GRAPHING MOTION

Unit Introduction

In the past two units, you learned about speed, velocity and acceleration. Your goals in this unit include learning how to interpret and create graphs that show motion.

Interpreting Motion Graphs

Motion can be represented by position vs. time and velocity vs. time graphs, as well as others. You can use the graphs to discover details about the speed, direction, and change in motion of an object.

If you are graphing speed on a position-time graph, a constant speed will be a straight line on a graph. The slope will indicate the relative speed of the object. If the line has a steep slope, the speed would be greater than a line that is not as steep. If the object speeds up or slows down, you can tell by the changing slope of the line. On the image above, an object with positive acceleration has a line that is curving upward as its speed increases over time. If an object is not moving, the line will be horizontal on the graph.

Here, the graph shows an object speeding up, or with positive acceleration.

Here is an example of an object with a constant speed shown on a graph.

A velocity-time graph is used to show acceleration of an object. If the line segment is pointing up from the x-axis, it is positive acceleration. If the line segment is horizontal, there is no acceleration (constant velocity). If the line segment is pointing downward, it shows negative acceleration. Here is a velocity-time graph with explanations at various points along the line segments.

Now that you may be beginning to recognize what motion looks like on a graph, watch this video clip to learn more about how to interpret information on a position-time graph. Take notes and work out problems along with the video. Submit your work as question #10 in the assessment portion of the unit.

Printable: KAHN video guide.

Once you have finished watching the video, read this article on position-time graphs. Work out the two examples and submit your work as question #11 in the assessment portion of the unit.

Now, read the article at the following website to practice interpreting velocity-time graphs. Work out the two examples and submit your work as question #12 in the assessment portion of the unit.

Creating Motion Graphs

When certain guidelines are followed, it is simple to create motion graphs. Let’s start with a position-time graph.

Recall that a position-time graph is a way to show the average speed of an object. You can also interpret instantaneous speed, average velocity, instantaneous velocity, distance, and displacement from the graph.

Use a sheet of graph paper (or print one) to create some position-time graphs. You will work only in the positive quadrant of the graphs. Label the y-axis as “position (m)” and the x-axis as “time (s).” Write the numbers 0-10 along each axis, using one number per square.

Imagine for this first example that Todd is running at a steady pace of 1 m/s. On the graph, take your pencil and start at 0,0. Go up to 1 meter on the y-axis, then go over to 1 second on the x-axis and draw a dot. This represents that in one second, Todd is running one meter.

Now, since you know Todd’s running pace, you can fill in the rest of the data in the table.

Time (seconds)	Position (meters)
0	0
1	1
2
3
4
5
6
7
8
9
10

What did you come up with? If your numbers are the same in both columns, you are correct! Todd’s pace is a steady 1 m/s, so there is no variation and his position increases by 1 meter for every 1 second.

Time (seconds)	Position (meters)
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
10	10

Now, you can plot the remaining points on your graph. From the first point, go up one and over one and place a point at 2, 2. Do the same for the rest of the points. Finally, connect the points by drawing a straight line segment. Use a ruler or a straight edge to make it straight and neat. Your graph should look like the one below.

Here is a new set of data for you to create another position-time graph.

Time (seconds)	Position (meters)
0	0
1	0.75
2	1.5
3	2.25
4	3.0
5	3.75
6	4.5
7	5.25
8	6.0
9	6.75
10	7.5

Since the positions are not in whole-meter increments, you need to approximate where the points will land on the graph. Try to be consistent when plotting your points. Your second graph should look like this one.

The next graph will be a velocity-time graph to show acceleration of an object. Write “velocity (m/s)” along the y-axis and “time (s)” along the x-axis. For every square going up the y-axis, increase by 5 m/s. For every square going across the x-axis, increase by 1 s.

Imagine a car is accelerating at a rate of 5 m/s² from rest over a period of 10 seconds. In the first second, the car reaches a velocity of 5 m/s. This is how it will look on a graph.

The rest of the data is provided in the table below. Use it to complete your velocity-time graph.

time (s)	velocity (m/s)
0	0
1	5
2	10
3	15
4	20
5	25
6	30
7	35
8	40
9	45
10	50

Plot the remaining points on your graph and draw a straight line to connect the points. When you are finished, your graph should look like this one.

You may be starting to wonder how a position graph differs from a velocity graph, since the lines look very similar. Let’s take the data from the velocity graph and reinterpret the data to more closely examine the speed of the car over time, rather than the acceleration. To do that, we will use the data to create a position graph. If the car accelerates by 5 meters per second every second, fill in the next table accordingly. Remember: velocity is increasing every second, so the position will be increasing exponentially.

Time (s)	Velocity (m/s)	Position (m)
0	0	0
1	5	5
2	10	15
3	15
4	20
5	25
6	30
7	35
8	40
9	45
10	50

You may have noticed a pattern that if you add the number in the velocity column to the past position, you get the new position. So, if the last position was 15m, and the next velocity is 15m/s, the new position is 30m. Check your work with the table below to see if you got the correct answers.

Time (s)	Velocity (m/s)	Position (m)
0	0	0
1	5	5
2	10	15
3	15	30
4	20	50
5	25	75
6	30	105
7	35	140
8	40	180
9	45	225
10	50	275

Since the position values increase greatly, create the y-axis for position up to 300m. The time on the x-axis goes up to 10s. Plot your data points. Notice that the shape will be a line that curves upward to indicate the increase in velocity, or positive acceleration.

Once the line is drawn, your graph will look something like this one.

Now that you have some practice on creating position-time and velocity-time graphs, you can complete the next activity.