Suppose that the position of one particle at time t is given by x1=3 sint y1=2 cost 0 ⩽t ⩽

Suppose that the position of one particle at time t is given...

Suppose that the position of one particle at time $t$ is given by $$\quad x_{1}=3 \sin t \quad y_{1}=2 \cos t \quad 0 \leqslant t \leqslant 2 \pi$$ and the position of a second particle is given by $$x_{2}=-3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$ (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. (c) Describe what happens if the second particle is given by $$x_{2}=3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$

Suppose that the position of one particle at time $t$ is given
by
$$\quad x_{1}=3 \sin t \quad y_{1}=2 \cos t \quad 0 \leqslant t \leqslant 2 \pi$$
and the position of a second particle is given by
$$x_{2}=-3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$
(a) Graph the paths of both particles. How many points
of intersection are there?
(b) Are any of these points of intersection collision points?
In other words, are the particles ever at the same place
at the same time? If so, find the collision points.
(c) Describe what happens if the second particle is given by
$$x_{2}=3+\cos t \quad y_{2}=1+\sin t \quad 0 \leqslant t \leqslant 2 \pi$$

Key Concepts

Time Parameter Synchronization

Time parameter synchronization refers to the alignment or matching of the time variables for different particles when analyzing their motion. Even if the paths intersect spatially, it is essential to check whether the parameter values (typically time) at these points are equal, as this determines whether a true collision occurs where both particles meet at the exact same moment.

Parametric Equations

Parametric equations describe curves by expressing the x and y coordinates as functions of an independent parameter, often time. This framework is especially useful in modeling the motion of particles along a path and in representing conic sections such as ellipses and circles, where each coordinate function (for example, using sine and cosine) defines the position continuously over time.

Curve Intersection

This concept involves finding common points between two distinct curves independent of how they are traversed as functions of time. Determining intersections requires solving the equations defined by the x and y coordinate functions so that they yield the same spatial coordinates, but this analysis does not necessarily account for the timing of the encounters.

Collision Points

Collision points are specific types of intersections where not only do the spatial paths cross, but the moving objects (or particles) also occupy the same point at the same time. This requires simultaneously solving the coordinate functions along with ensuring that the time parameter at which each particle reaches the common point is identical.

Graphing Parametric Paths

Graphing parametric paths is a key technique for visualizing the trajectories generated by the parametric equations. It helps in visual assessment of the shape of the curves—such as determining whether the path is elliptical, circular, or otherwise—and aids in predicting possible intersection points and understanding the overall motion dynamics.