Overview
This lecture explains how to solve intersection and collision problems for objects moving along 3D trajectories, emphasizing the need for algebraic methods over graphing.
Intersection and Collision in 3D
- In 3D, visualizing trajectory intersections is difficult, so algebraic equations are used instead.
- Two paths (trajectories) are defined using vector functions: r(t) and u(s).
- Intersection means both paths occupy the same (x, y, z) position, possibly at different times.
- Collision means both paths occupy the same (x, y, z) position at the same time.
Setting Up the Problem
- Assign different parameters: t for r(t) and s for u(s).
- Set up the system:
- ( r_x(t) = u_x(s) )
- ( r_y(t) = u_y(s) )
- ( r_z(t) = u_z(s) )
- Solve for values of s and t that make all three coordinates equal.
Solving the Example
- Use the x-equation to relate t and s; substitute into the y-equation.
- Reduce the system to one variable and solve the resulting quadratic equation.
- Two solution pairs are found: (s=0, t=1) and (s=½, t=2).
- Check these pairs in the z-equation to confirm valid intersections.
Interpreting Results
- For (s=0, t=1): Both paths pass through (1, 1, 1).
- For (s=½, t=2): Both paths pass through (2, 4, 8).
- Therefore, the trajectories intersect at two distinct points in space at different times.
- No collision occurs since there are no solutions where s = t for the intersection points.
Key Terms & Definitions
- Intersection — When two trajectories pass through the same point in space, possibly at different times.
- Collision — When two trajectories pass through the same point in space at the same time.
- Trajectory — The path an object moves along, described by a function of time.
- Parameterization — Using variables (e.g., t, s) to describe positions along paths.
Action Items / Next Steps
- Practice setting up intersection and collision equations for other 3D trajectory problems.
- Review the distinction between intersection and collision conditions.
- Reflect on how algebraic and graphical methods both describe 3D motion.