Displacement, Velocity, and Acceleration: Describing Motion with Functions

This video focuses on describing motion using equations for position and velocity as functions of time, and developing an understanding of the rate of change.
Functions are mathematical stories that describe how several different physical quantities are related.
The three kinematic equations are functions that show relationships between position, velocity, acceleration, and time.
Through careful use of algebra, these functions can be rearranged to describe variables in various ways.
These relationships primarily describe motion with constant acceleration.
Notation for Kinematic Equations:
- X0 (or X{initial}, X{starting}, X{naught}) represents the position when time t=0.
- v0 (or v{initial}, v_{starting}) represents the velocity when time t=0.
- \Delta X is the difference between final and initial position (\Delta X = X{final} - X{initial}).
- Subscripts like x (e.g., x0, v{0x}) on AP Physics 1 equation sheets remind us that motion is described in a particular direction.

Example: Cart Rolling Along a Track
- Position data of a cart was recorded at various times.
- To simplify calculations, the initial time (3.97 s) was subtracted from all time data points to start at t=0.
- Graphing: Position (Y-axis) versus Time (X-axis) was plotted.
- A line of best fit was sketched, revealing a linear pattern.
Equation of a Line and its Physical Meaning
- Recall the algebraic equation for a line: Y = mX + B.
- In the physics context of the position-time graph:
  - Y-axis variable: Position (X).
  - Y-intercept (B): Represents the initial position (X_0).
    - For the cart example, X_0 \approx 0.3 m (where the line crosses the position axis).
  - Slope (m): Represents the velocity (v).
    - For the cart example, v \approx 0.47 m/s, indicating meters per second.
  - X-axis variable: Time (t).
- The resulting equation describing the cart's motion is: X = X_0 + vt or specifically, X = 0.3 \text{ m} + 0.47 \text{ m/s} \cdot t.
Predictive Power of the Equation
- The derived equation allows for predicting the cart's position at any given time.
- Comparing measured positions with calculated (predicted) positions shows a strong alignment, demonstrating the equation's accuracy in describing the motion.
- Algebraic rearrangement allows the same function to answer different questions, e.g.,