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2.3 Position-Time GraphsPosition-Time Graphs

● Develop position-time graphs for moving objects.

● Use a position-time graph to interpret an object’s position or displacement.

● Make motion diagrams, pictorial representations, and position-time graphs that are equivalent representations describing an object’s motion.

In this section you will:

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2.3 Position-Time GraphsPosition-Time Graphs

Position-Time Graphs

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2.3 Position-Time GraphsPosition-Time Graphs

Graphs of an object’s position and time contain useful information about an object’s position at various times. It can be helpful in determining the displacement of an object during various time intervals.

Using a Graph to Find Out Where and When

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2.3 Position-Time GraphsPosition-Time Graphs

Using a Graph to Find Out Where and When

The data in the table can be presented by plotting the time data on a horizontal axis and the position data on a vertical axis, which is called a position-time graph.

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2.3 Position-Time GraphsPosition-Time Graphs

To draw the graph, plot the object’s recorded positions. Then, draw a line that best fits the recorded points. This line represents the most likely positions of the runner at the times between the recorded data points.

Using a Graph to Find Out Where and When

The symbol x represents the instantaneous position of the object—the position at a particular instant.

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2.3 Position-Time GraphsPosition-Time Graphs

Words, pictorial representations, motion diagrams, data tables, and position-time graphs are all representations that are equivalent. They all contain the same information about an object’s motion.

Depending on what you want to find out about an object’s motion, some of the representations will be more useful than others.

Equivalent Representations

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2.3 Position-Time GraphsPosition-Time Graphs

Considering the Motion of Multiple Objects

In the graph, when and where does runner B pass runner A?

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2.3 Position-Time GraphsPosition-Time Graphs

Step 1: Analyze the Problem

Considering the Motion of Multiple Objects

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Considering the Motion of Multiple Objects

Restate the question.

At what time do A and B have the same position?

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2.3 Position-Time GraphsPosition-Time Graphs

Step 2: Solve for the Unknown

Considering the Motion of Multiple Objects

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2.3 Position-Time GraphsPosition-Time Graphs

In the figure, examine the graph to find the intersection of the line representing the motion of A with the line representing the motion of B.

Considering the Motion of Multiple Objects

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2.3 Position-Time GraphsPosition-Time Graphs

These lines intersect at 45.0 s and at about 190 m.

Considering the Motion of Multiple Objects

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2.3 Position-Time GraphsPosition-Time Graphs

B passes A about 190 m beyond the origin, 45.0 s after A has passed the origin.

Considering the Motion of Multiple Objects

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2.3 Position-Time GraphsPosition-Time Graphs

The steps covered were:

The steps covered were:

Considering the Motion of Multiple Objects

Step 1: Analyze the Problem

Restate the questions.

Step 2: Solve for the Unknown

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2.3 Section CheckSection Check

A position-time graph of an athlete winning the 100-m run is shown. Estimate the time taken by the athlete to reach 65 m.

Question 1

A. 6.0 s

B. 6.5 s

C. 5.5 s

D. 7.0 s

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Reason: Draw a horizontal line from the position of 65 m to the line of best fit. Draw a vertical line to touch the time axis from the point of intersection of the horizontal line and line of best fit. Note the time where the vertical line crosses the time axis. This is the estimated time taken by the athlete to reach 65 m.

Answer 1

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A position-time graph of an athlete winning the 100-m run is shown. What was the instantaneous position of the athlete at 2.5 s?

Question 2

A. 15 m

B. 20 m

C. 25 m

D. 30 m

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Reason: Draw a vertical line from the position of 2.5 m to the line of best fit. Draw a horizontal line to touch the position axis from the point of intersection of the vertical line and line of best fit. Note the position where the horizontal line crosses the position axis. This is the instantaneous position of the athlete at 2.5 s.

Answer 2

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From the following position-time graph of two brothers running a 100-m dash, at what time do both brothers have the same position? The smaller brother started the race from the 20-m mark.

Question 3

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Answer 3

Answer: The two brothers meet at 6 s. In the figure, we find the intersection of lines representing the motion of one brother with the line representing the motion of other brother. These lines intersect at 6 s and at 60 m.

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2.4 How Fast?How Fast?