Sample Physics Problem - Modeling

Background

Though many areas of physics lend themselves to visualization, Newtonian physics stands out as an excellent example of how 3-D and 2-D visualizations can help support learning about physical principles. Formulas representing the principles of Newtonian mechanics often use coordinate space values both as independent and dependent variables. These values can not only be represented in traditional graphs, but also as symbolic models. In this module, we will take the example of a cannonball being shot from a cannon and model it in a 3-D modeling package, TrueSpace.

Data Gathering

Before modeling, data needs to be generated based on the initial conditions of the cannonball and its resulting trajectory. The following equations were used to generate this data:

The cannonball is initially launched at a 45 degree angle at an initial velocity of 20 m/s. Based on this initial condition, the following values for the x and y coordinates and the x and y velocities were generated:
 time x-position y-position x-velocity y-velocity
0.00 0.00 0.00 14.1 14.1
0.48 6.77 5.67 14.1 9.40
0.96 13.54 9.31 14.1 4.69
1.44 20.3 10.1 14.1 0.00
1.92 27.1 9.31 14.1 -4.69
2.40 33.8 5.67 14.1 -9.40
2.88 40.6 0.00 14.1 -14.1

Modeling

Once data is generated for representative points along the trajectory, a animated model can be created to represent the problem. Each coordinate point will represent the ball at a specific point in time. TrueSpace does not accurately represent all of the physical properties of the model; for, example, mass of the cannonball and its elasticity can not be represented. What can be accurately represented, however, is the location in space at each of the representative points. Using these representative points, TrueSpace will extrapolate the location in time and space of the ball between each given point. The number of intermediate points calculated can be set by the user. Though this extrapolation may not be entirely accurate, for a model that will only be analyzed visually, it will suffice. Needless to say, the more representative points which are calculated and entered into the model, the more accurate the visualization will be.

Start the visualization by modeling a cannon. Primitives, such as cylinders, spheres, and tori can be used to create a fairly representative cannon. The 'glue' tool can be used to create a parent child hierarchy of parts and allow the cannon to be easily moved as one object. Your cannon might look something like this:

Next model a cannonball to fit within the cannon. For ease of calculating the movement of the cannonball, place the center of the ball at 0,0,0. Though it is somewhat counter intuitive, the cannon can now be moved over the ball. The cannon barrel can be aimed by rotating it to 45 degrees above the horizon. Your positioning can be double-checked after the ball is moved to its second coordinate point:

The animation is created by capturing a 'keyframe' of the ball as it is moved to the representative locations in space. Using the animation controls in your modeling package, a keyframe is captured for the ball located at each of the X.Y coordinates you previously calculated. You may find it easier to use multiple orthogonal viewports to help you visualize where the ball is moving to.

Since the x and y coordinates were calculated at equal time intervals, any equal spacing of frames should work for a representative animation. Ten frames between each keyframe creates an animation of reasonable length and size.With the knowledge that the animation can be generated at either 24 or 30 frames per second, you can also calculate time intervals and use keyframe spacing to generate a truly realistic animation of the initial conditions.

With the animation created, it can be rendered can saved as an animation file from a number of different viewpoints. Different viewpoints can be used to isolate various dimensions by creating a projection which collapses one of the three dimensions.

These animations can be played with the MediaPlayer accessory included with Windows or 'avi aware programs like PowerPoint.

Further Work

Many similar Newtonian physics problems can be generated using similar techniques. Though the example above only made use of 2 dimensions, you can easily incorporate a third dimension. For example, a crosswind could be added to the above problem.
 

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