|Module 15 - Particle Motion and Parametric Models|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self Test|
|Lesson 15.1: Motion Along a Line|
In this lesson you will model the motion of a particle that moves along the x-axis using parametric equations. The motion of the particle will be illustrated using the animation feature of the TI-83. By developing different parametric equations to model the same movement, you will see that parametric equations that model a particle's movement are not unique.
Suppose a particle moves along the x-axis so that its position is given by the equation below, where t represents time in seconds.
You can use your TI-83 to illustrate the motion of the particle by defining its movement with parametric equations, which were explored in Module 4. Displaying the parametric equations in animated graph mode will help determine when the particle is at rest, when it is moving right, and when it is moving left.
To graph parametric equations on the TI-83 you need to change the graphing mode.
The particle's horizontal position along the x-axis is given by x(t) = 2t3 - 9t2 + 12t + 1 and its vertical position is constantly zero because the particle is moving along the x-axis.
Setting the Graph Style to Animate
When an equation is graphed in Animate style, a small circular icon moves along the path defined by the equation. You can view an animation of the particle's motion by changing the Graph style of the parametric equations.
Setting the Viewing Window
With the values shown, t will initially be 0 and then increase by steps of one tenth until it is 4. For each value of t the position of the particle, as determined by the corresponding x and y-values, will be plotted.
You should see a circular icon move along the x-axis. This circle illustrates the path of the particle over the time interval from t = 0 to t = 4 as defined by x(t) = 2t3 - 9t2 + 12t + 1.
The screen above shows an intermediate view of the animation when the graph is displayed.
You can see the animation again by pressing [DRAW] and selecting 1:ClrDraw.
Animation Using the Trace Feature
The Trace feature can produce a similar animated effect but the speed of the animation is under your control.
Try pressing the left or right arrow and holding it down.
Notice that the right arrow key moves the particle forward in time, which may not always coincide with motion to the right, and the left arrow key moves the cursor back in time, which may not always coincide with motion to the left. The value of t and the x- and y-coordinates are shown at the bottom of the screen as you trace the particle's motion.
15.1.1 Use the cursor movement keys to estimate the time interval(s) when the particle is moving right. Click here for the answer.
15.1.2 Estimate when the particle is moving left. Click here for the answer.
15.1.3 When does the particle appear to change direction? Click here for the answer.
Illustrating the Path Over Time
The animation on the x-axis is a realistic model for the motion of the particle. However, it's often helpful to have a static graph that can be used to help visualize and study motion. Make the following additions in the Y= editor to better illustrate the motion of the particle over the four-second interval.
Set X2T = X1T by following the procedure below.
Setting the Path Style
The graphs defined in X1T, Y1T and in X2T, Y2T can be drawn at the same time by setting the Graph mode to Simultaneous.
Displaying the Graphs
Notice the advantage that this visualization has over the original "Path" style. Any trail left on a 1-dimensional motion hides changes in direction and doesn't give any information about when the particle is in what position.
15.1.4 Identify the features of the graph that indicate when the particle appears to change direction, and then approximate the times when the particle changes direction. Click here for the answer.
A Different Model of the Particle's Motion
Different parametric equations can be used to make another model of the same particle motion. The functions below represent the same motion shown earlier, but time is now shown on the horizontal axis and the particle's position is represented vertically.
In the graph above, the y-values represent the particle's position and the x-values represent the corresponding times. This model has the advantage of having time as the x-variable and position as the y-variable, which is comparable to the representation normally used in the FUNCTION Graph mode.
Instantaneous Velocity of the Particle
The particle's instantaneous velocity function can be displayed along with its position function. Defining new parametric equations where the x-values represent time and the y-values represent the particle's instantaneous velocity (the derivative of the function in Y1T) can do this.
15.1.5 Describe how these graphs show when the particle is moving left, moving right, and at rest. Click here for the answer.
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