1. Four examples...
Modellus 4
2008 http://modellus.fct.unl.pt
The Dino starts moving to the right with acceleration
pointing to the left…
Before starting the motion, it is possible to define the
initial position, the initial velocity and the acceleration, Interactive Modelling
with Mathematics
dragging the corresponding Vectors.
In the graphs drawn, the second graph represents the
derivative of the first and the third graph represents
the derivative of the second…
Modellus is designed to introduce students
and teachers in scientific computation,
namely through the analysis and exploration of
mathematical models based on functions, iterations
and differential equations. For example, it allows
the construction and analysis of models illustrating
Newton’s reasoning when comparing the motion of
A stroboscopic photo of a collision was placed as a
a projectile and the motion of a satellite.
background in the workspace... The development of Modellus 4 is being done with generous support from the Portu-
guese Ministry of Education (DGIDC), the Portuguese Foundation for Science and
Three Vectors were created to measure, in an arbitrary Technology (FCT), the European Community, the Institute of Physics (UK) and the
scale, the linear momentum of each object, before and research unit UIED-FCTUNL.
after the collision…
Dragging the Vectors it is easy to check the
conservation of the linear momentum...
The Modellus installation program includes many
examples and others are regularly added to the web
site http://modellus.fct.unl.pt.
Besides the Modellus files, the web site also contains
several documents for students and teachers from
basic learning levels to higher education. These
The side of a square was defined… documents are regularly updated.
The area and the perimeter were calculated…
The square was represented by Geometrical Objects In the figure, an example that illustrates the use of
(Segments), which may be linked in succession… sinusoidal functions in an oscilloscope. With this model
Several Pens were created to represent relations it is possible to analyse the frequency, the amplitude
between area and perimeter, etc… and other aspects of periodic sinusoidal signals.
Model of the motion of a bouncing ball: the vertical
A model was created using a system of ordinary trajectory of the ball and several graphs for physical
differential equations (which represent the instantaneous quantities as functions of time may be visualised
rate of change of products and reactants…). simultaneously. It is also possible to attribute a certain
The model assumes plausible reaction velocities… initial velocity to the ball and to study the ideal case
Several Level Indicators (Bars) and Pens were created to when there is no energy dissipation.
represent parameters and initial values…
Play / Pause executes the model…
Using the mouse it is possible to dynamically change
the concentration values and observe how the system
behaves when there is a change in the concentration of
Modellu
Modellus 4
the chemical species…
2008 http://m de
http:// odellus.fct.unl.pt
http://modellus.fct.unl.pt
/ ct t

2. Modellu
Modellus 4
2008 http://m de
http:// odellus.fct.unl.pt
http://modellus.fct.unl.pt
/ ct t
5
1 How to build a Once the Vector representing
the initial velocity is defined,
using its components, the tip of
model? the Vector may be dragged to
Start by defining the functions that describe one or
5 attribute adequate values to the
components of the initial velocity.
several properties/quantities of the phenomenon or
mathematical object under study… 1 Alternatively, these values may
be directly introduced in the
For example, the equations highlighted in the figure
Mathematical Model window or in
refer to the motion of a projectile when air resistance
the Parameters ribbon.
is negligible.
2 6
It is possible to create
objects in the workspace to
represent the variables of the The Play / Pause button starts the simulation of
model, either using the icons the projectile motion. The domain and step of the
or the right button of the independent variable t are defined in the Independent
mouse.
2 Variable ribbon: by default, the domain is [0; 50] and
the step is 0.1. These values may be changed.
For example, a Particle may
be created to represent the 6 In this example the motion was interrupted when
projectile. t = 7.90…
3 7 7
3
To each object of In the example shown in the figure, it was also created
the workspace it is a Pen to trace the graph of y as a function of t in the
possible to assign workspace.
defining properties.
For example, x and Each object in the workspace may have its own scale,
y are associated to independently of the other objects, as well as other
the Particle as the features (colour, etc.). The Autoscale button may be
particle coordinates. used to adjust a scale that is a function of the minimum
and maximum values of the quantities defining each
object.
4 8 8
Since in the model Each model may have as many objects as necessary
the components of to better visualize the phenomenon, the physical
the initial velocity are quantities involved or the corresponding mathematical
indicated by v0x and objects.
v0y, it is possible to
4 represent the initial To the projectile model were added functions describing
velocity by a Vector. the potential energy, the kinetic energy, etc, as
functions of time, as well as the velocity and the
acceleration in each instant.
Modellu
Modellus 4
2008 http://m de
http:// odellus.fct.unl.pt
http://modellus.fct.unl.pt
/ ct t