Position, Velocity, and Acceleration: Horizontal Motion

Position Function (S(t)): The original function that models an object's location at any given time. In the example, the Ewok's position on a log for time t > 0 is modeled by S(t) = t^2 - 4t + 3 feet.
Velocity Function (V(t) = S'(t)): The first derivative of the position function. It describes the rate of change of position with respect to time.
Acceleration Function (A(t) = V'(t) = S''(t)): The first derivative of the velocity function or the second derivative of the position function. It describes the rate of change of velocity.

Definition: Displacement measures the net change in position from the starting point to the ending point, regardless of the path taken. It is not the total distance traveled.
Calculation for an interval [t1, t2]: S(t2) - S(t1).
Example (Ewok during the first two seconds, [0, 2] seconds):
- Initial position at t=0: S(0) = 0^2 - 4(0) + 3 = 3 feet.
- Position at t=2: S(2) = 2^2 - 4(2) + 3 = 4 - 8 + 3 = -1 feet.
- Displacement: S(2) - S(0) = -1 - 3 = -4 feet.
- Interpretation: The Ewok moved 4 feet in the negative direction from its starting point of 3 to its ending point of -1.
Note: Total distance traveled is a different concept, typically calculated using integrals later in the course.

Definition: Average velocity over an interval is calculated as the slope of the secant line between two points on the position function graph. It represents the overall rate of change of position over that interval.
Calculation for an interval [t1, t2]: rac{S(t2) - S(t1)}{t2 - t1}.
Example (Ewok during the first four seconds, [0, 4] seconds):
- Initial position at t=0: S(0) = 3 feet.
- Position at t=4: S(4) = 4^2 - 4(4) + 3 = 16 - 16 + 3 = 3 feet.
- Average Velocity: rac{S(4) - S(0)}{4 - 0} = rac{3 - 3}{4 - 0} = rac{0}{4} = 0 feet per second.
- Important Note for Exams: Always show the full slope calculation, including the subtraction in the denominator (e.g., 4-0), even if the answer is obvious.
- Interpretation: An average velocity of 0 feet per second does not mean the object was stationary; it simply means it returned to its starting position over the interval.