⋅ dr = − dE p
Let us take a free material point, immersed in a force field F = F (r) , which describes the trajectory
identified by r = r(t ), with velocity v = rÝ. If the hypothesis F ⋅ v = − E& p (r ) is valid, we find
F ⋅ vdt = − E& (r )dt , that is
F ⋅ dr = −dE p (r ) .
(8.1) expresses the variation in potential energy along an elementary segment of trajectory. If we
take a finished arc of the trajectory, with ends rA ed rB , the variation in the potential energy is given
F ⋅ dr = − ∫ dE p = E p (r A ) − E p (rB ) .
If we consider all the possible trajectories, with ends rA and rB , then (8.2) states that the integral
F ⋅ dr , although calculated along different integration paths, always assumes the same value
expressed by − ∆E p . Therefore (8.2) suggests the well known definition of conservative force; if it
is an exact differential, the force F is conservative, that is,
F ⋅ dr == E p (r A ) − E p (rB ) , whatever
may be the path which has rA ed rB as its ends.
10. Constrained motion, friction forces and the concept of work
Let us consider the material point on which not only active forces but also constraining reactions
act. We shall indicate these constraining reactions with the symbol Φ . The law governing these
forces is not generally known in advance, that is, before finding the law of motion. However, if the
constraining reactions are normal at the trajectory of the material point, that is, normal to the
velocity vector, the mechanical energy is always a motion constant. But if there are constraining
reactions acting on the material point which possess an anti-parallel component to the velocity
vector, the mechanical energy is not preserved. In may cases which occur in practice the antiparallel component is due to the friction or resistant forces. If we calculate the quantity of
mechanical energy dissipated due to the friction forces, this leads to the natural definition of work
done by a force.
11. Constraining reactions normal to the velocity vector
Let us assume that the resultant force acts on the material point
R = F + Φ,
where F represents the active force, while Φ represents the constraining reaction. If
Φ⋅v = 0,
the mechanical energy is always a motion constant. In fact, we can write
(F + Φ ) ⋅ v = F ⋅ v = dp ⋅ v .
Consequently, if F ⋅ v = − E& p , we can conclude that the principle of conservation of mechanical
energy Em = Ec + E p remains valid.
We observe that Φ can represent the tension of the unstretchable wire, T , in the motion of the
mathematical pendulum, or the normal constraining reaction of the supporting plane, N , in the case
of motion along an inclined plane without friction.
12. Friction forces
Let us now suppose that on the material point a resultant force is acting, given by
R = F + Φ + Fr ,
where F represents the active force, Φ the constraining reaction perpendicular to the velocity
vector and Fr = Fr (− vˆ ) a friction force, anti-parallel to the velocity vector. In the case of dynamic
grazing friction, for example, we have Fr = µ d N (− vˆ ) , in the case of viscous friction Fr = bv (− vˆ ).
Remembering that R = p& , from (10.1) we find
⋅ v − F ⋅ v = Fr ⋅ v ,
that is, thanks to (2.5) and to (3.1),
E m = Fr ⋅ v .
Therefore the mechanical energy is not preserved, in fact the product Fr ⋅ v = − Fr v expresses the
rapidity with which it is dissipated. The variation of energy in a time interval [tA , tB ] is defined by
means of the integral of the first and the second member (10.3), that is,
∆E m = ∫ Fr ⋅ vdt .
Since v dt = dr , from (10.3) we obtain dE m = Fr ⋅ dr and we can re-write (10.4) in the form
∆E m = ∫ Fr ⋅ dr ,
where rA and rB are the ends of the trajectory arc described by the material point in the time
interval [tA , tB ].
We should point out that, in general, (10.4) or (10.5) do not allow us to calculate ∆Em and
therefore of the integral which appears as the second member, if the dependency of the velocity on
the time v = v(t ) is not explicitly known. Therefore, for the actual calculation of the dissipated
energy, we must first solve the differential equation
dp ˆ
ˆ + F ⋅T
⋅T = F⋅T
obtained from (10.1) by multiplying in scalar fashion the first and second member by the versor T
ˆ = 0.
tangent in every point of the trajectory, replacing R = pÝ and remembering that Φ ⋅ T
13. Work
Traditionally, the integral
Fr ⋅ dr is called work done by the resistant force Fr and is indicated by
the symbol
L = ∫ Fr ⋅ dr .
The physical meaning of (11.1) is supplied by (10.4) or by (10.6). We would add that this definition
of work also clarifies the etymological meaning of the term , which in Italian derives from the Latin
labor and, apart from work, action, enterprise can also denote labor, toil, worry.
14. Conclusions
The teaching of physics to young people must include activities suitable to develop awareness of
the role of mathematics in physics and the ways in which it is used in different circumstances. In
physics, it is the integration of the two planes, phenomenal and formal, which creates the meanings;
it builds those concepts which allow us to describe and interpret the world, using models able to
explain classes of phenomena through theory. The well-known limits in managing mathematics are
mainly of a conceptual character and due to the fact that students are not used to looking for the
significance of the calculus which they have learnt to do. Attention must therefore be paid to this
problem and time must be spent on didactic research in order to construct proposals able to
encourage the acquisition of a thorough knowledge of quantitative features of physics.
Starting from the following considerations:
- The principles of conservation are part of the deepest roots of physics and, in particular, are
decisive in the development of mechanics.
- Every principle of conservation entails the existence of one or more constants of motion.
- Among these, mechanical energy is particularly important, which allows us to identify interesting
characteristics of motion of a material point, establishing a relationship between position and
We have developed a proposal which leads to identify mechanical energy and its conservation
starting from Newton's second law, without the prior introduction of the concept of work. Its
coherence and completeness are recognized by the possibility of obtaining the expressions of
mechanical energy in the same way when the forces in play are of various types: from gravitational
forces to Coulomb forces, from weight to elastic force, from Lorenz force to restraining reactions.
The definition of conservative force can be recognized in terms of a corollary with respect to the
prior introduction of a potential energy. The case of forces of friction introduces the nonconservation of mechanical energy and the concept of work. The itinerary proposed, which does not
require mathematical knowledge and/or ability above those which every student learns in a
secondary school specializing in scientific subjects, allows the student to familiarize himself with
some important ways of looking at physics through mathematics, such as using the scalar product,
vectorial de-composition and the properties of temporal derivatives with respect thereto, the search
for primitive functions, the identification of quantities which are conserved.
If we do not choose it as an alternative to the traditional way of introducing mechanical energy and
its conservation, it represents a powerful opportunity for reflecting on concepts which connect the
way of looking at the world through forces with the way of looking at the world through energy.
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