This tutorial describes how to use the Newton Tool to solve for the motion of a particle experiencing a constant acceleration. In addition, this tutorial illustrates how the adaptive step factor can change the data generated by the Newton Tool. If you have not already done so, launch the Newton Tool.
Acceleration Panel
There are many physical examples for constant accelerations. We typically treat gravity as a constant acceleration of -9.8 (m/s2). Another example is a car with an acceleration of 10 mph/s.
For this particular tutorial the acceleration equation is simple. We will solve for motion of a particle accelerating at 10 m/s2 from rest, so enter 10 as the equation as shown below.
Initial Conditions
To illustrate how the adaptive step factor affects the results, begin by using the initial conditions as shown below.
For each trial we'll use all values the same except the 'Adaptive Step Factor', which we will change to see how this affects the plots.
Plots
Notice, for this first trial we're using the default 'Adaptive Step Factor' of 1. Choose to plot x position versus Time. Also, on the 'Plot Results' panel, check the box to display the data points so the difference between the plots will be more concrete. After checking this box, you have the option of turning off the line that connects the data points as well.
The plot generated when the adaptive step is 1 looks like this:
Step Factor = 1
The above plot is very smooth and has a lot of data points. Now rerun the calculation with all settings the same as before except the adaptive step factor. Set that to 10. Then generate a new x position vs. Time plot:
Step Factor = 10
Now rerun the calculation and use a step size factor of 100. Generate the following plot:
Step Factor = 100
Finally, rerun the calculation with an adaptive step size factor of 1000. Generate the following plot, which is significantly less smooth than the others:
Step Factor = 1000
From this example it is easy to see how the 'Adaptive Step Factor' changes the number of data points generated and how smooth the plot is. But how much accuracy has been lost?
Table
To see how accurate the algorithm is, click the View Data button in the Step = 1000 plot panel. The following table appears:
Since the acceleration, a, is a constant, we know that the solution for x(t) is x(t) = (1/2)·a·t2. With a = 10 m/s, we see that x(t) = 5·t2. The x position in the table agrees quite well with that calculated from this formula. That is because the Runge-Kutta method we employ is highly accurate, especially for problems with solutions that are simple powers of time, like this one. The conclusion is that increasing the adaptive step size factor will decrease the number of points calculated, which decreases the smoothness of any resulting plots, but it does not typically limit the accuracy much, unless the step size factor is increased to unreasonably large values (> 1000).
Challenge
Can you find an acceleration equation that does not remain as accurate as this example did over a wide 'Adaptive Step Size' range? What is different about the equation you chose?
