Math Problem Statement
Solution
The problem involves analyzing the motion of two particles, each following a specific parametric path over the interval .
The given positions are:
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Particle 1:
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Particle 2:
The questions asked are:
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(a) Graphing the paths: The graph options are provided, and it seems the correct answer (based on a check mark) is the third plot.
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(b) Finding intersections and checking for collisions (same position and time):
- There are two intersection points given: and .
- It is confirmed that there are no collision points, as none of these points match at the same -value for both particles.
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(c) Modifying the second particle's path to:
- Finding the new center of the circular path of the second particle, which is at .
Let me know if you need details on each part or have any specific questions.
Additional Questions for Practice:
- How would you determine the parametric equations for the velocity of each particle?
- What would be the parametric form of the tangent to each path at any given point ?
- How can you mathematically determine the exact -values at which intersections occur?
- What changes in the equation for Particle 2 would shift its path vertically by 3 units?
- Can you determine the speeds of the particles at any point on their paths?
Tip:
When analyzing intersections in parametric equations, ensure both the positions and corresponding parameter values (like ) match to confirm a collision.
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Math Problem Analysis
Mathematical Concepts
Parametric Equations
Graphing Parametric Curves
Intersection of Curves
Formulas
Position of Particle 1: x1 = 3sin(t), y1 = 2cos(t)
Position of Particle 2: x2 = -3 + cos(t), y2 = 1 + sin(t)
Theorems
Conditions for Curve Intersection
Collision Condition (Same Position and Time)
Suitable Grade Level
Grades 11-12
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