Position-time graphs are essential tools for understanding motion. They show an object's location relative to a reference point over time, with position on the y-axis and time on the x-axis. The graph's shape reveals key details about the object's movement.

These graphs help calculate average and instantaneous velocity. Average velocity is found by dividing the change in position by the change in time between two points. Instantaneous velocity is determined using tangent lines at specific points on the curve.

Position vs. Time Graphs

Interpretation of position-time graphs

• Position-time graphs visually represent an object's position relative to a reference point over a given time period
  • Position is measured along the vertical y-axis (meters, feet, miles)
  • Time is measured along the horizontal x-axis (seconds, minutes, hours)
• The shape and slope of the graph line indicate the type and characteristics of the object's motion
  • Straight line with positive slope signifies the object is moving at a constant velocity in the positive direction (away from the reference point)
  • Straight line with negative slope signifies the object is moving at a constant velocity in the negative direction (towards the reference point)
  • Horizontal line with zero slope indicates the object is stationary or at rest, not changing its position over time
  • Curved line suggests the object is accelerating or decelerating, changing its velocity over time
    • Concave up curve with increasing slope shows the object is speeding up or accelerating (car merging onto a highway)
    • Concave down curve with decreasing slope shows the object is slowing down or decelerating (car braking to a stop)
• The coordinate system used in position-time graphs provides a framework for analyzing motion

Average velocity from position-time graphs

• Average velocity $v_{avg}$ is calculated by dividing the change in position $\Delta x$ by the change in time $\Delta t$: $v_{avg} = \frac{\Delta x}{\Delta t}$
  • Change in position $\Delta x$ is the difference between the final position $x_2$ and initial position $x_1$: $\Delta x = x_2 - x_1$
  • Change in time $\Delta t$ is the difference between the final time $t_2$ and initial time $t_1$: $\Delta t = t_2 - t_1$
• To calculate average velocity using a position-time graph, select any two points on the graph line $(x_1, t_1)$ and $(x_2, t_2)$
  1. Determine the change in position $\Delta x$ by subtracting the initial x-coordinate from the final x-coordinate: $\Delta x = x_2 - x_1$
  2. Determine the change in time $\Delta t$ by subtracting the initial time from the final time: $\Delta t = t_2 - t_1$
  3. Divide the change in position by the change in time to obtain the average velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$
• The average velocity equals the slope of the straight line connecting the two selected points on the graph (rise over run)

Instantaneous velocity using tangent lines

• Instantaneous velocity is the velocity of an object at a specific moment or instant in time (speedometer reading)
• To determine instantaneous velocity from a position-time graph, find the slope of the tangent line at the point of interest
  • A tangent line is a straight line that touches the curve at a single point without intersecting it
  • The slope of the tangent line represents the instantaneous velocity at that specific time
• Steps to find instantaneous velocity using a tangent line:
  1. Identify the point on the graph where you want to determine the instantaneous velocity
  2. Draw a tangent line to the curve at that exact point
  3. Select two points on the tangent line $(x_1, t_1)$ and $(x_2, t_2)$
  4. Calculate the change in position $\Delta x = x_2 - x_1$ and change in time $\Delta t = t_2 - t_1$
  5. Divide the change in position by the change in time to get the instantaneous velocity $v_{inst} = \frac{\Delta x}{\Delta t}$
• Instantaneous velocity measures how fast an object's position is changing at a particular instant (rate of change of position with respect to time)

Kinematics and Motion Analysis

• Position-time graphs are fundamental tools in kinematics for analyzing motion
• Graphical interpretation of position-time graphs allows for the study of an object's trajectory over time
• Motion analysis using position-time graphs provides insights into an object's behavior and helps predict its future positions