Overview
This lecture continues the discussion of kinematics in two and three dimensions, focusing on vector mathematics and motion under constant acceleration.
Introduction to Kinematics in 2D & 3D
- Kinematics in 2D/3D describes real-world motions like snowboarding, projectiles, and water flow.
- Gravitational field and constant acceleration scenarios expand from 1D to higher dimensions.
Vectors: Definitions and Operations
- Vectors have both magnitude (length) and direction, unlike scalars.
- Vector addition in 1D uses simple arithmetic, but 2D/3D requires geometric rules.
- The tail-to-head method sums vectors by connecting one’s tail to another’s head.
- The parallelogram rule is an alternative vector addition approach.
- Vector addition is commutative (order doesn't matter) and associative (grouping doesn't matter).
- Subtracting vectors uses the negative/opposite direction.
- Multiplying a vector by a scalar changes its magnitude but not its direction.
Components and Unit Vectors
- A vector can be decomposed into perpendicular x, y (and z) components.
- The magnitude of each component: ( v_x = |v| \cos\theta ), ( v_y = |v| \sin\theta ).
- Unit vectors ((\hat{i}, \hat{j}, \hat{k})) define directions along axes: any vector ( \vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} ).
- Vector sums: components add separately, e.g., ( (v_{1x} + v_{2x})\hat{i} ).
Vector Kinematics
- Displacement in 2D/3D: ( \Delta \vec{r} = \vec{r}_2 - \vec{r}_1 ), components change accordingly.
- Average velocity: ( \Delta \vec{r} / \Delta t ).
- Instantaneous velocity is the tangent vector at a point on a path, found by differentiation.
- Acceleration: time-derivative of velocity, can also be written in components.
- Under constant acceleration, position and velocity extend from 1D equations to vector forms.
Projectile Motion & Parabolic Trajectories
- Objects projected at an angle decompose motion into independent x (constant velocity) and y (accelerated) components.
- Projectiles follow a parabolic path described by eliminating ( t ) between ( x(t) ) and ( y(t) ).
Relative Motion & Velocity
- Relative velocity: ( \vec{v}_{AB} = \vec{v}_A - \vec{v}_B ); direction/subscript conventions matter.
- For moving platforms (train, boat, river), sum velocities carefully by matching indices._
Vector Multiplication (Advanced Note)
- Two types: dot product (scalar result) and cross product (vector result); both are beyond this course's scope for now.
Key Terms & Definitions
- Vector — A quantity with both magnitude and direction.
- Scalar — A quantity with magnitude only, no direction.
- Unit Vector — A vector of magnitude 1 pointing along a coordinate axis.
- Projectile Motion — Motion under gravity in 2D, following a parabolic path.
- Relative Velocity — The velocity of one object as observed from another moving object.
Action Items / Next Steps
- Complete suggested textbook problems for chapter 3 (kinematics in 2D/3D).
- Review the reading assignment from Giancoli, Chapter 3.
- Practice vector addition, component decomposition, and solving projectile motion problems.