Question

Describe the motion of a particle with position (x, y) as t varies in the given interval.

Answer 1

Step 1:

The parametric equations are and and .

Consider .

Similarly .

Trigonometric identity : .

Substitute and in the above identity.

The above equation is in form of general form of ellipse.

So the particle moves in elliptical path.

Step 2:

Draw a table for different values of ranging from .

Determine the direction of the curve.

Graph

(1) Draw the coordinate plane.

(2) Plot the points obtained in the table.

(3) Determine the directions of the curve.

Observe the graph:

From to , the ellipse completes it first revolution in clockwise.

Similarly the ellipse completes it second and third revolution at and .

The particle moves in 3 times around the elliptical path.

Solution:

The particle moves in 3 times around the elliptical path.

The equation is .