Content uploaded by Michael Pohlig
Author content
All content in this area was uploaded by Michael Pohlig on Oct 06, 2021
Content may be subject to copyright.
September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in
Teaching˙Relativity˙Computer˙aided˙modeling page 1
1
Teaching relativity: Computer-aided modeling
F. Herrmann and M. Pohlig∗
Institute for Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT),
Karlsruhe, Baden-W¨urttemberg, Germany
∗E-mail: pohlig@kit.edu
www.kit.edu
Mathematical derivations alone do not necessarily lead to physical understanding. Tools
that can replace the mathematical treatment of a physical process and at the same time
increase the physical understanding are computer-aided modeling programs, also called
system dynamics software. Examples of such software are Stella, Berkeley Madonna,
Wensim, Dynasys, Powersim or COACH 7. They solve differential equations and sys-
tems of differential equations with numerical methods. One works with a graphical user
interface. We want to show how such a software can be used to get from a non-relativistic
model to a relativistic model with only minimal modifications. Equating mass and en-
ergy alone, ensures that the model provides essential statements of relativistic dynamics:
the existence of a terminal velocity for all physical motions, the relativistic dependence
of the velocity of a body on its momentum, the relativistic relation between momentum
and energy of a body.
Keywords: Teaching relativity, computer-aided modeling, system dynamic software,
Coach 7.
1. Introduction
There are two reasons why learning relativity is difficult. First, there is a widespread
belief that relativity is essentially a physics of reference frame changes. Second,
mathematics, which seems to be indispensable for a first approach to relativity ac-
cording to common usage, is a major, even often insurmountable, hurdle for many
students. Therefore, the structure of our course follows two didactic recommenda-
tions: 1. avoid reference frame changes 2. reduce the use of mathematics to the
most necessary. We fulfill the first recommendation by teaching dynamics before
kinematics. Instead of postulating that the speed of light is invariant under refer-
ence frame changes, we declare from the very beginning that energy and mass are
the same physical quantity. We call it energy when its value is measured in joules
and mass when measured in kilograms. The second recommendation can be fulfilled
if we use a suitable modeling software, a so-called system dynamics software (SDS).
In a SDS, physical quantities and their relations are described by graphical symbols
and are intuitively understandable. Since its usage is self-explanatory to a large
extent, we do not need to go into details of the handling here.
As a representative of such a software we use COACH 71.
September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in
Teaching˙Relativity˙Computer˙aided˙modeling page 2
2
2. Teaching Relativity using a system dynamics software
2.1. Momentum flow into a body
We create a model for a body which is accelerated from rest. Its momentum, which
is zero at the beginning, increases as a constant momentum current flows into it
(as a force is exerted on it). In a first step we treat this model non-relativistically.
In the second step, the model is transformed into a relativistic model by a small
modification.
In a SDS model, momentum pand other substance-like (extensive) quantities
such as electric charge Q, energy E, or entropy Sare represented by boxes. All
these quantities change their values by an inflow or an outflow. (Entropy can also
change its value by being produced). In the model, these currents are represented
by thick arrows, Figure 1. Since in our lessons momentum is not introduced as a
derived quantity, it gets its own unit of measurement, the Huygens, abbreviated Hy.
The unit of its current is Huygens per second (Hy/s). The unit Hy is SI-compatible.
We thus have 1Hy/s = 1 N.
Fig. 1. A temporally constant momentum current Fleads to a linear increase of the momentum
p.
Since the body is accelerated from rest, its momentum at the beginning of its
motion is 0 Hy. We assume the momentum current to be constant in time, we set
e.g. F= 1 N. When the model is started, the simulation begins running and the
actual momentum is calculated for previously defined time steps.
Fig. 2. The velocity is calculated from momentum and mass.
In order for our model to be able to tell us something about the velocity of the
body, we must add the mass mof the body to the model, in Figure 2 represented
by a circle. We choose m= 1 kg for a first simulation and m= 2 kg for the second
one. The velocity is calculated using v=p/m. This relationship is stored in circular
symbol for the velocity v. Thin, red arrows are pointing from the symbols for pand
September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in
Teaching˙Relativity˙Computer˙aided˙modeling page 3
3
mto the symbol of v. This ensures that the actual values of pand mare available
for the calculation of vat any instant of time during the simulation. In concrete
terms, this means that while the simulation is running, the momentum is divided
by the mass at fixed time steps, and thus the actual velocity vis calculated.
Fig. 3. v−pdiagrams or bodies of masses 1kg and 2 kg.
The dependencies of the various variables occurring in the model can be dis-
played graphically in output windows. Figure 3 shows the v-pdiagrams for two
bodies of masses 1 kg and 2 kg.
2.2. Together with momentum energy is flowing into the body
We now add the kinetic energy of the body to our model, Figure 4..
Fig. 4. The energy current Pinto the body is calculated from the velocity vof the body and the
momentum current F(force).
Also the energy Eis represented by a box. Since the body is accelerated from
rest, not only the momentum but also the kinetic energy must have the initial value
Eint = 0. Like the momentum current, the energy current (power) into the body
is represented by a thick arrow. It can be calculated as P=vF . The software
provides the energy-momentum diagrams for m= 1 kg and m= 2 kg , Figure 5.
These graphs could be described by the formula
September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in
Teaching˙Relativity˙Computer˙aided˙modeling page 4
4
Ekin =p2
2m.(1)
However, this equation has not been used by the SDS.
Fig. 5. E−pdiagrams generated with the SDS for m= 1 kg and m= 2 kg.
2.3. From the classical to the relativistic model
The two models described up to now serve as basic models for others, such as for
”free fall in the gravitational field of the earth”, for ”falling with friction” and
others.
If momentum is replaced by other substance-like quantities such as electric
charge or entropy, analogous models are obtained from electricity and thermody-
namics, respectively. In this way, students learn to use the modeling software as
a tool in other subfields of physics. So when they create relativistic models, they
are already familiar with the tool. We now want to create a relativistic model by
Fig. 6. Relativistic model: The constant mass is replaced by the total energy of the body.
modifying our non-relativistic model appropriately, Figure 4. For this purpose, only
a small modification has to be made. The mass, which was originally a constant, is
September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in
Teaching˙Relativity˙Computer˙aided˙modeling page 5
5
now identified with the energy2according to
E=k·m(2)
Here, the energy is no longer the kinetic energy but the total energy of the body.
Figure 6 shows the modified model. The initial value of the energy is no longer 0
J, but it is equal to the value of the energy that the body has at rest. We choose
Eint = 10 J and for another simulation Eint = 5 J. To transform the unit kilograms
into joules, we have to multiply the mass by k. We first set its value arbitrarily to
k= 16 J/kg. The actual true value of kwill be discussed later. The velocity of the
body is now calculated according to
v=p
m=kp
E.(3)
Fig. 7. v−pdiagrams obtained with the relativistic model.
As Figure 7 shows, the v-pdiagrams are different from those in Figures 3. It
can be seen that the velocities of both bodies approach a common terminal velocity
of 4 m/s. One ”plays” with further, freely chosen values for the rest energy and
always finds the same terminal velocity. Furthermore, the v-pdiagrams show that
light bodies reach the terminal velocity quicker than heavy ones. Finally, one rec-
ognizes that for sufficiently small momentum values the velocity of a body increases
linearly with momentum. This confirms what is already known from non-relativistic
mechanics.
The diagrams are graphs of the relation
v(p) = p
m=kp
E=kp
pE0
2+kp2.(4)
However, we don’t need to know this equation, or to derive it, in order to be able
to read the important properties of this relationship from the diagrams.
September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in
Teaching˙Relativity˙Computer˙aided˙modeling page 6
6
The diagrams show that for relativistic motions the following rules hold:
- small momentum values: v∼p
- large momentum values: v=√k
By choosing other initial values for the energy and other values for k, one easily
gets convinced of the generality of these rules. For the diagrams in Figure 8, the
values 9 J/kg, 16 J/kg and 25 J/kg were selected for k. The terminal velocity results
to be 3 m/s, 4 m/s and 5 m/s, respectively.
Fig. 8. v−pdiagrams for different kvalues.
We see that the value of the terminal velocity is just the square root of k. While
kused to be merely a conversion factor between kilogram and joule, it now acquires
a physical meaning. Moreover, from the fact that a conversion factor for units is
universal, the terminal velocity also has a universal value. It must be a universal
constant. For this reason, we give the square root of k, i.e. the terminal velocity, a
symbol of its own, namely
c:= √k(5)
Up to now, we had chosen the value of kand thus also that of carbitrarily. But
which value has nature given to kresp. c? Experiments show that k= 9 ·1016 J/kg
and therefore √k=c= 3 ·108m/s.
Another relationship that our model provides in the form of a diagram is that
between energy and momentum, Figure 9. The diagrams shown are graphs of the
equation:
E(p) = qE0
2+kp2=qE0
2+c2p2(6)
For small momentum values we get the non-relativistic relation between kinetic
energy and momentum, shifted by the rest energy of the body:
E(p) = p2
2m+E0(7)
September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in
Teaching˙Relativity˙Computer˙aided˙modeling page 7
7
Fig. 9. E−prelation in the relativistic model. The rest energy was set to 5 J and in the second
run to 10 J.
Moreover, we see that the ratio of energy and momentum is the same for large
values of the momentum.
E(p) = √k·p=c·p(8)
3. Conclusion
A non-relativistic SDS model which describes the behavior of a body whose mo-
mentum increases linearly, i.e. into which a constant momentum current is flowing,
becomes a relativistic model by a small modification. The change consists in equat-
ing the quantities mass and energy. The new model then provides diagrams that
are well-known from relativistic physics. The software provides them without using
the respective equations. From these diagrams, important results of relativity can
be interpreted and understood. The use of computer-aided modeling in teaching
has the advantage of getting rid of the mathematical ballast and that one can con-
centrate on the physical content. Our experience shows that even younger students
can easily learn a software like COACH 7 and use it to work out challenging results
in relativistic dynamics.
References
1. Coach 7 for Desktop and Tablet: CMA (Centre for Microcomputer Application
https://cma-science.nl/downloads_en accessed July 26 2021
2. Herrmann, F. et al., The Karlsruhe Physics Course: Mechanics: for the upper sec-
ondary school, (2019)
http://www.physikdidaktik.uni-karlsruhe.de/index_en.html accessed August
2021.