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Position-Time Graphs - Numerical Analysis - help9

The slope of a position-time graph at any given instant in time or during any given time interval is equal to the velocity of the object at that time or during that time interval. Being a vector, velocity has a direction. For graphs, that direction is often associated with the positive and negative sign on the slope.

There are four nearly identical versions of this question. Each version involves a slope calculation for an upward-sloped line at a specified time. The actual graph and the time value varies from version to version. Two of the versions are shown below.

Version 1

This position-time graph describes an object's motion. Use it to determine the velocity (in m/s) of the object at a time of 4.0 seconds. Enter the appropriate + or – sign.

Version 2
This position-time graph describes an object's motion. Use it to determine the velocity (in m/s) of the object at a time of 6.0 seconds. Enter the appropriate + or – sign.

The slope of the line on a position-time graph is the velocity. So this question involves calculating slope. The question asks you to determine the slope at a specified time. Since the line is straight, its slope is a constant value. The slope at 4.0 seconds is the same as the slope at 8.0 seconds or at any other time. So the specified time is not important.

To determine slope, you will need to determine the "x, y" coordinates of two points on the line. Pick points for which you are certain of what the coordinates are. The first point (at t = 0 s) and the last point on the graph would be great choices. Once you've selected the two points and determined their coordinates, calculate the ratio of the y-coordinate difference divided by the x-coordinate difference. That is, determine the ratio of ∆position to ∆time.

Avoid falling prey to the common error of failing to compute a ratio of two changes. Too many students are prone to simply finding the coordinates of a single point and calculating the ratio of the y-coordinate to the x-coordinate. This will only work if the line on the graph passes through the point (0 s, 0 m). The line on this graph does not do that. The process of determining this slope always involves finding the change in the vertical coordinate divided by the change in the horizontal coordinate for any two points located on the line. You MUST use two points.

Meaning of Slope for a p-t Graph

Determining Slope on a Position-Time Graph