Movement of a point over time question

[frumpydolphin 0]

Background

- System of points a, b, c and q

- a, b, and c exert a push on q moving it to the end of a vector that is 2 times the vector from a, b, or c to q.

- a, b, and c do this at the same time, after each cycle the x and y values of q are the average of the end points of the vectors.

- This happens n times

 

First derivation

The final x value is the sum of the averages of all the vectors endpoints divided by the number of iterations so n.

Same goes for y

 

 

I further thought why keep it at intervals when it can be a constant force.

From this I derived

The integral from 1 to n of the average of the averages of the vector endpoints in respect to the x value of q.

Same for y

Hope I wasn't too vague with it but am I at least close to right?

Also how do I insert equations?

This wasn't homework by the way don't rage at me.

 

[mathematic 83]

You need to clarify your description.

[frumpydolphin 0]

Ya, so there is a point(q) starting with coordinates x_​q and y_q​. Three other points a, b, and c have there x and y coordinates labelled in the same manner​. A, b, and c exert a "push" on q each second suppose, and move it to the endpoint of a vector 2 times the magnitude of vector a,b, or c to q. The direction remains the same and the origin of the vector is still a,b or c. The average of the endpoints created by these vectors, 1 for a, 1 for b, and1 for c, gives q's position after 1 second. After n times though q's position would be the average of all these averages.

I derived the sum to n from 1 of the small averages divided by the number of seconds, so n, to get the final x and y values of q. (really need to figure out how to use equations)

Hope that helps but that is only the first part.

[mathematic 83]

 

 

A, b, and c exert a "push" on q each second suppose, and move it to the endpoint of a vector 2 times the magnitude of vector a,b, or c to q.

All at the same time or 3 separate cases?

[frumpydolphin 0]

Same time

[mathematic 83]

 

 

The direction remains the same and the origin of the vector is still a,b or c.

The direction of what? Origin of what? Origin a,b, or c - very ambiguous.

[frumpydolphin 0]

Geez I suck at this, a, b, and c act as the origin of the vectors, the direction is the direction of the vector from a, b, or c towards q

[mathematic 83]

Your overall description is confusing. a,b,c are different vectors with differing origins and directions. Your description talks about one origin and one direction. I can't put it together.

[frumpydolphin 0]

Well, I can't think of a good way to explain so sorry for time waste. Thanks for trying to help though.

[mathematic 83]

Try to describe the scenario using equations. It should clarify for yourself what you intend.

[renerpho 9]

I understand it as follows:

You start with a vector [math]\vec{q}(0)[/math], and (simultaneously) apply forces [math]\frac{2}{3} (\vec{a}-\vec{q}(0))[/math], [math]\frac{2}{3} (\vec{b}-\vec{q}(0))[/math] and [math]\frac{2}{3} (\vec{c}-\vec{q}(0))[/math]. You repeat [math]n[/math] times. That means, in the n-th step you apply forces [math]\frac{2}{3} (\vec{a}-\vec{q}(n-1))[/math], [math]\frac{2}{3} (\vec{b}-\vec{q}(n-1))[/math] and [math]\frac{2}{3} (\vec{c}-\vec{q}(n-1))[/math] on [math]\vec{q}(n-1)[/math] to get [math]\vec{q}(n)[/math].

You can give an explicit formula for the result: [math]\vec{q}(n)=\vec{q}(0) + \frac{2n}{3} (\vec{a}+\vec{b}+\vec{c})[/math]. All the intermediate terms cancel out nicely.

Edited November 21, 2016 by renerpho

[renerpho 9]

I'm not entirely sure about the correct sign (the question is not clear about that). So it is [math]\vec{q}(n)=\vec{q}(0) \pm \frac{2n}{3} (\vec{a}+\vec{b}+\vec{c})[/math], depending on the direction in which the force is acting.

I will leave vector arrows aside from now on; the following refers to vectors in [math]R^{2}[/math].

 

The continuous system can be modelled analytically, too. Note that [math]q=q(t)[/math].

You have [math]F(t,q)=q \pm \frac{2}{3} (a+b+c)[/math]. From Newton's second law, we have [math]F(t,q)=m \cdot \ddot{q}[/math], where [math]m[/math] is the mass of the test particle in [math]q[/math]. So the differential equation for the system becomes [math]m \cdot \ddot{q} - q = \pm \frac{2}{3} (a+b+c)[/math].

The two equations to solve are

[latex]\begin{pmatrix} m \ddot{x}-x \\ m \ddot{y}-y \end{pmatrix}=\pm \frac{2}{3} \begin{pmatrix} a{_x}+b{_x}+c{_x} \\ \ a{_y}+b{_y}+c{_y} \end{pmatrix}[/latex].

The solution of this system is

[latex]\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} v{_1} \cdot e^{\frac {t}{\sqrt{m}}}+v{_2} \cdot e^{-\frac {t}{\sqrt{m}}}+v{_3} \\ w{_1} \cdot e^{\frac {t}{\sqrt{m}}}+w{_2} \cdot e^{-\frac {t}{\sqrt{m}}}+w{_3} \end{pmatrix}[/latex], with constants [math]v{_i},w{_i}[/math] depending on the initial choice of [math]a,b,c,q(0)[/math].

 

The question does not mention the mass of the test particle. But as it is based on forces, the mass is an important factor for the continuous system.

Edited November 21, 2016 by renerpho