Overview
This lecture explores how to determine if two objects following vector-valued trajectories in the same space collide, merely intersect, or have a near-miss, using algebraic methods.
Trajectory Basics and Graphing
- Two trajectories are defined: ( r(t) = (t^2, t) ) (a parabola) and ( u(t) = (3-2t, 4t-3) ) (a straight line).
- Substituting values for ( t ) reveals the shapes: ( r(t) ) traces a parabola, and ( u(t) ) traces a straight line in xy-space.
- Direction of travel is determined by increasing ( t ) values and plotting sample points.
Collisions vs. Intersections
- A collision occurs if both trajectories occupy the same point in space at the same time (( t )).
- An intersection (without collision) happens if both trajectories share the same coordinates but at different times (( t ) for one, ( s ) for the other).
Algebraic Approach to Collisions
- To test for collision, set both x- and y-coordinates of ( r(t) ) and ( u(t) ) equal at the same time:
- ( t^2 = 3 - 2t )
- ( t = 4t - 3 )
- Solving yields ( t = 1 ), which when checked satisfies both equations, confirming a collision at point (1, 1) at ( t = 1 ).
Algebraic Approach to Intersection (No Collision)
- Set equalities for location but allow different times (( t ) for ( r ), ( s ) for ( u )):
- ( t^2 = 3 - 2s )
- ( t = 4s - 3 )
- Solve for ( s ) in terms of ( t ): ( s = (t + 3)/4 ).
- Substitute back and solve quadratic; solutions: ( t = 1 ) (collision), ( t = -\frac{3}{2} ) (intersection).
- At ( t = -\frac{3}{2} ), corresponding ( s = \frac{3}{8} ), both trajectories pass through point (( \frac{9}{4}, -\frac{3}{2} )) at different times.
Visualization and Validation
- Animation or plotting can confirm calculations: two points of intersection, only one is a collision.
- Proper algebra distinguishes between spatial intersections and actual collisions.
Key Terms & Definitions
- Trajectory — The path an object follows in space, described here by a vector-valued function.
- Collision — Both objects reach the same location at the same time.
- Intersection (No Collision) — Both objects pass through the same point in space but at different times.
- Vector-valued curve — A function that assigns a position vector to each value of time.
Action Items / Next Steps
- Review the algebraic setup for collision vs. intersection with different trajectories.
- Practice similar problems, setting coordinates equal at the same or different times.