Help to model a phisical problem in Matlab

Team,

I'm sorry but I'm too rusted to solve this alone. If anyone can help me :)

It’s a physical problem, I have a plastic pipe coming out vertically from an inlet. At the same time the end of the pipe is tied to a wire that is coming out below.

The speed of the output of the plastic pipe and of the wire is proportional and fixed: Vplastic = Ratio * Vwire.

In addition to that there is a point I need the curve to pass by at a certain time as well as the end point at the end of the movement should be on the X axis.

I need to model how the ending point of the plastic tube will evolve through time.

So here is the starting schema:

Here is an example (fake curve… it is what I need to find):

These are the conditions I have:

1. Vplastic = Ratio * Vwire
2. Vplastic = Length of arc at the end divided by time
3. Vwire = Max length divided by time, note that is possible that the PassBy point has a length superior to the final length. In this case the Vwire = Length at PassBy + abs value of (length PassBy – End Length)
4. T0 plastic = Point (0, 2) or parametric
5. T0 wire = Point (0, 0) or parametric
6. Tangent of curve at T0 plastic is vertical
7. For any T the length of the arc is proportional to the length of the wire. Is related to point 1)
8. Tangent of curve at and point can with any inclination (in the diagram is vertical by chance)
9. Tend = Point (0, 26) or parametric.

Answer 1

[darova]

Why about spline? I placed two points close to each other to manipulate with tangent angle

x0 = [0   0 2 3 4 4];                   % some data
y0 = [0 0.1 2 2 0.1 0];
[t0,r0] = cart2pol(x0-mean(x0),y0);     % ceonvert to polar (move the data to center)
t1 = linspace(0,pi);                    % more points
r1 = spline(t0,r0,t1);                  % interpolation
[x1,y1] = pol2cart(t1,r1);              % convert ot cartesian
x1 = x1+mean(x0);                       % move new data to original position
plot(x0,y0,'.-r')
line(x1,y1)
axis equal
legend('original curve','interpolated')

[Massimo Savazzi]

I used a spline just to get a curve that was not a circle or an ellipse :)

What I need to find is the curve itself... I have the conditions but not it's "equation" or model