Find parametric equations and a parameter interval for the motion of a particle starting at the point ( 2 2,0) and tracing the top half of the circle x squared plus y squared equals 4 x2+y2=4 four four times. Find parametric equations for the particle's motion. Let the parameter interval for the motion of the particle be 0 less than or equals ≤t less than or equals ≤ 4 4 pi π.

1. Parametrization: and

2. In case that , the desired parametrization is

Step-by-step explanation:

Consider the particle at the point (2,0) and the circle of equation . Recall that the general equation of a circle of radius r is given by . Then, in our case, we know that the circle has radius 2.

One classic way to parametrize the movement of a particle that starts at point (r,0) and moves in a counterclockwise manner over a circular path of radius r is given by the following parametrization . Since, for all t we have that

If we want to draw only the upper half of the circle, we must have .

So, with r=2 the desired parametrization is and . Recall that in this parametrization when t=0 the particle is at (2,0) and when t=pi the particle is at (-2,0).

In the case that we want the parameter s but keeping the same particle's motion, we must do a transformation. We know that if parameter t is in the interval we get the desired motion. Note that in this case we are multiplying this interval by 4. So, we have that s = 4t.  If we solve for the parameter t, we get that t=s/4. Then, with the parameter s in the interval we get the parametrization which is obtained by replacing t in the previous parametrization.

Note that since when we have that and that when s=0, we have t=0, then the motion of the particle is the same (it changes only the velocity in which the particle moves a cross the path).

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step-by-step explanation:

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