Question

Suppose that the position of one particle at time is given by x1=3sin t, y1 = 2 cos t, 0 ≤ t ≤ 2π and the position of a second particle is given by x2 = -3 + cos t, y2 = 1 + sin t, 0 ≤ t ≤ 2π. 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.

Answer 1

After equating the position equations for both particles and attempting to solve for time t, it appears that no solution satisfies both conditions simultaneously within the interval 0 ≤ t ≤ 2π, indicating there are no collision points.

Step-by-step explanation:

To determine if there are any collision points where two particles are at the same place at the same time, we set the position equations of both particles equal to each other and solve for time t.

For particle 1, the position is given by:

• x1(t) = 3sin(t)
• y1(t) = 2cos(t)

For particle 2, the position is given by:

• x2(t) = -3 + cos(t)
• y2(t) = 1 + sin(t)

To find collision points, we equate the coordinates:

3sin(t) = -3 + cos(t) --> Equation 1

2cos(t) = 1 + sin(t) --> Equation 2

Solve these two equations simultaneously. Suppose we find a t that satisfies both equations within the given range, then that t will represent a collision point.

However, these equations do not seem to have a solution that meets both conditions simultaneously within the specified interval of t, which suggests the two particles do not collide.