Question

Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 3 + 2 cos(t), y = 3 + 2 sin(t), π/2 ≤ t ≤ 3π/2

Answer 1

The particle starts at (3,5) and travels counter clockwise along a semi-circle with a radius of 2 and centered at (3,3) through the point (1,3) and stops at point (3,1). First, let's look at the two equations: x = 3 + 2 cos(t) y = 3 + 2 sin(t) If you ignore the limits specified for t, you'll realize that they describe a circle with a radius of 2, centered at (3, 3). Now that we know that the functions describe a circle, let's look at the limits specified for t and see what they imply. Since the value is from π/2 to 3π/2, it will span a distance of π radians, so we know the particle will trace half of a full circle, or a semi-circle. Since the starting point is π/2, the starting coordinate of the particle will be: (3 + 2 cos(π/2), 3 + 2 sin(π/2)) = (3 + 2*0, 3 + 2*1) = (3+0, 3+2) = (3,5) And the ending point of the particle will obviously be (3,1). But to demonstrate: (3 + 2 cos(3π/2), 3 + 2 sin(3π/2)) = (3 + 2*0, 3 + 2*(-1)) = (3+0, 3+ (-2)) = (3,1) Now let's see where the particle is in the middle. (3 + 2 cos(π), 3 + 2 sin(π)) = (3 + 2*(-1), 3 + 2*0) = (3 + (-2), 3 + 0) = (1, 3) So the final description is: The particle starts at (3,5) and travels counter clockwise along a semi-circle with a radius of 2 and centered at (3,3) through the point (1,3) and stops at point (3,1).