Position–time graphs Knowledge-Based Study

Study notes

1. What is a Position–Time Graph?

   A position–time graph ($x$–$t$ graph) visually represents the location of an object moving along a straight line over an interval of time.

   • Horizontal Axis ($t$): Represents elapsed time, measured in seconds (s). Time always advances forward from left to right.
   • Vertical Axis ($x$): Represents position relative to a chosen reference point or origin ($x = 0$), measured in meters (m).

   A position above the time axis indicates a positive position, whereas a position below indicates a negative position. It is essential to remember that a position–time graph does not show the actual physical trajectory of an object through two-dimensional space; it tracks one-dimensional motion over time.

   The Physical Meaning of Slope

   The rate of change of an object's position with respect to time defines its velocity. On a position–time graph, this rate corresponds directly to the slope of the line:

   $$\text{Slope} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1} = v$$

   Interpreting Different Slopes

   • Positive Slope: The object is moving in the positive direction (away from the origin or toward greater positive coordinate values).
   • Negative Slope: The object is moving in the negative direction (toward the origin or into negative values).
   • Zero Slope (Horizontal Line): The object remains at the same position, meaning it is at rest ($v = 0$).
   • Steeper Slope: Indicates a greater magnitude of velocity (higher speed), regardless of sign.

   Straight Lines and Curves

   The shape of the graph reveals whether an object moves with constant velocity or is accelerating.

   1. Constant Velocity (Uniform Motion)

   When an object moves at a steady speed without changing direction, equal displacements occur in equal time intervals. This yields a straight line with a constant slope.

   2. Changing Velocity (Accelerated Motion)

   If velocity changes over time, the slope of the position–time graph changes continuously, resulting in a curved line:

   • Instantaneous Velocity: Given by the slope of the tangent line drawn to the curve at a specific instant.
   • Average Velocity: Given by the slope of the secant line connecting two points at times $t_1$ and $t_2$.
   • Curving Upward: Indicates that the slope is becoming more positive (positive acceleration).
   • Flattening Out: Indicates that the magnitude of velocity is decreasing (the object is slowing down).

   Reading Key Motion Quantities

   Position–time graphs provide direct information about displacement and direction changes:

   • Displacement ($\Delta x$): Found by subtracting the initial position from the final position: $$\Delta x = x_{\text{final}} - x_{\text{initial}}$$
   • Total Distance: Found by summing the absolute values of displacements for each distinct segment of motion.
   • Change in Direction: Occurs when the slope switches signs (from positive to negative or from negative to positive). On the graph, this appears as a peak or a trough, where the tangent is momentarily horizontal ($v = 0$).
   • Passing the Origin: Occurs whenever the line crosses the horizontal time axis ($x = 0$), which does not necessarily mean the object has stopped.