Casyopee - Model and locus from a parameter
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Suppose that some points depend on a parameter. It is interesting to obtain the locus of one point and to model dependencies between variables involving this parameter.
This is an example.

Consider a "Target" moving at uniform speed 2 on a straight line, and a "Chaser" moving at uniform speed 3, making an angle a in degrees relatively to the trajectory of the "Target ".

The two mobiles have an initial distance of 2, the "Chaser" being on a perpendicular to the trajectory of the "Target".

How to model these mobiles? For what values ​​of a, does the " Chaser " reach the "Target"?

We first create two parameters, t for time, and a for the angle in degrees, as shown in the adjacent picture. We create T, coordinated point (2t, 0) for the "Target" and C for the "Chaser" of coordinate : (3cos(ap/180) t ; 2-3sin(ap/180) t).

The trajectory of C is a straight line. This can be confirmed by using the menu entry "locus".

Please click on the tool in the menu (1 in adjacent figure) and then to (2), then the parameter (3).

Casyopée confirms that the locus is included in a line and proposes to create it.

 

This done, it is interesting to give different values ​​to a.

With a value of 30 °, the chaser passes before the target without hitting it. With a value of 60 °, it passes behind.

Casyopée’s automatic modeling will allow studying the variation of the distance between the two mobiles.

We create, c0 and c1, calculations corresponding to the time and distance.

The function f is obtained by automatic modeling. Its graphical representation appears strictly above the x-axis except for a value of a close to 48 °. In this case, it looks like the graph of a piecewise linear function and the distance is close to zero for a value of t close to 0.9.

Another consideration may be based on the observation that C is at a distance 3t from its initial position, that is to say on a circle of radius 3t centered on this position. By creating this circle, we can animate t so that it passes through point T. It is again a value of t close to 0.9 and an angle close to 48 °.

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By trigonometry we find latex: frac{2}{3} for the cosinus of a and latex: frac{sqrt{5}}{3}for the sinus. For these values, the formula of f is

latex:text{f}(t)= sqrt{5cdot t^2-frac{12cdot sqrt{5}}{3}cdot t+4} and simplifies into latex:text{f}(t)= |sqrt{5}cdot t-2)|

The above conjectures are proved.

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Creation date : 01/12/2014 - 17h06
Category : - help
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