Position, displacement, and distance are fundamental concepts in kinematics. They describe an object's location and movement in space. Understanding these terms is crucial for analyzing motion and solving physics problems.

Average velocity provides a simplified view of an object's motion over time. It's calculated by dividing displacement by time interval. This concept is key for understanding more complex motion scenarios and forms the basis for studying acceleration.

Position, Displacement, and Distance

Position, displacement, and distance

• Position specifies an object's location at a specific time relative to a chosen reference point (origin) and is represented by a coordinate such as x in one-dimensional motion
• Displacement measures the change in an object's position calculated as the final position minus the initial position $\Delta x = x_f - x_i$ and is a vector quantity having both magnitude and direction, depending only on the initial and final positions, not the path taken
• Distance traveled is the total length of the path an object travels regardless of direction, always positive or zero never negative, a scalar quantity having only magnitude, and depends on the entire path taken not just the initial and final positions (hiking trail, road trip)

Calculations with position-time data

• Displacement can be calculated given initial position $x_i$ at time $t_i$ and final position $x_f$ at time $t_f$ using $\Delta x = x_f - x_i$ and can be positive, negative, or zero (train moving forward and backward, elevator moving up and down)
• Distance traveled is calculated by summing the absolute values of individual displacements or for continuous motion, integrating the absolute value of velocity over time $d = \int_{t_i}^{t_f} |v(t)| dt$ (running laps around a track, driving a delivery route)

Kinematics and Motion Analysis

• Kinematics is the branch of physics that describes the motion of objects without considering the forces causing the motion
• A coordinate system is used to define positions and displacements in space
• The frame of reference is the perspective from which observations are made, affecting how motion is described
• A trajectory represents the path of a moving object through space over time
• Motion diagrams visually represent an object's position at regular time intervals, helping to analyze its motion

Average Velocity

Average velocity computation

• Average velocity is the rate of change of position over a given time interval calculated as displacement divided by time $v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$, a vector quantity having both magnitude and direction (car traveling on a highway, boat crossing a lake)
• Average velocity indicates the net rate and direction of motion over a time interval but does not provide information about instantaneous velocity or speed, useful for understanding the net motion of an object over a period of time (plane flying from one city to another, cyclist riding across town)

Analysis of motion scenarios

1. Identify the initial and final positions and times

2. Calculate displacement using $\Delta x = x_f - x_i$

3. Determine the distance traveled by summing the absolute values of individual displacements or integrating the absolute value of velocity over time

4. Compute average velocity using $v_{avg} = \frac{\Delta x}{\Delta t}$

5. Interpret the signs of displacement and average velocity to understand the direction of motion (positive for forward/upward, negative for backward/downward)

6. Analyze the relationship between displacement, distance traveled, and average velocity to gain insights into the motion scenario (comparing a direct route to a scenic route, examining the efficiency of different modes of transportation)