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September 8, 2021 10:49 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in

Teaching˙Relativity˙Computer˙aided˙modeling page 1

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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.

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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

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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

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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

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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.

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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)

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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.