Can Vector-Valued Functions Accurately Model Particle Motion in Two Dimensions?

[opticaltempest]

I am trying to model the position of a particle as a function of time. The particle is being modeled in the x-y plane. I will be using Maple 10 to do the regression, graphing, etc.

Here is a movie of the particle in motion. It's actually an ant. I plan on modeling the position of the ant as a function of time in order to estimate the velocity and acceleration of the ant.

http://www.zippyvideos.com/6826494495890326/ant/"

I plotted where the ant crosses a vertical line. Each grid is 1cm by 1cm. When the ant crosses a vertical line in the video, I estimated its vertical distance from the 0cm line, noted its exact horizontal distance from the 0cm line, and noted the exact time that frame occurred in the video. Each row in the spreadsheet corresponds to when the ant was on a vertical line.

http://img227.imageshack.us/img227/7903/pointplotwithcoordinatetk5.png

http://img135.imageshack.us/img135/3810/dataqq0.jpg 1. I am trying to model this as a vector-valued function. Does it need to be modeled as a vector-valued function? I believe I must use a vector-valued function because without using vectors how can I completely describe both the horizontal and vertical motion as a function of time?

2. How do I model a vector-valued function? Should I model the vertical distance from the origin as a function of time in order to get the x-component of the vector? Then model the horizontal distance as a function of time to get the y-component of the vector? Then combine these two to form a r(t)=[x-component function]i+[y-component function]j ?

3. Would a graph of this vector-valued function appear anything like my plotted points?

Thanks

 

[mezarashi]

Yes, I believe you are correct in all your doubts. Where you let r represent the position vector of the particle from 0,0.

r_vector(t) = x(t) i + y(t) j

Differentiating this, you will get the velocity vector. Once more and it will be the acceleration vector.

 

[kyle8921]

for your questions, let me provide some guidance on how to approach modeling this particle's motion.

1. It is important to model this as a vector-valued function because it allows you to fully describe the motion of the particle in both the horizontal and vertical directions. Without using vectors, you would be limited to only describing the motion in one direction at a time.

2. To model a vector-valued function, you can use the horizontal and vertical distances as functions of time, as you suggested. This will give you the x and y components of the vector. You can then combine these two components to form a vector-valued function r(t)=[x(t), y(t)].

3. A graph of this vector-valued function will not necessarily look like your plotted points because the plotted points only show the position of the particle at specific points in time. The vector-valued function will show the continuous motion of the particle over time. However, you can plot the vector-valued function on a graph to see how the particle's position changes over time in both the x and y directions.

In terms of using Maple 10 for regression and graphing, it is a powerful tool for modeling and analyzing data. You can use Maple's regression tools to determine the best-fit curve for your data and then graph the vector-valued function using Maple's graphing capabilities. Just be sure to include all of your data points and any assumptions or limitations in your model to accurately represent the motion of the particle.