Final answer:
The motion describes an elliptical path with semi-major and semi-minor axes of 7 and 3 units respectively. The particle completes one and a half cycles in the interval −π≤3π with uniform speed.
Step-by-step explanation:
The motion of a particle with position (x,y) as t varies in the interval −π≤t≤3π, where x=7sin(t) and y=3cos(t), describes an elliptical path in the xy-plane. This can be identified by recognizing the parametric equations for an ellipse, with the x component as a sine function and the y component as a cosine function. The particle will make a full cycle as 't' goes from −π to 2π, and then repeat the same path as 't' goes from 2π to 3π since the sine and cosine functions have a period of 2π. At any time 't', the position of the particle is determined by plugging the value of 't' into the parametric equations.
The radii of the elliptical path are half the coefficients of the sine and cosine functions, so the semi-major axis has a length of 7 units (along the x-direction) and the semi-minor axis has a length of 3 units (along the y-direction). Observing this elliptical motion from −π to 3π would show the particle completing one and a half cycles of this elliptical path. As this equation represents uniform motion, there is no acceleration in the direction of motion; the speed (not velocity) of the particle remains constant.