Overview
This lecture introduces velocity in two-dimensional (2D) motion, focusing on vector representation, component analysis, vector addition, and the independence of x and y motion.
Velocity Vectors
- Velocity vectors represent the speed and direction of an object at a specific instant.
- The vector's length shows the magnitude (speed); direction shows motion.
- For constant velocity, the velocity vector remains unchanged at every location.
Components of 2D Velocity
- 2D velocity has x (horizontal) and y (vertical) components: VX and VY.
- Use trigonometry to find components: VX = V * cos(θ), VY = V * sin(θ), where θ is the angle from the x-axis.
- If velocity is 2 m/s at 60°, VX = 1 m/s and VY = 1.7 m/s.
- Component signs depend on direction (right/up = positive, left/down = negative).
Negative Velocity Components
- Positive x: right; negative x: left.
- Positive y: up; negative y: down.
- Magnitude of velocity is always positive; only components can be negative.
Adding Velocity Vectors
- To combine velocities (e.g., motion plus wind), add the x and y components separately.
- Resultant (total) velocity magnitude: V = √(VX² + VY²).
- Direction (angle): θ = arctan(VY/VX).
Independence of X and Y Motion
- X (horizontal) and Y (vertical) motions are independent; changing one does not affect the other.
- Example: A wind changes only VY, not VX.
Applying Components: Example with a Boat
- Convert angled motion into VX and VY to analyze each direction separately.
- Use VY to calculate time to cross a river (distance / VY).
- Use VX and time to calculate how far the boat moves along the river.
Summary Points
- Velocity vectors show speed and direction at a moment.
- Components are found with sine and cosine functions.
- Resultant vectors use Pythagorean theorem and inverse tangent for magnitude and angle.
- Magnitude is always positive; component signs show direction.
- X and Y motions are analyzed independently for problem-solving.
Key Terms & Definitions
- Velocity Vector — shows object's speed and direction at an instant.
- Component — part of a vector along the x- or y-axis.
- Magnitude — the size or length of a vector (speed).
- Resultant Vector — combined effect of two or more vectors.
- Instantaneous Velocity — velocity at a specific moment in time.
Action Items / Next Steps
- Practice breaking 2D velocities into components and recombining them.
- Complete assigned problems on vector addition and component analysis.
- Review trigonometric relationships for converting between magnitude/angle and components.