Overview
This lecture covers essential concepts, skills, and tips for solving two-dimensional (2D) motion and vector problems, providing step-by-step solutions to various example problems.
Key Concepts for 2D Motion & Vectors
- Geometry knowledge is needed, especially right triangles using Pythagorean theorem and trigonometric functions (sine, cosine, tangent).
- Familiarity with Cartesian coordinate systems (x and y axes, positive/negative directions) is required; sometimes problems use rotated or flipped axes.
- Vectors can be described by magnitude and angle; know how to draw, find components, magnitude, and angle, and how to add them graphically and mathematically.
- Use 1D kinematic equations (for linear motion), applied separately to x and y components in 2D problems.
Tips for Solving 2D Motion & Vector Problems
- Always draw a clear picture of the scenario and vectors, but rely on calculations rather than estimates from your drawing.
- Pay close attention to how angles are described; use conventional angles measured counterclockwise from the positive x-axis when applicable.
- Convert 2D problems into separate 1D problems for the x and y directions using components.
- Double-check calculator settings (degrees vs. radians) for trigonometric calculations.
Example Problem Solutions
Adding Two Vectors (A + B)
- Write down given magnitudes and angles, identify if angles are relative to the x-axis.
- Find x and y components using: ax = a·cos(θ), ay = a·sin(θ); same for b.
- Add components: cx = ax + bx, cy = ay + by.
- Find magnitude: √(cx² + cy²); angle: arctan(cy/cx), adjust reference angle as needed.
Displacement Below the Horizontal (Zipline)
- Draw initial and final points, label horizontal and vertical distances.
- Form a right triangle with legs (horizontal, vertical); angle below horizontal = arctan(vertical/horizontal).
- Solution: angle = 6.2° below horizontal.
Relative Motion (Plane in Wind)
- Draw plane's velocity south and wind's velocity west as perpendicular vectors.
- Use Pythagorean theorem for resultant (ground) speed: √(800² + 120²) = 809 km/h.
- Angle off original direction: arctan(120/800) = 8.5° west of south.
Coordinates and Vector Motion (Hockey Puck)
- Initial position: (6, 4) m; velocity: 16 m/s at 72° to x-axis; time: 2 s.
- Displacement magnitude: velocity × time = 32 m.
- Find x, y components: Δx = 32·cos(72°), Δy = 32·sin(72°).
- Final coordinates: (6 + Δx, 4 + Δy) = (15.9, 34.4) m.
Key Terms & Definitions
- Vector — a quantity with both magnitude and direction.
- Component — the projection of a vector along an axis (e.g., x or y).
- Magnitude — the length or size of a vector.
- Reference Angle — the smallest angle between a vector and a given axis.
- Displacement — the change in position of an object, expressed as a vector.
Action Items / Next Steps
- Review right triangle trigonometry and Pythagorean theorem.
- Practice breaking vectors into components and adding them.
- Complete any assigned homework on 2D kinematic problems.
- Double check calculator mode before doing trig calculations.