INVESTIGATION B2

B2 Speed, Acceleration, and Free Fall Key Question: How does gravity affect the motion of a falling object?

Earth’s gravity pulls objects downward toward the center Materials for each group of Earth. In Investigation B1, students learned to measure yy Physics stand* the speed of a falling marble. In this investigation, students apply their knowledge to examine the speed yy Gravity Drop catcher and dropper of the marble at several points along its path. With this yy CPO Timer with 2 photogates* information, they determine how the speed of the falling yy Plumb line marble changes as it falls. Students then learn to graph yy Steel marble their results, find slope values, and create a mathematical model to confirm their measurements. yy Metric ruler* yy Calculator* Learning Goals *provided by the teacher ✔✔Make a graph of the motion of a falling marble. ✔✔Interpret motion graphs. Online Resources Available at curiosityplace.com ✔✔Explain the difference between speed yy Equipment Videos: Gravity Drop, CPO Timer and acceleration. yy Skill and Practice Sheets yy Whiteboard Resources GETTING STARTED yy Animation: Acceleration Graph Time 100 minutes yy Science Content Video: Speed vs. Time Graphs yy Student Reading: Acceleration Setup and Materials 1. Make copies of investigation sheets for students. 2. Watch the equipment video. 3. Review all safety procedures with students.

NGSS Connection This investigation builds conceptual understanding and skills for the following performance expectation. HS-PS2-4. Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.

Science and Engineering Practices Disciplinary Core Ideas Crosscutting Concepts

Using Mathematics and Computational Thinking PS2.B: Types of Interactions Patterns

Gravity Drop 29 Speed, Acceleration, and Free Fall

Vocabulary In this investigation, students use the Gravity Drop to acceleration – the change in speed over time investigate the speed of the marble at different points along its downward path. The act of dropping the marble experiment – a procedure carried out under controlled past the photogates is a single event in time. Students conditions to test a hypothesis cannot measure every change in the marble’s motion free fall – occurs when an object is accelerating due to that occurs during this event. However, by repeating the force of gravity and no other forces are acting upon similar events, such as dropping the marble from the that object same height several times, students can gather data and hypothesis– a possible explanation that can be tested by infer what happens to the marble’s motion each time comparison with scientific evidence it falls. percent error – the difference between an approximate These inferences are captured by creating a graph or measured value and an exact or known value, that relates the observations of the marble’s speed at expressed as a percent of the known value photogate B to the time for which the marble has been slope – the ratio of the vertical distance to the horizontal falling. In this way, the graph that students create is a distance that separates any two points on a line model of the change in speed, or the acceleration, speed – a measure of the distance traveled in a given a marble will likely experience during its downward amount of time travel. Looking at this model, students can verify their trend line – a line that represents the general hypothesis and make predictions about future drops relationship among data on a graph of the marble. The process of repeatedly dropping the marble is an experiment designed to test student hypotheses about the motion of a falling object. BACKGROUND Once students have measured the speed of the falling marble at several points, they can plot their data and fit a Objects in free fall on Earth accelerate downward, trend line to their graphs. From this trend line, students increasing their speed by 9.8 m/s2 every second. can estimate the marble’s acceleration. Then, the The graph below shows data for the speed over time investigation asks students to use the algebraic equation: of an object in free fall. We know that the object has a constant acceleration because the shape of the graph y variable slope x variable y intercept is a straight line. Constant acceleration means that an object’s speed changes by the same amount over a given y = mx + b time interval. vaB =+tvAB A Free fall speed vs. time

50 Time (s) Speed (m/s) By substituting values for velocity, time, and slope into 40 the equation, students can calculate the speed of the 0 0 marble at different points. Finally, students can compare 30 1 9.8 their calculated values with experimentally-found values 2 19.6 20 and calculate percent error to check the accuracy of Speed (m/s) 3 29.4 their data. 10 4 39.2 5 49.0 0 0 1 2 3 4 5 Time (s)

30 B2

5E LESSON PLAN

Engage Draw the following three graphs on the board:

Graph A Graph B Graph C

Speed (cm/s)

Science Content Video Animation Time (s) Time (s) Time (s) Speed vs. Time Graphs Acceleration Graph

Explain to students that these graphs show the motion of an object over time. Ask students to describe the motion Elaborate of the object in each graph. Then ask students to suggest In working with the gravity drop, students learn quickly different objects whose motion might be similar to these that they need to pay attention to their dropping graphs. Answers will likely vary. Accept all reasonable technique. More often than not, students will obtain answers. Students should be able to identify that speed results that indicate some experimental error, which is increasing in graph A, staying constant in graph B, can generally be attributed to a misalignment between and decreasing in graph C. This Engage activity can be the diameter of the marble and the beams of the used as a formative assessment to determine student photogates. The percent difference between your readiness for the graphing activities in this investigation. acceleration value and 9.8 m/s2 indicates the percent error in the experiment. This is explained in Part 7 of Explore the investigation. Have students complete Investigation B2, Speed, Identifying and controlling errors in measurement is an Acceleration, and Free Fall. Students apply their important scientific practice. Consider using Part 7 to knowledge to examine the speed of the marble at conduct a class discussion about error and experimental several points along its path. With this information, they design. Alternatively, you may have students conduct determine how the speed of the falling marble changes trials of this investigation with the specific goal of over time. examining error in their technique.

Explain Evaluate Revisit the Key Question to give students an opportunity yy During the investigation, use the checkpoint to reflect on their learning experience and verbalize questions as opportunities for ongoing assessment. understandings about the science concepts explored in yy After completing the investigation, have students the investigation. Curiosityplace.com resources, including answer the assessment questions on the Evaluate student readings, videos, animations, and whiteboard student sheet to check understanding of the resources, as well as readings from your current concepts presented. science textbook, are other tools to facilitate student communication about new ideas.

Gravity Drop 31 Speed, Acceleration, and Free Fall

Explore INVESTIGATION B2 Explore INVESTIGATION B2

Name ______Date ______ Setting up the experiment 1. Attach the catcher at the first hole of the physics stand. B2 Speed, Acceleration, and Free Fall Materials: ✔ Physics stand 2. Attach the dropper at the nineteenth hole of the physics stand. How does gravity affect the motion of a falling object? ✔ Gravity Drop catcher and 3. Attach photogate A at the seventeenth hole of the physics stand. Earth’s gravity pulls objects downward toward the center of Earth. In dropper 4. Attach photogate B so that it is 5 cm below photogate A. This will be Investigation B1, you learned how to measure the speed of a falling marble. ✔ CPO Timer and 2 photogates at the sixteenth hole of the physics stand. In this investigation, you will apply your knowledge to examine the speed ✔ Plumb line of the marble at several points along its path. With this information, you can 5. Connect photogates A and B to the timer. Set the timer to interval ✔ Steel marble determine how the speed of the marble changes as it falls. mode. ✔ Metric ruler 6. Use the plumb line to align the dropper and catcher. Adjust the ✔ Calculator  Formulating a hypothesis physics stand if necessary. 7. Use the steel marble to conduct a test drop and ensure that the a. If you dropped a stone off a bridge into a river, how would you describe the motion of the falling stone? dropper and catcher are aligned. If necessary, repeat Steps 6 and Does the speed of the stone change during the fall? 7 until you obtain a good drop. Check the display on the timer to ensure that the photogates are recording time intervals, or “times,” accurately.

 Conducting the experiment 1. Begin Trial 1. Drop the steel marble until you get a good drop. In

Table 1, record the time registered by photogate A (tA ). Then record

the time registered by photogate B (tB ). Finally, record the time registered when the marble passed from photogate A to photogate

B (tAB ). As the stone falls, its speed steadily increases until it hits the water. 2. For Trial 2, move photogate B so that it is 10 cm below photogate A. Again, drop the steel marble until you get a good drop and b. What about a marble in free fall? An object is in free fall if it is accelerating due to the force of gravity then record the times registered by the photogates. and no other forces are acting on it. In this investigation, you will measure the speed of a steel marble at certain places along its path. Do you think the speed of the marble will change as it falls? If so, how will 3. Complete trials 3-15. Continue to drop the marble and record data the speed change? Be sure to include the term free fall in your answer. using the same process as in Step 2. Be sure to move photogate B Sample answer: The speed of a marble in free fall will steadily to the distances indicated in Table 1 for each trial. increase until it lands in the catcher. Your answer to question b. is your hypothesis because you are making a prediction about the way the marble will move before you measure its motion. A scientist uses a hypothesis to guide his or her investigation. A hypothesis is tested with an experiment.

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Guiding the INVESTIGATION Guiding the INVESTIGATION

 Formulating a hypothesis  Setting up the experiment The investigation asks students to think about This investigation assumes that students learned to changes in motion before introducing the term align the dropper and catcher, and properly release acceleration. This is done for two reasons: First, to the marble from the dropper, during Investigation ensure that when students learn the concept of B1. If students have not completed B1, or if time acceleration, their learning is firmly grounded in a has passed since that investigation, they may have physical event. Second, the term acceleration has difficulty completing Part 2. If students exhibit connections to non-scientific uses in speech. For difficulty, consider having students review B1. instance, students may have heard of the accelerator in a car. If students associate terms like accelerator with speed, then this language association may strengthen the misconception that acceleration is the same as speed. Be alert as students develop their hypotheses. If they already think acceleration is speed, it will likely show up in the hypotheses they develop.

32 B2

Explore INVESTIGATION B2 Explore INVESTIGATION B2 Sample data: b. Compare the times registered at photogate A and the times registered at photogate B. Are they the same or different? Do you see a pattern? If so, what is it? Table 1: Falling marble data The times at A are all nearly the same. This makes sense, because Trial Distance Diameter Time A Speed at A Time B Speed at B Time from A to B, between of marble tA vA photogate A was in the same place for all drops. The times at B get tB vB photogates (cm) t (s) (cm/s) (s) (cm/s) AB shorter and shorter as photogate B moves farther and farther away (cm) (s) from the dropper. 1 5 1.90 0.0182 104.40 0.0129 147.29 0.0422 2 10 1.90 0.0183 103.83 0.0108 175.93 0.0745  Calculating speed 3 15 1.90 0.0181 104.97 0.0090 211.11 0.1049 The speed of the marble at either photogate is calculated by dividing the marble’s diameter by the time the light 4 20 1.90 0.0183 103.83 0.0079 240.51 0.1293 beam was broken as the marble dropped through the photogate. The diameter of the marble is 1.90 centimeters. 5 25 1.90 0.0185 102.70 0.0075 253.33 0.1523 Using the speed equations below, calculate the speed of the marble at photogates A and at B. Record your 6 30 1.90 0.0185 102.70 0.0070 271.43 0.1728 results in Table 1. 1.9 cm 1.9 cm Speed at A, v == Speed at B, v 7 35 1.90 0.0185 102.70 0.0066 287.88 0.19 A B Time at A, tA Time at B, tB 8 40 1.90 0.0183 103.83 0.0056 339.29 0.2075 9 45 1.90 0.0187 101.60 0.0060 316.67 0.2261 Speed values are recorded in Table 1. 10 50 1.90 0.0188 101.06 0.0057 333.33 0.2407 11 55 1.90 0.0186 102.15 0.0052 365.39 0.2563  Graphing the data 12 60 1.90 0.0185 102.70 0.0049 387.76 0.2713 Create a graph of the data. A graph gives you a “picture” of the motion of the marble as it falls. 13 65 1.90 0.0187 101.60 0.0051 372.55 0.2843 Speed vs. time for a falling marble 14 70 1.90 0.0186 102.15 0.0047 404.26 0.2986 What does a graph 15 75 1.90 of the motion of 0.0188 101.06 0.0046 413.04 0.3109 a falling marble . look like? Using your data, answer the following:

a. What happens to the time it takes for the marble to fall through photogate B as the photogate is moved Speed of the marble at B (cm/s) Falling time A to B (s) closer and closer to the catcher? 1. Make a graph with speed of the marble at B on the y-axis and the time from A to B on the x-axis. As photogate B moves closer and closer to the catcher, it is moving 2. When you finish plotting the data, draw a trend line. A trend line is a line that represents the general farther and farther away from the starting point of the dropped relationship between data points on a graph. Begin the trend line at the same x-coordinate as the first data marble. The time gets shorter and shorter because it takes less time point in the graph. End the line at the last x-coordinate in the graph. Position and angle your trend line so that it passes through or between as many of the data points as possible. for the marble to pass through the photogate as the marble travels downward.

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TEACHING TIP

Speed and velocity In this investigation, units of speed, or velocity, are given in centimeters per second (cm/s). Physics often draws a distinction between speed and velocity where speed is considered a scalar, or rate, while velocity is considered a vector. A vector is a measurement that implies both a rate and an orientation or direction. For this investigation, this distinction isn’t necessary because the Gravity Drop doesn’t change the direction, or orientation, of the marble’s travel. Both values are generally represented with the symbol v in calculations, and that is the convention used here.

Gravity Drop 33 Speed, Acceleration, and Free Fall

Explore INVESTIGATION B2 Explore INVESTIGATION B2

 Drawing conclusions

a. Was the hypothesis you formulated in Part 1 correct? Explain. Sample answer: The speed-versus-time graph shows that my hypothesis was correct. As the time from A to B increases (as photogate B is moved down the pole), the speed at B increases.

b. The terms speed and acceleration are often used to describe motion. The term acceleration means a change in speed over time. Based on this definition, was the marble accelerating as it fell? Explain your answer. Yes, the marble was accelerating as it was falling because the speed of the marble was changing.

c. What did you notice about the speeds at A? The speed at A stayed almost the same for each trial. y -label:

x-label: d. If there were changes in the speeds found at photogate A, what might this tell you? Answers may vary. Variation in the speeds found at A would indicate See Part 5 sample graph. that students used an inconsistent dropping technique.

a. What is the general shape of the data in the graph? What does the graph show you about the motion of  Developing a model the falling marble? In science, graphs and equations can be used to develop a model of the forces acting on an object like the marble in the Gravity Drop. One advantage of having these models is that they can be used to predict future Looking from left to right, in general, the data falls along a linear events. For instance, imagine if you wanted to know what the speed of the marble might be if it fell farther path. The data shows that the speed of the marble increases at a than the total height of the physics stand. With a model that accurately simulated the Gravity Drop, you could predict that speed. Another advantage of a model is that it can confirm an outcome for a set of conditions that steady rate over time. you have actually measured. For instance, you could use a model of the Gravity Drop to predict the speed of the marble under the same conditions as one of the trials you conducted earlier. Then it would be possible b. Looking at the graph, what is happening to the motion of the marble in free fall? to compare the measured values to the modeled values of speed, and this comparison may help to identify The graph shows that as the time increases from A to B, the speed measurement errors. The values predicted by a model should generally match those measured in an experiment, but if one or more measurements varies significantly from the model, and that model has a good fit with the of the marble increases at photogate B. This means that the marble other data, then it is likely some sort of measurement error has occurred. Based on this, you might change your is speeding up over time. procedure to avoid that type of error. Developing a model of the Gravity Drop begins with the trend line. The trend line gives you a way to estimate the marble’s acceleration using the concept of slope. Slope is the ratio of the vertical distance to the horizontal distance that separates any two points on a line. Sometimes slope is called “rise over run” because when it is shown as a fraction, the vertical displacement between the two points, or “rise,” appears as the numerator and the horizontal displacement, or “run,” is the denominator. In the trend line you made on your speed vs. time graph, slope is the marble’s acceleration. Before you estimate the slope of your trend line, let’s look at a sample graph. The graph below is made from four points of data.

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Part 5 sample graph

Speed of marble at B vs. time from A to B

500 450 400 350 300 250 200 150 100 50 Speed of the Marble at B (cm/s)

0.000 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Falling Time A to B (s)

34 B2

Explore INVESTIGATION B2 Explore INVESTIGATION B2

Starting with the horizontal dimension of the trend line, we find that the line extends from roughly 0.040 to Part 7 sample graph 0.135. On the graph, these coordinates are rearranged so that the resulting difference is a positive number, and 300 the result is 0.095. This is the horizontal distance, or “run,” value in the slope. 250 The next value to be found is the vertical distance, or “rise.” The lowest point of the trend line is found around 140 and the highest is found at about 240, giving us a rise of 100. Here is our resulting slope: 200 100 cm/s 150 = 1052.6 cm/s2 0.095 s

Speed (cm/s) 100 a. What is the value for the slope of the line on your speed vs. time graph? Show your work on your own 50 graph.

0 0 0.05 0.10 0.15 0.20 See 7a sample graph. Time (s) b. The slope of the line on any speed vs. time graph is equal to the object’s ______.acceleration

Distance between Time from Using the graph to check measured values Speed at B  Trial photogates A to B (cm/s) As mentioned in Part 7, a graph can be used to check the measurement accuracy for a system like the Gravity Drop. (cm) (s) The equation for a line on a graph is: y variable slope x variable y intercept 1 5 0.0422 147.286822 2 10 0.0745 175.925926 y = mx + b 3 15 0.1 172.727273 If you substitute the specific variables from your speed vs. time graph into the linear equation, the equation will appear like this: 4 20 0.13564 240.536777 vaB =+tvAB A To estimate the slope of the trend line, compare the positions of any two points along the line. Often, choosing the points where the trend line intersects grid lines from the vertical and horizontal axes of the line is the This equation uses slightly different symbols than are found in the linear equation above. In science, speed and simplest way to accomplish this. velocity are often shown with the symbol v. Meanwhile, acceleration is shown with the letter a. This equation can be used to predict the speed of the marble as time passes. Part 7 sample graph

300 Using your values for the y-intercept ( vA ), slope (a), and time ( tAB ), calculate the speeds of the marble at B

(vB ). For each trial, or row, of Table 1, you can use the values of speed at A, slope, and time from A to B to 250 calculate the speed at B. Record your calculated speeds in Table 2.

200 240 – 140 = 100 See Part 8, Trial 1 sample answer.

150 0.135 – 0.040 = 0.095

Speed (cm/s) 100

50

0 0 0.05 0.10 0.15 0.20 Time (s)

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7a sample graph Part 8, Trial 1 sample answer

According to Table 1, Speed vs. time 500 vA = 104.40 cm/s 265.8733 cm/s (0.3109, 413.5793) 450 slope = = 989.48 cm/s2 t = 0.0422 s 0.2687 s AB 400 According to Part 7, 350 a = 989.40 cm/s2 300 (413.5793 – 147.7061 = 265.8733) Solving for vB using the values above, 250 vB = atAB + vA 200 (0.3109 – 0.0422 = 0.2687) v = (989.40 cm/s2 )(0.0422 s) +104.40 cm/s 150 B

100 vB = 146.133 cm/s (0.0422,147.7061) 50

Speed of the marble at B (cm/s) 0 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Falling time A to B (s)

Gravity Drop 35 Speed, Acceleration, and Free Fall

Explore INVESTIGATION B2 Explore INVESTIGATION B2

Table 2: Calculating speed a. How do your calculated values compare with the line that represents the measured values? Answers will vary. Sample answer: The calculated values fall on the Time from Acceleration Speed at A Calculated speed at B Measured speed Trial A to B line that was drawn through the experimental values. (cm/s2) (cm/s) (cm/s) at B (s) b. What are some possible explanations for any differences between the measured and the calculated values 1 989.046 0.0422 104.3956 146.133345 147.2868 of velocity? Answers will vary. Sample answer: The calculated and the accepted 2 989.046 0.0745 103.8251 177.5090626 175.9259 value are very close. The acceleration value I calculated is 3 989.046 0.1049 104.9724 208.7232997 211.1111 989.046 cm/s2. This value is higher than the accepted value for the 4 989.046 0.1293 103.8251 231.7087827 240.5063 acceleration of gravity, 980.0 cm/s2. There is about a 1% difference between these two values. 5 989.046 0.1523 102.7027 253.3344064 253.3333

6 989.046 0.1728 102.7027 273.6098492 271.4286  Percent error 2 7 989.046 0.19 102.7027 290.6214401 287.8788 We said above that gravity on Earth causes objects to accelerate at about 9.8 m/s . How much error was there between your measurements of gravity using the Gravity Drop and this figure? To answer this question, you 8 989.046 0.2075 103.8251 309.0521788 339.2857 can calculate percent error. Percent error is often used in science to determine the accuracy of a test. Percent error can be found using this equation: 9 989.046 0.2261 101.6043 325.2275756 316.6667 Exact value - found value 10 989.046 0.2407 101.0638 339.1271987 333.3333 ×=100 % error Exact value

11 989.046 0.2563 102.1505 355.643024 365.3846 a. In this case, the acceleration of gravity, 9.8 m/s2 is the exact value because it has been found through 12 989.046 0.2713 102.7027 371.0308788 387.7551 many independent experiments. Meanwhile, the value of slope you found experimentally is the “found value.” Calculate the percent error of your findings. 13 989.046 0.2843 101.6043 382.790052 372.549 trend line slope = 989.48 cm/s2 14 989.046 0.2986 102.1505 397.4796692 404.2553 Converting cm/s22 to m/s : 15 989.046 0.3109 101.0638 408.558227 413.0435 1m/s2 989.48 cm/s2 ×= 9.8948 m/s2 1. Plot the times and calculated speeds on your speed vs. time graph. Use a different color to distinguish the 100 cm/s2 calculated speeds from the measured speeds. 2 See Part 8 sample graph. Expressing slope as a percentage of gravity (9.8 m/s ) : 9.8 m/s22− 9.8948 m/s ×100 = 0.967347%, about 1% 9.8 m/s2

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Part 8 sample graph

Speed vs. time with calculated speed at B and measured speed at B

450 400 350

300 Speed of marble at B 250 vs. time from A to B 200 Calculated speed at B 150 100 50 Speed of the Marble at B (cm/s) 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Time A to B (s)

36 B2 Notes and Reflections Evaluate INVESTIGATION B2

Name ______Date ______

1. In your own words, explain the difference between speed and acceleration. Answers may vary. Students should understand that speed is the rate of change of position for an object. Acceleration is the rate of change of speed for an object.

2. Four photogates are attached to four timers that each measure the time for the marble to fall through the light beam (shown at right). Which photogate goes with which timer? 1: D 2: B 3: C 4: A

3. The data table below shows results from a typical Gravity Drop experiment. Which trend line in the graph indicates the speed versus time relationship indicated in the table below? Choose A, B, C, or D. Note: speeds are in m/s.

Speed vs. time Distance A to B Speed at A Speed at B Time A to B 4.50 (m) (m/s) (m/s) (s) 4.00 A B 0.10 1.005 1.769 0.0783 3.50 C 0.20 1.000 2.247 0.1305 3.00 0.30 1.000 2.653 0.1724 2.50 D 0.40 1.005 3.081 0.2086 2.00 Speed at B (m/s) 0.50 1.000 3.351 0.2409 1.50 1.00 0.60 1.000 3.604 0.2703 0.00 0.10 0.20 0.30 0.70 1.000 3.898 0.2977 Time from A to B (s)

Line B represents the correct relationship.

4. A student completes a single trial using the Gravity Drop and two photogates. The student obtains the following data: Distance between Trial photogates (cm) Acceleration (cm/s2) Time from A to B (s) Speed at A (cm/s) Predicted Speed at B (cm/s) 1 5 989.046 0.5 104.3956 598.9185976

Use the information in the table to predict the speed of the marble at B. Be sure to show your work on the reverse side of this sheet. See Question 4 sample answer.

Copyright © CPO Science B2 Speed, Acceleration, and Free Fall Can be duplicated for classroom use Gravity Drop

Question 4 sample answer WRAPPING UP According to Table 1, Have your students reflect on what they vA = 104.40 cm/s learned from the investigation by answering the t = 0.0422 s AB following questions: According to Part 7, 1. What is the difference between an object’s a = 989.40 cm/s2 acceleration and an object’s speed? 2. What is the shape of a graph that describes the Solving for vB using the values above, motion of a falling marble? v = at + v B AB A 3. How can a graph be used to predict the speed of 2 vB = (989.40 cm/s )(0.0422 s) +104.40 cm/s an object in free fall?

vB = 146.133 cm/s

Gravity Drop 37 Speed, Acceleration, and Free Fall

Notes and Reflections

38