AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.2 — Parametric Functions Modeling Planar Motion
Notes — Position, Path, and Direction
💡 Learning Objectives (4.2.A)
By the end of this lesson you will be able to:
- Use parametric equations to model the motion of an object in the plane
- Identify position, path, and direction from a parametric description
- Determine when an object reaches a specific location
- Recognize when a particle is moving left, right, up, or down based on x(t) and y(t)
1. Position vs. Path
When parametric equations model motion, t represents time. The pair (x(t), y(t)) gives the POSITION of the object at time t. As t advances, the object MOVES — and the set of all positions visited is called the PATH.
The path is just a curve in the plane. The motion adds DIRECTION and TIMING — two different objects might travel the same path at different speeds or in opposite directions.
2. Reading a Motion Description
📘 Example — A particle moving along a parabola
A particle has position (x(t), y(t)) = (t, t² − 4) for t ∈ [−3, 3].
- At t = 0: position (0, −4) — the vertex of the parabola
- At t = 2: position (2, 0)
- At t = −2: position (−2, 0)
- As t increases, x always increases (since x = t), so the particle moves RIGHT throughout
- The y-component goes down then up, so the particle moves down on (−3, 0), then up on (0, 3)
3. Direction of Motion at a Given Instant
To decide whether a particle is moving up/down or left/right at a given t, look at how each coordinate is CHANGING:
- If x(t) is increasing as t grows: particle moves RIGHT
- If x(t) is decreasing: particle moves LEFT
- If y(t) is increasing: particle moves UP
- If y(t) is decreasing: particle moves DOWN
To check whether a coordinate is increasing or decreasing, evaluate just before and just after the t in question, or analyze x(t) and y(t) as ordinary functions.
4. When Does the Particle Reach a Specific Point?
To find the time at which a particle is at a particular (a, b), set up the system x(t) = a AND y(t) = b. The values of t that satisfy BOTH are the times the particle is at that point. (No solution to the system means the particle never visits that point.)
📘 Example — When does the particle reach (3, 5)?
A particle has (x(t), y(t)) = (t + 1, t² − 4).
- Solve x(t) = 3: t + 1 = 3 ⇒ t = 2
- Check y(t) at t = 2: y(2) = 4 − 4 = 0, NOT 5
- Conclusion: the particle never reaches (3, 5).
5. Self-Intersections of the Path
Sometimes a path crosses itself — different times produce the same position. Find these by solving (x(t₁), y(t₁)) = (x(t₂), y(t₂)) with t₁ ≠ t₂.
📘 Example — Finding a self-intersection
A particle has (x(t), y(t)) = (t² − 1, t³ − 3t).
- Need t₁² − 1 = t₂² − 1 and t₁³ − 3t₁ = t₂³ − 3t₂
- First eq: t₁² = t₂², so t₂ = ±t₁. Take t₂ = −t₁ (otherwise same time).
- Second eq: t₁³ − 3t₁ = −t₁³ + 3t₁ ⇒ 2t₁³ − 6t₁ = 0 ⇒ 2t₁(t₁² − 3) = 0
- So t₁ = 0 (trivial) or t₁ = ±√3. With t₁ = √3, t₂ = −√3, both give (2, 0).
- The path crosses itself at (2, 0).
6. Modeling Real Motion
Some classic motion examples:
- Projectile: x(t) = v₀ cos(α) · t, y(t) = v₀ sin(α) · t − (1/2)g t²
- Uniform circular motion: x(t) = R cos(ωt), y(t) = R sin(ωt)
- Linear motion: x(t) = x₀ + at, y(t) = y₀ + bt
7. Reading from a Graph of x(t) and y(t)
On AP-style problems, you often see graphs of x(t) and y(t) on separate sets of axes (rather than the path itself). To extract motion information:
- Where x(t) crosses zero: the particle is on the y-axis
- Where y(t) crosses zero: the particle is on the x-axis
- Where x(t) has a maximum or minimum: x momentarily stops; particle moves purely vertically
- Where y(t) has a maximum or minimum: y momentarily stops; particle moves purely horizontally
8. Summary
- Parametric motion: (x(t), y(t)) gives position; the path is the set of all visited points
- Direction at any t depends on whether x and y are increasing or decreasing
- To find when the particle is at (a, b), solve x(t) = a AND y(t) = b together
- Self-intersections occur when two different t-values produce the same position
- Graphs of x(t) and y(t) reveal motion features: zeros, extrema, monotonicity