# Graphing Motion

## Inquiry Lab Kit for AP® Physics 1

### Materials Included In Kit

Digital publication content

Data collection interface and software
Meter sticks
Motion detectors

### Safety Precautions

The materials in this lab are considered nonhazardous. Follow all laboratory safety guidelines.

### Disposal

All materials may be saved and stored for future use.

### Lab Hints

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Design and Procedure for students to discuss and design the guided-inquiry procedures. Prelab Questions may be completed before lab begins the first day.
• Students should hold a textbook directly in front of them as they walk toward and away from the motion detector to reduce noise resulting from sound waves bouncing off uneven surfaces. Moreover, facing the detector, particularly when walking backwards, led to better graphs because a straight path was easier to maintain.
• If motion detectors are not available, discuss the type of walking necessary to generate the graphs given in the Introductory Activity and Guided-Inquiry Design and Procedure section of the investigation. Ask students to walk in front of the class and discuss whether the walking would generate the given graph. There are sufficient exercises in this investigation that absence of a motion detector does not preclude its completion nor diminish its utility and effectiveness. Moreover, videos that capture the types of motion described are readily available on the Internet for viewing and analysis.
• To generate reasonable graphs, it is necessary to consider time scales. Walking distances are fairly short and so data collection times, set in the Vernier interface, should be kept fairly short. For example, it is very difficult to maintain a constant, very slow velocity while walking a distance of 3 m in 20 seconds. Instead, data can be collected over 3–5 seconds and the resulting graphs are in general much smoother. Another consideration is the range of the motion detector. Beyond 4 meters, there may be a significant increase in the signal to noise ratio.
• The Vernier LabQuest 2 was used to collect the data presented herein. Other data collection interfaces may also be used.
• To generate reasonable graphs, it is necessary to consider time scales. Walking distances are fairly short and so data collection times, set in the Vernier interface, should be kept fairly short. For example, it is very difficult to maintain a constant, very slow velocity while walking a distance of 3 m in 20 seconds. Instead, data can be collected over 3–5 seconds and the resulting graphs are in general much smoother. Another consideration is the range of the motion detector. Beyond 4 meters, there may be a significant increase in the signal to noise ratio.
• The Introductory Activity may be conducted by student groups or presented as a teacher demonstration or cooperative class activity.

### Teacher Tips

• Students should achieve a detailed understanding of p–t, v–t and a–t graphs, particularly how one graph relates to the others via slope. For example, a p–t graph with a constant, nonzero slope indicates a constant, nonzero velocity. On a v–t graph, this will be represented as a straight horizontal line that intersects with the y-axis at a point equal to the velocity. Students should understand the three main types of motion: constant speed, and positive and negative acceleration. Students should be able to draw v–t and a–t graphs from p–t graphs. The reverse exercise is more difficult and suitable for post-lab exercises.
• Spheres or gliders may be rolled up and down inclined planes or air tracks to illustrate the types of motion described herein.
• The Vernier LabQuest 2 will recognize a motion detector when connected to it via a USB cable and automatically display position data. Use the “pen” attached to the LabQuest to select the time box and a virtual keyboard will appear. Use the virtual keyboard to change time settings in the LabQuest. In addition, use the attached pen to shuttle between the home screen and graph mode, which displays p–t and v–t graphs as they are generated.
• Note that a person’s initial distance from the motion detector corresponds to the initial point on a p–t graph’s y-axis. Thus, as a person walks toward the detector at constant speed, a p–t graph will show a line with a constant negative slope. The magnitude of the slope is equal to the person’s velocity.
• Graphs will look dramatically different with scale changes. A noisy graph can be smoothed by increasing the scale on the y-axis.

### Further Extensions

Opportunities for Inquiry

Conduct this experiment by analyzing the motion of a cart moving up or down an inclined plane toward a motion detector, or on the horizontal. In addition, the p–t, v–t and a–t graphs of a bouncing ball may be analyzed. This lab can also be run as a challenge lab in which students compete to see who can most closely match the graphs.

Alignment to the Curriculum for AP® Physics 1

Enduring Understandings and Essential Knowledge
All forces share certain common characteristics when considered by observers in inertial reference frames. (3A)
3A1: An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration.

Learning Objectives
3A1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations.
3A1.2 The student is able to design an experimental investigation of the motion of an object.
3A1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations.

Science Practices
1.4 The student can use representations and models to analyze situations or solve problems qualitatively and quantitatively.
1.5 The student can reexpress key elements of natural phenomena across multiple representations in the domain.
3.1 The student can pose scientific questions.
4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
4.3 The student can collect data to answer a particular scientific question.
4.4 The student can evaluate sources of data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
5.2 The student can refine observations and measurements based on data analysis.
5.3 The student can evaluate the evidence provided by data sets in relation to a particular scientific question.
6.1 The student can justify claims with evidence.
6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.

### Science & Engineering Practices

Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking

### Disciplinary Core Ideas

HS-PS2.A: Forces and Motion

### Crosscutting Concepts

Patterns
Scale, proportion, and quantity
Systems and system models

### Performance Expectations

HS-PS2-1. Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.
HS-PS2-2. Use mathematical representations to support the claim that the total momentum of a system of objects is conserved when there is no net force on the system.

1. Draw p–t and v–t graphs showing a person at rest, or stopped, 1 m from a motion detector.
2. Draw p–t and v–t graphs showing a person at rest, or stopped, 2 m from a motion detector.
3. Draw p–t and v–t graphs showing a person moving at a constant speed away from a motion detector.
4. Draw p–t and v–t graphs showing a person moving at a constant speed toward a motion detector.
5. Draw p–t and v–t graphs showing a person moving at a constant speed toward a motion detector, but faster than that in step 4.
6. Draw p–t, v–t and a–t graphs showing a person moving with constant acceleration away from a motion detector.
7. Draw p–t, v–t and a–t graphs showing a person moving with constant negative acceleration away from a motion detector.
8. Draw a p–t graph representing a person, initially at the 1-m mark, walking away from a motion detector to the 3-m mark in 6 seconds, pausing for ten seconds, and then walking 2 m toward the motion detector in 3 seconds.

### Sample Data

Introductory Activity

Collect p–t and v–t data for the types of motion described.

1. Stand motionless at the 2-m line for 10 seconds.
{13782_Data_Figure_23}
Note the straight line has origin at (0,2) and a slope close to zero, indicating the person is not moving. The corresponding v–t graph shows a straight line originating at (0,0). The motion detector is sensitive enough to resolve minor movements and reflection distance. As a result, the v–t curve is not perfectly linear.
2. Quickly walk away from the motion detector at a constant speed for 4 seconds.
{13782_Data_Figure_24}
Note from the p–t graph that the person begins about 0.2 m from the detector and walks with a constant velocity away from the motion detector for 4 seconds. The near constant slope on the p–t graph indicates the person is walking at a near constant speed, the magnitude of which is small (about 1 m/s) and given by the v–t graph. The person is walking away from the motion detector as evidenced by the p–t curve’s constant positive slope and the v–t graphs positive y-intercept.
3. Quickly walk towards the motion detector at a constant speed for 2 seconds.
{13782_Data_Figure_25}
In contrast to constant velocity motion away from the detector, constant velocity motion towards the detector is represented by a line with a constant negative slope on a p–t graph. The slope of the line is equal to the person’s speed, or velocity. Because the person moves at a constant speed toward the motion detector, the v–t graph shows a line with a constant zero slope in a negative quadrant of the y-axis. The blips at the beginning of the curves are attributable to non-uniform motion that arises as walking begins simultaneous to data collection.
4. Slowly walk toward the motion detector at a constant speed for 10 seconds.
{13782_Data_Figure_26}
The p–t graph shows a line with a constant negative slope because the person walked with a constant speed towards the motion detector. The slope is slightly smaller in magnitude than the slope of the line shown on the p–t curve in Part C, due to the slower speed. As a result of this slower speed, the v–t graph shown below gives a line with a constant zero slope that runs nearly along the 0-line of the y-axis.

Guided-Inquiry Discussion Questions

1. What information can be found from the slopes of p–t, v–t and a–t graphs? How does the slope of a p–t graph relate to velocity? How does the slope of a v–t graph relate to acceleration?

The slope of a p–t graph is equal to velocity. The slope of a v–t graph is equal to acceleration. The slope of an a–t graph describes the rate of change of acceleration with time. An a–t graph with a straight line of zero slope indicates constant acceleration.

2. Does the origin of a p–t graph always have to be at (0,0)?

No. When using a motion detector, the point (0,0) refers to the position of the motion detector itself. The walker cannot occupy the same space as the motion detector and so the walker’s initial position, or value on the y-axis, must be nonzero. A walker standing 1 m from the motion detector as data collection begins will have a motion graph with origin (0,1).

3. Describe any difficulties you encountered while using the motion detector in the Introductory Activity.

Students may note noisy graphs, or blips. These occur when students walk outside the detector’s high-resolution range of about 4 m. In addition, noise arises when sound waves hit objects near students or the wall behind students. Most motion detectors are sensitive enough to register even the minor movements people make when standing still.

4. Describe why choosing an appropriate data collection window, or amount of time over which data is collected, is important when using a motion detector to monitor walking in a finite space.

Consider a person trying to maintain a constant velocity while walking 4 m or so over a time span of 2 minutes. The person would have to walk incredibly slowly at a constant speed. This type of motion would be extremely difficult to reproduce. In contrast, consider a person moving very quickly towards or away from the motion detector in a classroom. If the motion detector is set to record data over 30 seconds, for example, the person will run out of space well before the motion detector stops collecting data. We found that data collection in the 3–10-second time range and 0–4-m distance range was ideal.

5. Why is scale such an important thing to consider when analyzing motion graphs?

Consider the motion graphs shown below. Both represent a person standing motionless approximately 1 m from a motion detector for 10 seconds. The top has a position scale of 1.101 m–1.030 m. As a result, the graph appears noisy. In contrast, the lower graph’s scale has been adjusted to 0.5 m–1.5 m. The curve appears smooth because minor movements become insignificant on the larger scale. The scale can be adjusted in the data collection software.

6. Qualitatively describe the motion, or walking pattern, necessary to produce the following position vs. time graphs.
7. Draw v–t and a–t graphs for each of the graphs given in number 6.

Guided-Inquiry Design and Procedure

1. Attempt to match each of the graphs given in number 6 by recording a walker’s position vs. time data.
Note the p–t curve's parabolic shape, indicative of constant acceleration. As a result, the v–t graph shows velocity increasing with time as a positively-sloped line.
Note the nonlinear shape of the p–t graph. The nonlinear slope corresponds to accelerated motion. The negative values of the changing slope correspond to movement towards the detector. The v–t graph's negative slope corresponds to constant (approx.) velocity motion towards the detector.
Note that the curve on the v–t graph crosses the 0 line when the person is at rest, has a positive slope when motion is away from the detector, and a negative slope when motion is toward the detector.
2. Describe the difficulties you encountered trying to match the graphs.

Constant velocity graphs are notably easier to generate than graphs of accelerated motion. It is very difficult to generate an idealized constant acceleration v–t graph. Generally, bumpy graphs are the result of such attempts.

AP® Physics 1 Review Questions

1. Consider a ball rolled or kicked up a hill at a constant velocity. Draw p–t and v–t graphs that represent the ball’s motion.
2. Consider the same ball rolled down the hill from an initial, stationary state. Draw p–t and v–t graphs that represent the ball’s motion.
3. What is the difference between instantaneous and average velocity?

Instantaneous velocity is speed at a specific or single moment in time. Average velocity is the average of all instantaneous velocities over a defined time period.

4. Calculate the instantaneous velocity at 5 seconds as well as the average velocity from 8 to 11 seconds using the graph shown.
To calculate average velocity, mark the two given points (8 s and 11 s) on the curve and draw a line to the y-axis to indentify corresponding positions. Calculate the average slope of the curve based on the two given points:
To calculate instantaneous velocity, draw a tangent line to the desired point (5 s) on the p–t graph. The slope of the tangent line, calculated by choosing two points on the tangent line, is equal to the instantaneous velocity:
5. What information, if any, can be gathered from the areas under curves on p–t, v–t and a–t graphs?

To determine what quantity the area underneath a motion graph corresponds to, multiply the y-axis value by the x-axis value. For example, the p–t graph has meters (m) on the y-axis and seconds (s) on the x-axis. Multiplication of the two units gives m x s, which does not correspond to a measurable, physical property or quantity. In contrast, the v–t graph has m/s on the y-axis and s on the x-axis. Multiplication of the two units gives m/s x s = m. Thus, the area under a v–t graph corresponds to travel distance. An a–t graph has m/s2 on the y-axis and s on the x-axis. Multiplication of the two units gives m/s2 x s = m/s. Thus, the area under an a–t graph corresponds to velocity.

### References

AP Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.

# Graphing Motion

## Inquiry Lab Kit for AP® Physics 1

### Introduction

When you move through a crowded hallway, you constantly change your velocity and acceleration as you encounter obstacles in your path. What would a graph of your motion look like? What information could you learn from looking at a graph showing position, velocity or acceleration versus time? Discover the connection between position, velocity and acceleration by graphing your motion.

### Concepts

• Position vs. time graphs
• Velocity vs. time graphs
• Acceleration vs. time graphs
• Positive and negative acceleration
• Constant velocity

### Background

Walking is so common that examining it might seem superfluous. Graphical analysis of walking is a great way to achieve a detailed understanding of some fundamental physics concepts. For example, a person walking at a constant speed down a school corridor engages in motion described by the following position vs. time (p–t) graph (see Figure 1).

{13782_Background_Figure_1_Position vs. time}
Notice that p, the distance from a defined point such as a motion detector, increases with time in a direct proportion. The person’s motion can be further described with a velocity vs. time (v–t) graph (see Figure 2). The v–t graph is a straight line of zero slope indicating the person is not speeding up or slowing down. The magnitude of the constant velocity is equal to the y-intercept.
{13782_Background_Figure_2_Velocity vs. time}
In contrast to constant velocity motion, p–t and v–t graphs describing constant acceleration are shown below (see Figures 3 and 4).
{13782_Background_Figure_3_Position vs. time}
{13782_Background_Figure_4_Velocity vs. time}
Notice the p–t graph (see Figure 3) is not linear. Rather, a parabolic line curves upward with an increasingly steep slope because a person walking with a constant acceleration traverses a greater distance with each passing second. The v–t graph (see Figure 4) is linear with a constant slope equal to acceleration. Steep slopes indicate high accelerations and shallow slopes indicate low accelerations. Constant acceleration, on an acceleration vs. time (a–t) graph, gives a straight line with a slope of zero. The acceleration’s magnitude is equal to the a–t graph’s y-intercept (see Figure 5).
{13782_Background_Figure_5_Acceleration vs. time}
The graphs shown can be replicated by walking in the path of a motion detector. A motion detector emits ultrasonic sound waves that bounce off of an object or person. Reflected sound waves travel back to the sensor, which measures the total time of the wave’s travel and uses the speed of sound in air to calculate the deflecting object’s or person’s position. This principle is foundational to navigation and tracking technologies, such as radar and sonar. For example, it is used to track commercial airplanes during flight, map ocean floors and catch speeding drivers.

### Experiment Overview

This advanced inquiry investigation begins with pre-lab questions to provide a foundation for carrying out the introductory activity. The introductory activity introduces students to motion detectors and directs students to walk in specific ways to generate basic p–t and v–t graphs. The guided-inquiry portion of the investigation challenges students to consider how best to collect p–t and v–t data with a motion detector and to replicate higher-level graphs. Review questions summarize and reinforce the concepts learned in the investigation.

### Materials

Data collection interface and software
Meter stick
Motion detector

### Prelab Questions

1. Draw p–t and v–t graphs showing a person at rest, or stopped, 1 m from a motion detector.
2. Draw p–t and v–t graphs showing a person at rest, or stopped, 2 m from a motion detector.
3. Draw p–t and v–t graphs showing a person moving at a constant speed away from a motion detector.
4. Draw p–t and v–t graphs showing a person moving at a constant speed towards a motion detector.
5. Draw p–t and v–t graphs showing a person moving at a constant speed towards a motion detector, but faster than that in step 4.
6. Draw p–t, v–t and a–t graphs showing a person moving with constant positive acceleration away from a motion detector.
7. Draw p–t, v–t and a–t graphs showing a person moving with a constant negative acceleration away from a motion detector.
8. Draw a p–t graph representing a person, initially at the 1-m mark, walking away from a motion detector to the 3-m mark in 6 seconds, pausing for ten seconds, and then walking 2 m toward the motion detector in 3 seconds.

### Safety Precautions

The materials in this lab are considered nonhazardous. Follow all laboratory safety guidelines.

### Procedure

Introductory Activity

1. Place a motion detector on the edge of a tabletop or flat surface slightly above waist height. Remove potential sources of interference, such as chairs and desks, from the motion detector’s path of emitted sound waves.
2. Use a meter stick to mark distance in 1-m increments from the edge of the table or motion detector. Place masking tape at each 1-m interval, up to 4 m.
3. Connect the motion detector to a computer interface. Select graph mode.
4. Stand in the path of the motion detector at the 1-m line. Hold a book or other flat surface steady in front of your body to serve as a uniform deflector for the emitted sound waves. This will reduce noise in the motion graphs.
5. Begin collecting data.
6. Collect p–t and v–t data for the types of motion described below.
1. Stand motionless at the 2-m line for 10 seconds.
2. Quickly walk away from the motion detector at a constant speed for 4 seconds.
3. Quickly walk toward the motion detector at a constant speed for 2 seconds.
4. Slowly walk toward the motion detector at a constant speed for 10 seconds.

Guided-Inquiry Design and Procedure

Form a working group with other students to discuss the following questions.

1. What information can be found from the slopes of p–t, v–t and a–t graphs? How does the slope of a p–t graph relate to velocity? How does the slope of a v–t graph relate to acceleration?
2. Does the origin of a p–t graph always have to be at (0,0)?
3. Describe any difficulties you encountered while using the motion detector in the Introductory Activity.
4. Describe why choosing an appropriate data collection window, or amount of time over which data is collected, is important when using a motion detector to monitor walking in a finite space.
5. Why is scale such an important thing to consider when analyzing motion graphs?
6. Qualitatively describe the motion, or walking pattern, necessary to produce the following position vs. time graphs.
{13782_Procedure_Figure_6}
7. Draw v–t and a–t graphs for each of the graphs given in number 6.
8. Attempt to match each of the graphs given in number 6 by recording a walker’s position vs. time data. For each graph, choose appropriate time and distance scales and explain your choices.
9. Describe the difficulties you encountered trying to match the graphs.

### Student Worksheet PDF

13782_Student1.pdf

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