Final answer:
The vector-valued function r(t) = (4.0 cos 3t)i + (4.0 sin 3t)j describes the motion of a particle moving in a circle of radius 4.0 centered at the origin in the xy-plane. The function is periodic with a period of 2π and its second derivative is parallel to the original function, confirming circular motion.
Step-by-step explanation:
In this case, the position of the particle is given by the vector-valued function r(t) = (4.0 cos 3t)i + (4.0 sin 3t)j, where r is in meters and t is in seconds. To show that this function describes the motion of a particle moving in a circle, we need to check all the given options.
- Option 1: The function is periodic with a period of 2π, representing circular motion. This is true, as the cosine and sine functions are periodic with a period of 2π, and 3t ranges from 0 to 2π for one complete revolution.
- Option 2: The function exhibits constant magnitude, indicating circular motion. This is true, as the magnitude of r(t) is always 4.0, which is the radius of the circle.
- Option 3: The function's derivative is orthogonal to the original function, characteristic of circular motion. This is not true, as the derivative of r(t) is (-12.0 sin 3t)i + (12.0 cos 3t)j, which is not orthogonal to r(t).
- Option 4: The function's second derivative is parallel to the original function, confirming circular motion. This is true, as the second derivative of r(t) is (-36.0 cos 3t)i - (36.0 sin 3t)j, which is parallel to r(t).
Based on the above analysis, the correct options are Option 1 and Option 4.