Question
Student Exploration: Distance-Time and Velocity-Time Graphs
Directions: Follow the instructions to go through the simulation. Respond to the questions and prompts in the orange boxes.
[NOTE TO TEACHERS AND STUDENTS: This lesson was designed as a follow-up to the Distance-Time Graphs Gizmo. We recommend you complete that activity before this one.]
Vocabulary: displacement, distance traveled, slope, speed, velocity
Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
Dora runs one lap around the track, finishing where she started. Clark runs a 100-yard dash along the straight side of the track.
Which runner traveled a greater distance?
Which runner had a greater change in position, start to finish?
Gizmo Warm-up
The Distance-Time Graphs Gizmo shows a dynamic graph of the position of a runner over time. The Distance-Time and Velocity-Time Graphs Gizmo includes that same graph and adds two new ones: a velocity vs. time graph and a distance traveled vs. time graph.
The graph shown below (and in the Gizmo) shows a runner’s position (or distance from the starting line) over time. This is most commonly called a position-time graph.
Check that the Number of Points is 2. Turn on Show graph and Show animation for both Runner 1 and Runner 2.
Drag the points to create the graph shown to the right.
Runner 1’s line (the red one) should have endpoints at (0, 0) and (4, 40).
Runner 2’s line (the blue one) should have endpoints at (0, 40) and (4, 20).
Click the green Start button on the stopwatch. Watch the two runners carefully. In what two ways are the runners’ motions different?
Activity A:
Velocity-time graphs
Get the Gizmo ready:
Click the red Reset button on the stopwatch.
Change the Number of Points to 5.
Turn off Show graph and Show animation for Runner 2.
Speed is a measure of how fast an object moves, regardless of direction. Speed can never be negative. Velocity describes both speed and direction, and can be positive or negative.
In the Gizmo, make a position-time graph for Runner 1 with the following features:
There is at least one major change in speed.
There is at least one major change in direction.
Click the green Start button and watch the runner run. Adjust your graph if needed to meet the requirements.
✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch your graph to the right.
Where was the runner each second? Based on your graph, fill in all except the final column in the table below. (Leave the velocity column blank for now.) Label any numbers with units.
Time
Position at end of time interval (yds)
Distance moved this time interval (yds)
To the left or right?
Velocity this time interval (y/s)
0 – 1 sec
1 – 2 sec
2 – 3 sec
3 – 4 sec
To calculate the velocity for each time interval, first calculate the speed of the runner in that interval (speed = distance ÷ time). If the direction is left to right, velocity is positive. If the direction is right to left, velocity is negative.
Fill in the velocity column of the table above. Use units (y/s).
When this runner is running to the left (negative velocity), what does his position-time graph look like?
Slope is the steepness of a graph. To find the slope of a line, divide the change in y-value (rise) by the change in x-value (run). Like velocity, slope can be positive, zero, or negative.
Fill in the slope of each segment of your position-time graph, along with the runner’s velocity during each time interval, in the table below.
Time interval
Slope
Velocity (y/s)
0 sec – 1 sec
1 sec – 2 sec
2 sec – 3 sec
3 sec – 4 sec
Examine your velocities and the position-time graph you made. How is the slope of a position-time graph related to the velocity of the runner?
On the left side of the Gizmo, select the VELOCITY-TIME GRAPH tab. Use the green probes to compare the velocity-time graph to the position-time graph.
How does a velocity-time graph show that a runner is moving fast?
How does a velocity-time graph show that a runner is moving from left to right?
To the right is a position-time graph of a runner. ✏️ Hand draw in this space or click here to select EDIT to use the drawing tool.
First, sketch what you think his velocity-time graph will look like on the blank axes at the far right.
Then check your answer in the Gizmo.
Activity B:
Velocity and position
Get the Gizmo ready:
Set the Number of Points to 3.
Turn on Show graph and Show animation for both Runner 1 and Runner 2.
In the Gizmo, make the position-time graphs shown below. Click the green Start button and watch the runners run. ✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch what you think their velocity-time graphs look like on the second set of axes. (If you can, use a red line for runner 1, and a blue line for runner 2.)
Then select the VELOCITY-TIME GRAPH tab in the Gizmo. Sketch the actual graph on the third set of axes below.
Make any position -time graphs you want for Runners 1 and 2. ✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch them below. Then do the same thing – sketch what you think their velocity-time graphs look like, and then check.
Compare the velocity-time graphs to their related position-time graphs.
When do two different position-time graphs have matching velocity-time graphs?
What information is missing from a velocity-time graph?
Activity C:
Distance and displacement
Get the Gizmo ready:
Turn off Show graph and Show animation for Runner 2.
Create the position-time graph for Runner 1 shown at right. Then fill in the blanks below to describe what you think the runner will do, based on that graph.
The runner will run
yards in the first 2 seconds,
with a velocity of
y/s. His direction will be from
to
Then he will run
yards in the next 2 seconds,
with a velocity of
y/s. His direction will be from
to
Click the green Start button and watch the runner go. Were you correct?
Two students, Gina and Walter, are discussing the runner whose graph is shown above.
Gina says the runner moved more than 40 yards.
Walter says the runner moved less than 40 yards.
Who do you think is right?
Explain your answer
On top of the left half of the Gizmo, select the DISTANCE TRAVELED tab.
What was the total distance traveled by the runner after 4 seconds?
Displacement is equal to the difference between the starting and ending positions. Displacement to the right is positive while displacement to the left is negative.
What is the displacement shown by the graph at the top of the page
In the Gizmo, create a position-time graph of a runner with these characteristics:
travels a distance of 60 yards in 4 seconds
has a displacement of +10 yards
✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch your graph on the blank axes to the right.
Look at the graph you made in question 4. Think about the speed of that runner.
What was the runner’s speed for the first 2 seconds?
What was the runner’s speed for the last 2 seconds?
What was the runner’s average speed over all 4 seconds?
Now think about the velocity of the runner in question 4.
What was the runner’s velocity for the first 2 seconds?
What was the runner’s velocity for the last 2 seconds?
What was the runner’s average velocity over all 4 seconds?
Suppose you knew the time, displacement, and total distance traveled for a runner.
How would you calculate the runners’ average speed?
How would you calculate the runners’ average velocity?
To the right is a graph of a runner. Calculate the values below for this runner. Include appropriate units.
Distance traveled:
Displacement:
Average speed:
Average velocity:
Directions: Follow the instructions to go through the simulation. Respond to the questions and prompts in the orange boxes.
[NOTE TO TEACHERS AND STUDENTS: This lesson was designed as a follow-up to the Distance-Time Graphs Gizmo. We recommend you complete that activity before this one.]
Vocabulary: displacement, distance traveled, slope, speed, velocity
Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
Dora runs one lap around the track, finishing where she started. Clark runs a 100-yard dash along the straight side of the track.
Which runner traveled a greater distance?
Which runner had a greater change in position, start to finish?
Gizmo Warm-up
The Distance-Time Graphs Gizmo shows a dynamic graph of the position of a runner over time. The Distance-Time and Velocity-Time Graphs Gizmo includes that same graph and adds two new ones: a velocity vs. time graph and a distance traveled vs. time graph.
The graph shown below (and in the Gizmo) shows a runner’s position (or distance from the starting line) over time. This is most commonly called a position-time graph.
Check that the Number of Points is 2. Turn on Show graph and Show animation for both Runner 1 and Runner 2.
Drag the points to create the graph shown to the right.
Runner 1’s line (the red one) should have endpoints at (0, 0) and (4, 40).
Runner 2’s line (the blue one) should have endpoints at (0, 40) and (4, 20).
Click the green Start button on the stopwatch. Watch the two runners carefully. In what two ways are the runners’ motions different?
Activity A:
Velocity-time graphs
Get the Gizmo ready:
Click the red Reset button on the stopwatch.
Change the Number of Points to 5.
Turn off Show graph and Show animation for Runner 2.
Speed is a measure of how fast an object moves, regardless of direction. Speed can never be negative. Velocity describes both speed and direction, and can be positive or negative.
In the Gizmo, make a position-time graph for Runner 1 with the following features:
There is at least one major change in speed.
There is at least one major change in direction.
Click the green Start button and watch the runner run. Adjust your graph if needed to meet the requirements.
✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch your graph to the right.
Where was the runner each second? Based on your graph, fill in all except the final column in the table below. (Leave the velocity column blank for now.) Label any numbers with units.
Time
Position at end of time interval (yds)
Distance moved this time interval (yds)
To the left or right?
Velocity this time interval (y/s)
0 – 1 sec
1 – 2 sec
2 – 3 sec
3 – 4 sec
To calculate the velocity for each time interval, first calculate the speed of the runner in that interval (speed = distance ÷ time). If the direction is left to right, velocity is positive. If the direction is right to left, velocity is negative.
Fill in the velocity column of the table above. Use units (y/s).
When this runner is running to the left (negative velocity), what does his position-time graph look like?
Slope is the steepness of a graph. To find the slope of a line, divide the change in y-value (rise) by the change in x-value (run). Like velocity, slope can be positive, zero, or negative.
Fill in the slope of each segment of your position-time graph, along with the runner’s velocity during each time interval, in the table below.
Time interval
Slope
Velocity (y/s)
0 sec – 1 sec
1 sec – 2 sec
2 sec – 3 sec
3 sec – 4 sec
Examine your velocities and the position-time graph you made. How is the slope of a position-time graph related to the velocity of the runner?
On the left side of the Gizmo, select the VELOCITY-TIME GRAPH tab. Use the green probes to compare the velocity-time graph to the position-time graph.
How does a velocity-time graph show that a runner is moving fast?
How does a velocity-time graph show that a runner is moving from left to right?
To the right is a position-time graph of a runner. ✏️ Hand draw in this space or click here to select EDIT to use the drawing tool.
First, sketch what you think his velocity-time graph will look like on the blank axes at the far right.
Then check your answer in the Gizmo.
Activity B:
Velocity and position
Get the Gizmo ready:
Set the Number of Points to 3.
Turn on Show graph and Show animation for both Runner 1 and Runner 2.
In the Gizmo, make the position-time graphs shown below. Click the green Start button and watch the runners run. ✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch what you think their velocity-time graphs look like on the second set of axes. (If you can, use a red line for runner 1, and a blue line for runner 2.)
Then select the VELOCITY-TIME GRAPH tab in the Gizmo. Sketch the actual graph on the third set of axes below.
Make any position -time graphs you want for Runners 1 and 2. ✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch them below. Then do the same thing – sketch what you think their velocity-time graphs look like, and then check.
Compare the velocity-time graphs to their related position-time graphs.
When do two different position-time graphs have matching velocity-time graphs?
What information is missing from a velocity-time graph?
Activity C:
Distance and displacement
Get the Gizmo ready:
Turn off Show graph and Show animation for Runner 2.
Create the position-time graph for Runner 1 shown at right. Then fill in the blanks below to describe what you think the runner will do, based on that graph.
The runner will run
yards in the first 2 seconds,
with a velocity of
y/s. His direction will be from
to
Then he will run
yards in the next 2 seconds,
with a velocity of
y/s. His direction will be from
to
Click the green Start button and watch the runner go. Were you correct?
Two students, Gina and Walter, are discussing the runner whose graph is shown above.
Gina says the runner moved more than 40 yards.
Walter says the runner moved less than 40 yards.
Who do you think is right?
Explain your answer
On top of the left half of the Gizmo, select the DISTANCE TRAVELED tab.
What was the total distance traveled by the runner after 4 seconds?
Displacement is equal to the difference between the starting and ending positions. Displacement to the right is positive while displacement to the left is negative.
What is the displacement shown by the graph at the top of the page
In the Gizmo, create a position-time graph of a runner with these characteristics:
travels a distance of 60 yards in 4 seconds
has a displacement of +10 yards
✏️ Hand draw in this space or click here to select EDIT to use the drawing tool. Sketch your graph on the blank axes to the right.
Look at the graph you made in question 4. Think about the speed of that runner.
What was the runner’s speed for the first 2 seconds?
What was the runner’s speed for the last 2 seconds?
What was the runner’s average speed over all 4 seconds?
Now think about the velocity of the runner in question 4.
What was the runner’s velocity for the first 2 seconds?
What was the runner’s velocity for the last 2 seconds?
What was the runner’s average velocity over all 4 seconds?
Suppose you knew the time, displacement, and total distance traveled for a runner.
How would you calculate the runners’ average speed?
How would you calculate the runners’ average velocity?
To the right is a graph of a runner. Calculate the values below for this runner. Include appropriate units.
Distance traveled:
Displacement:
Average speed:
Average velocity:
Answers
GPT-4o mini
Sure, let’s break down each section as you follow through the simulation:
**Prior Knowledge Questions**
1. Which runner traveled a greater distance?
- Clark, who ran a 100-yard dash, traveled a greater distance than Dora, who ran one lap around the track (which would sum to 0 net distance if she ended where she started).
2. Which runner had a greater change in position from start to finish?
- Dora had a change in position of 0 yards (as she finished at her starting point), while Clark had a change of 100 yards. Therefore, Clark had a greater change in position.
---
**Gizmo Warm-up**
- **Ways the Runners’ Motions Differ**:
1. Runner 1 (red line) travels a greater distance over the same time compared to Runner 2 (blue line).
2. Runner 1 is moving consistently in one direction (increasing position), while Runner 2 starts at a higher position and moves back toward the starting point (decreasing position).
---
**Activity A:**
- **Position-Time Graph** (please sketch it according to your observations)
- **Table**:
- **Time Interval Analysis**: For each time interval (0–1 sec, 1–2 sec, etc.), fill in the position, distance moved, and direction.
- **Velocity Calculation:**
- Examples:
- 0 – 1 sec:
- Position: 0 to 10 yds
- Distance: 10 yds
- Direction: to the right
- Velocity: 10 yds/s
- (Continue filling for all intervals)
- **Velocity Analysis**:
- When the runner is moving left, the position-time graph slopes downward.
- The slope corresponds to velocity, being positive or negative based on direction.
- **Slope and Velocity**:
- The slope of the position-time graph is equal to the runner's velocity during that interval. If the slope is steep, the velocity is high.
- **Velocity-Time Graph Analysis**:
- Fast movement is indicated by a higher position on the velocity-time graph.
- Rightward movement is indicated by a positive value on the velocity-time graph.
---
**Activity B:**
- **Position-Time Graphs**: Sketch based on observed motion.
- **Velocity-Time Graphs**: Compare predictions against actual graphs.
- **Comparison**:
- Two position-time graphs can have matching velocity-time graphs if the speeds are the same but paths differ (e.g., same speed, different start/end positions).
- Missing information from velocity-time graphs includes specific distance traveled unless combined with time.
---
**Activity C:**
- **Position-Time Graph for Runner 1**: Sketch as instructed.
- **Prediction**:
- Fill in expected yardage and velocity.
- Example:
- “The runner will run 20 yards in the first 2 seconds, with velocity of 10 y/s.”
- Direction will be from left to right.
- “Then he will run 20 yards in the next 2 seconds, with the same velocity of 10 y/s from left to right.”
- **Discussion**:
- Gina is correct, as the total movement (distance) exceeds 40 yards.
- Distance traveled is the total 'path' covered, while displacement indicates net change in position.
- **Distance Traveled**: Measure from the graphs, depending on the runner's movements.
- **Displacement**: Calculate the difference in starting and ending positions.
- **Graph Creation**: Sketch as specified (60 yards in 4 seconds, +10 yards displacement).
- **Speed & Velocity Calculations**:
- Speed calculated as total distance over time.
- Velocity calculated as total displacement over time.
- Calculation Methodology:
- Average speed = Total distance / Total time.
- Average velocity = Total displacement / Total time.
- **Runner Analysis**:
- Distance traveled: Total length of runner’s path.
- Displacement: Difference between final and initial position.
- Average speed = (total distance)/(total time) with appropriate distance units (yards/seconds).
- Average velocity = (displacement)/(total time) with appropriate distance units (yards/seconds).
Feel free to fill in any sections with your observations and data from the Gizmo simulation!
**Prior Knowledge Questions**
1. Which runner traveled a greater distance?
- Clark, who ran a 100-yard dash, traveled a greater distance than Dora, who ran one lap around the track (which would sum to 0 net distance if she ended where she started).
2. Which runner had a greater change in position from start to finish?
- Dora had a change in position of 0 yards (as she finished at her starting point), while Clark had a change of 100 yards. Therefore, Clark had a greater change in position.
---
**Gizmo Warm-up**
- **Ways the Runners’ Motions Differ**:
1. Runner 1 (red line) travels a greater distance over the same time compared to Runner 2 (blue line).
2. Runner 1 is moving consistently in one direction (increasing position), while Runner 2 starts at a higher position and moves back toward the starting point (decreasing position).
---
**Activity A:**
- **Position-Time Graph** (please sketch it according to your observations)
- **Table**:
- **Time Interval Analysis**: For each time interval (0–1 sec, 1–2 sec, etc.), fill in the position, distance moved, and direction.
- **Velocity Calculation:**
- Examples:
- 0 – 1 sec:
- Position: 0 to 10 yds
- Distance: 10 yds
- Direction: to the right
- Velocity: 10 yds/s
- (Continue filling for all intervals)
- **Velocity Analysis**:
- When the runner is moving left, the position-time graph slopes downward.
- The slope corresponds to velocity, being positive or negative based on direction.
- **Slope and Velocity**:
- The slope of the position-time graph is equal to the runner's velocity during that interval. If the slope is steep, the velocity is high.
- **Velocity-Time Graph Analysis**:
- Fast movement is indicated by a higher position on the velocity-time graph.
- Rightward movement is indicated by a positive value on the velocity-time graph.
---
**Activity B:**
- **Position-Time Graphs**: Sketch based on observed motion.
- **Velocity-Time Graphs**: Compare predictions against actual graphs.
- **Comparison**:
- Two position-time graphs can have matching velocity-time graphs if the speeds are the same but paths differ (e.g., same speed, different start/end positions).
- Missing information from velocity-time graphs includes specific distance traveled unless combined with time.
---
**Activity C:**
- **Position-Time Graph for Runner 1**: Sketch as instructed.
- **Prediction**:
- Fill in expected yardage and velocity.
- Example:
- “The runner will run 20 yards in the first 2 seconds, with velocity of 10 y/s.”
- Direction will be from left to right.
- “Then he will run 20 yards in the next 2 seconds, with the same velocity of 10 y/s from left to right.”
- **Discussion**:
- Gina is correct, as the total movement (distance) exceeds 40 yards.
- Distance traveled is the total 'path' covered, while displacement indicates net change in position.
- **Distance Traveled**: Measure from the graphs, depending on the runner's movements.
- **Displacement**: Calculate the difference in starting and ending positions.
- **Graph Creation**: Sketch as specified (60 yards in 4 seconds, +10 yards displacement).
- **Speed & Velocity Calculations**:
- Speed calculated as total distance over time.
- Velocity calculated as total displacement over time.
- Calculation Methodology:
- Average speed = Total distance / Total time.
- Average velocity = Total displacement / Total time.
- **Runner Analysis**:
- Distance traveled: Total length of runner’s path.
- Displacement: Difference between final and initial position.
- Average speed = (total distance)/(total time) with appropriate distance units (yards/seconds).
- Average velocity = (displacement)/(total time) with appropriate distance units (yards/seconds).
Feel free to fill in any sections with your observations and data from the Gizmo simulation!