Analyzing Motion from Position-Time Graphs: Calculating Instantaneous Speeds

Motion Analysis Using Position-Time Graphs

Understanding Position-Time Graphs

A position-time graph displays how an object's position (x) changes over time (t). The slope of the graph at any point represents the object's instantaneous velocity at that time.

• Position (x): The location of the object along a straight line, measured in meters (m).

• Time (t): The elapsed time, measured in seconds (s).

• Instantaneous Velocity (v): The rate of change of position at a specific instant, given by the slope of the tangent to the curve at that point.

Calculating Instantaneous Speed from a Position-Time Graph

To find the instantaneous speed at a given time, determine the slope of the graph at that point. For straight-line segments, the speed is constant and can be calculated as:

$v = \frac{\Delta x}{\Delta t}$

• Δx: Change in position (meters)

• Δt: Change in time (seconds)

Example: Interpreting the Provided Graph

The graph shows the object's position at various times. To find the speeds at t1 = 2.0 s, t2 = 4.0 s, and t3 = 13 s:

1. Identify the relevant segment: For each time, determine which straight-line segment of the graph contains that time.

2. Calculate the slope: Use the endpoints of the segment to calculate the speed.

Step-by-Step Calculation

Time Interval (s)	Position Interval (m)	Speed Calculation	Speed (m/s)
0 to 2	0 to 7	$v_1 = \frac{7 - 0}{2 - 0}$	3.5
2 to 6	7 to 7	$v_2 = \frac{7 - 7}{6 - 2}$	0
10 to 14	10 to 7	$v_3 = \frac{7 - 10}{14 - 10}$	-0.75

Note: For t1 = 2.0 s, the object is in the interval from 0 to 2 s, moving from 0 m to 7 m. For t2 = 4.0 s, the object is in the interval from 2 to 6 s, where position remains constant at 7 m. For t3 = 13 s, the object is in the interval from 10 to 14 s, moving from 10 m to 7 m.

Summary Table of Results

Time (s)	Speed (m/s)	Explanation
2.0	3.5	Object is moving forward; positive slope.
4.0	0	Object is stationary; flat segment.
13.0	0.75	Object is moving backward; negative slope. (Speed is the magnitude: 0.75 m/s)

Key Points

• Speed is always a positive quantity (magnitude of velocity).

• On a position-time graph, steeper slopes indicate higher speeds.

• Flat segments (zero slope) indicate the object is at rest.

• Negative slopes indicate motion in the opposite direction; speed is still positive.

Example Application

If a car's position-time graph shows a horizontal line, the car is stopped. If the line slopes upward, the car is moving forward; if it slopes downward, the car is moving backward.

Additional info: The calculation for v3 uses the magnitude of the velocity (speed), so the answer is 0.75 m/s, not -0.75 m/s.