Modelling Motion Using Dynamic Geometry - Geoff Wake
There are odd moments that spark or rekindle an interest in a particular topic in mathematics - times when something you see, hear or read allows connections to be made or a germ of an idea to form. I had such a moment when Jill Bruce showed me some of the work she was doing just before her untimely and tragic death in June. She was developing ideas using dynamic geometry software to allow engineering students to gain an insight into mathematics and how it can be applied in their discipline. What she showed me gave me the inspiration to develop the ideas that the following article reports.
The potential of dynamic geometry software extends far beyond its use as a pure geometric tool. It gives us another powerful medium to use whenever we are working in a domain where a spatial image is useful.
Here I give some initial thoughts about how such software can be used to develop and explore graphs when modelling motion. The method described allows one to gain a dynamic visual image when using a step-by-step method to solve differential equations. However, these initial explorations are perhaps best thought of in terms of kinematics before extending to more general differential equations; most students working at advanced level will have some feeling for the types of motion under investigation.
Figure 1
Figure 1 shows velocity-time and displacement-time graphs for a body moving with constant acceleration. To be more precise the body has an initial displacement x0, initial velocity v0 and acceleration a, in the positive direction of x as indicated in Figure 2. The graphs could perhaps be validated by appealing to qualitative, or even quantitative, investigation of say the motion of a ball rolling down a slope - Galileo’s experiment.
Figure 2
The primary aim of the graphs produced using dynamic geometry software is to examine form rather than numerical values. Because they were built using a step-by-step approach with such software, it is possible quickly to determine what would happen if...
- the initial displacement were zero
- the initial velocity were negative
- the acceleration were negative
and so on.
Figure 3
The key principles used are that acceleration is given by the gradient of a velocity-time graph, velocity is given by the gradient of a displacement-time graph and that over short time intervals these can be taken to be constant.
In this initial description the acceleration is taken to be constant. Figure 3 shows a segment, a, that is used to control the magnitude of this. The length of a is used to construct the height of a right-angled triangle the base of which is taken to be unit time (Figure 3). The hypotenuse of the triangle therefore gives the direction that a line representing velocity on a velocity-time graph would need to take. The gradient of this line will be
This is then used to construct a segment, starting at velocity v0, which has this gradient (the current acceleration) for a step in time (controlled by the segment labelled “step&rdsquo;). So at the end of a “step&rdsquo; in time the velocity of the body is:
as indicated on the vertical axis of the graph. The process of construction can be repeated as often as wished, using a macro or script if desired, so that a velocity- time graph such as that in Figure 1 is built.
Figure 4
The next step is to build, in a similar way, a displacement-time graph. This uses successive velocities from the velocity-time graph to construct segments, which have the correct direction, for each successive small step in time. Again the end product is illustrated in Figure 1.
Figure 5a
Figure 5b
Figure 6
Once the graphs have been built in this way it is possible to investigate the effect of altering initial values, the magnitude of the acceleration and/or the time step length. Some possibilities are shown in Figures 5a and 5b, but the power of the dynamic nature of the graphs is lost here on the page. Similar graphs can be built using a graphic calculator programme with the final output looking similar. However, such an approach does not allow the same immediacy of response when variables are altered, and using dynamic geometry software allows one to visualise more clearly how the graphs are built.
Care needs to be taken when investigating, for example, the motion of a ball falling freely under gravity. Here the positive or negative direction of displacement, velocity and acceleration needs to be carefully considered. Figure 6 shows graphs resulting when such motion is modelled. Here the upward vertical displacement, y, is taken to be positive.
Figure 7
As a next step I have looked at modelling projectile motion in two dimensions. By combining in vector form, horizontal and vertical components of displacement, the path of such a projectile can be plotted as in Figure 7. Already, the relatively simple, but powerful step-by-step method can be used to model quite complex motion.
Now I am ready to consider motion where the acceleration is not constant. The first motion of this type that I modelled is resisted motion where a= a1 - cv with a1 and c being constants. The magnitude of a1 and c can be controlled by altering the length of segments on my dynamic geometry sketches so that one can investigate their influence on the motion. Resulting graphs are shown in Figure 8.
Figure 8
Figure 9
The second motion with non-constant acceleration that I investigated is simple harmonic motion where a = -kx. Figure 9 shows a resulting set of graphs.
The figures shown here were drawn using The Geometer’s Sketchpad, but could be developed using any similar software. These are only initial ideas of how we might use this software to gain a new, and hopefully clearer, insight into another branch of mathematics. I hope that other readers may be able to use and develop them further.
Geoff Wake works at the Mechanics in Action Project, University of Manchester.
Useful Reading
Nuffield Advanced Mathematics Book 2 (1994), Longman
Nuffield Advanced Mathematics Mechanics 1 (1994), Longman
