key question: how does gravity affect the motion of a ... · key question: how does gravity affect...

B2INVESTIGATION

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

Materials for each group

y Physics stand*

y Gravity Drop catcher and dropper

y CPO Timer with 2 photogates*

y Plumb line

y Steel marble

y Metric ruler*

y Calculator**provided by the teacher

Online ResourcesAvailable at curiosityplace.com

y Equipment Videos: Gravity Drop, CPO Timer

y Skill and Practice Sheets

y Whiteboard Resources

y Animation: Acceleration Graph

y Science Content Video: Speed vs. Time Graphs

y Student Reading: Acceleration

Earth’s gravity pulls objects downward toward the center of Earth. In Investigation B1, students learned to measure the speed of a falling marble. In this investigation, students apply their knowledge to examine the speed of the marble at several points along its path. With this information, they determine how the speed of the falling marble changes as it falls. Students then learn to graph their results, find slope values, and create a mathematical model to confirm their measurements.

Learning Goals ✔Make a graph of the motion of a falling marble.

✔ Interpret motion graphs.

✔ Explain the difference between speed and acceleration.

GETTING STARTED

Time 100 minutes

Setup and Materials1. 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

Vocabularyacceleration – the change in speed over time

experiment – a procedure carried out under controlled conditions to test a hypothesis

free fall – occurs when an object is accelerating due to the force of gravity and no other forces are acting upon that object

hypothesis– a possible explanation that can be tested by comparison with scientific evidence

percent error – the difference between an approximate or measured value and an exact or known value, expressed as a percent of the known value

slope – the ratio of the vertical distance to the horizontal distance that separates any two points on a line

speed – a measure of the distance traveled in a given amount of time

trend line – a line that represents the general relationship among data on a graph

BACKGROUND

Objects in free fall on Earth accelerate downward, increasing their speed by 9.8 m/s2 every second.

The graph below shows data for the speed over time of an object in free fall. We know that the object has a constant acceleration because the shape of the graph is a straight line. Constant acceleration means that an object’s speed changes by the same amount over a given time interval.

0 1 2 3 4 50

10

20

30

40

50

Free fall speed vs. time

Time (s)

Sp

eed

(m/s

)

Time (s) Speed (m/s)

0

1

2

3

4

5

0

9.8

19.6

29.4

39.2

49.0

In this investigation, students use the Gravity Drop to investigate the speed of the marble at different points along its downward path. The act of dropping the marble past the photogates is a single event in time. Students cannot measure every change in the marble’s motion that occurs during this event. However, by repeating similar events, such as dropping the marble from the same height several times, students can gather data and infer what happens to the marble’s motion each time it falls.

These inferences are captured by creating a graph that relates the observations of the marble’s speed at photogate B to the time for which the marble has been falling. In this way, the graph that students create is a model of the change in speed, or the acceleration, a marble will likely experience during its downward travel. Looking at this model, students can verify their hypothesis and make predictions about future drops 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.

Once students have measured the speed of the falling marble at several points, they can plot their data and fit a trend line to their graphs. From this trend line, students can estimate the marble’s acceleration. Then, the investigation asks students to use the algebraic equation:

By substituting values for velocity, time, and slope into the equation, students can calculate the speed of the marble at different points. Finally, students can compare their calculated values with experimentally-found values and calculate percent error to check the accuracy of their data.

y = mx + by variable y interceptslope x variable

= +v at vB AB A

30

B25E LESSON PLAN

EngageDraw the following three graphs on the board:

Graph A

Speed(cm/s)

Time (s) Time (s) Time (s)

Graph B Graph C

Explain to students that these graphs show the motion of an object over time. Ask students to describe the motion of the object in each graph. Then ask students to suggest different objects whose motion might be similar to these graphs. Answers will likely vary. Accept all reasonable answers. Students should be able to identify that speed is increasing in graph A, staying constant in graph B, and decreasing in graph C. This Engage activity can be used as a formative assessment to determine student readiness for the graphing activities in this investigation.

ExploreHave students complete Investigation B2, Speed, Acceleration, and Free Fall. Students apply their knowledge to examine the speed of the marble at several points along its path. With this information, they determine how the speed of the falling marble changes over time.

ExplainRevisit the Key Question to give students an opportunity to reflect on their learning experience and verbalize understandings about the science concepts explored in the investigation. Curiosityplace.com resources, including student readings, videos, animations, and whiteboard resources, as well as readings from your current science textbook, are other tools to facilitate student communication about new ideas.

Science Content Video Speed vs. Time Graphs

Animation Acceleration Graph

ElaborateIn working with the gravity drop, students learn quickly that they need to pay attention to their dropping technique. More often than not, students will obtain results that indicate some experimental error, which can generally be attributed to a misalignment between the diameter of the marble and the beams of the photogates. The percent difference between your acceleration value and 9.8 m/s2 indicates the percent error in the experiment. This is explained in Part 7 of the investigation.

Identifying and controlling errors in measurement is an important scientific practice. Consider using Part 7 to conduct a class discussion about error and experimental design. Alternatively, you may have students conduct trials of this investigation with the specific goal of examining error in their technique.

Evaluate y During the investigation, use the checkpoint

questions as opportunities for ongoing assessment.

y After completing the investigation, have students answer the assessment questions on the Evaluate student sheet to check understanding of the concepts presented.

Gravity Drop 31


B2 Speed, Acceleration, and Free FallGravity Drop

Copyright © CPO ScienceCan be duplicated for classroom use

B2 Explore INVESTIGATION

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B2 Speed, Acceleration, and Free FallHow does gravity affect the motion of a falling object?

Earth’s gravity pulls objects downward toward the center of Earth. In Investigation B1, you learned how to measure the speed of a falling marble. In this investigation, you will apply your knowledge to examine the speed of the marble at several points along its path. With this information, you can determine how the speed of the marble changes as it falls.

Formulating a hypothesis

a. If you dropped a stone off a bridge into a river, how would you describe the motion of the falling stone? Does the speed of the stone change during the fall?

As the stone falls, its speed steadily increases until it hits the water.

b. What about a marble in free fall? An object is in free fall if it is accelerating due to the force of gravity 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 the speed change? Be sure to include the term free fall in your answer.

Sample answer: The speed of a marble in free fall will steadily 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.

Materials: ✔ Physics stand

✔ Gravity Drop catcher and dropper

✔ CPO Timer and 2 photogates

✔ Plumb line

✔ Steel marble

✔ Metric ruler

✔ Calculator

Name ____________________________________________ Date ________________________




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Setting up the experiment

1. Attach the catcher at the first hole of the physics stand.

2. Attach the dropper at the nineteenth hole of the physics stand.

3. Attach photogate A at the seventeenth hole of the physics stand.

4. Attach photogate B so that it is 5 cm below photogate A. This will be at the sixteenth hole of the physics stand.

5. Connect photogates A and B to the timer. Set the timer to interval mode.

6. Use the plumb line to align the dropper and catcher. Adjust the physics stand if necessary.

7. Use the steel marble to conduct a test drop and ensure that the dropper and catcher are aligned. If necessary, repeat Steps 6 and 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 ).

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 then record the times registered by the photogates.

3. Complete trials 3-15. Continue to drop the marble and record data using the same process as in Step 2. Be sure to move photogate B to the distances indicated in Table 1 for each trial.

Formulating a hypothesis The investigation asks students to think about changes in motion before introducing the term acceleration. This is done for two reasons: First, to ensure that when students learn the concept of acceleration, their learning is firmly grounded in a physical event. Second, the term acceleration has connections to non-scientific uses in speech. For 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.

Guiding the INVESTIGATION

Setting up the experiment This investigation assumes that students learned to align the dropper and catcher, and properly release the marble from the dropper, during Investigation B1. If students have not completed B1, or if time has passed since that investigation, they may have difficulty completing Part 2. If students exhibit difficulty, consider having students review B1.

Guiding the INVESTIGATION

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B2




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Sample data:Table 1: Falling marble data

Trial Distance between

photogates (cm)

Diameter of marble

(cm)

Time A tA

(s)

Speed at A vA

(cm/s)

Time B

tB

(s)

Speed at B

vB

(cm/s)

Time from A to B,

tAB

(s)1 5 1.90 0.0182 104.40 0.0129 147.29 0.04222 10 1.90 0.0183 103.83 0.0108 175.93 0.07453 15 1.90 0.0181 104.97 0.0090 211.11 0.10494 20 1.90 0.0183 103.83 0.0079 240.51 0.12935 25 1.90 0.0185 102.70 0.0075 253.33 0.15236 30 1.90 0.0185 102.70 0.0070 271.43 0.17287 35 1.90 0.0185 102.70 0.0066 287.88 0.198 40 1.90 0.0183 103.83 0.0056 339.29 0.20759 45 1.90 0.0187 101.60 0.0060 316.67 0.2261

10 50 1.90 0.0188 101.06 0.0057 333.33 0.240711 55 1.90 0.0186 102.15 0.0052 365.39 0.256312 60 1.90 0.0185 102.70 0.0049 387.76 0.271313 65 1.90 0.0187 101.60 0.0051 372.55 0.284314 70 1.90 0.0186 102.15 0.0047 404.26 0.298615 75 1.90 0.0188 101.06 0.0046 413.04 0.3109

.

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 closer and closer to the catcher?

As photogate B moves closer and closer to the catcher, it is movingfarther and farther away from the starting point of the dropped marble. The time gets shorter and shorter because it takes less time for the marble to pass through the photogate as the marble travels downward.




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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?

The times at A are all nearly the same. This makes sense, because photogate A was in the same place for all drops. The times at B get shorter and shorter as photogate B moves farther and farther away from the dropper.

Calculating speed

The speed of the marble at either photogate is calculated by dividing the marble’s diameter by the time the light beam was broken as the marble dropped through the photogate. The diameter of the marble is 1.90 centimeters.

Using the speed equations below, calculate the speed of the marble at photogates A and at B. Record your results in Table 1.

= =vt

vt

Speed at A, 1.9 cmTime at A,

Speed at B, 1.9 cmTime at B, A

AB

B

Speed values are recorded in Table 1.

Graphing the data

Create a graph of the data. A graph gives you a “picture” of the motion of the marble as it falls.

Spee

d of

the

mar

ble

at B

(cm

/s)

Falling time A to B (s)

Speed vs. time for a falling marble

What does a graphof the motion ofa falling marble

look like?

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.

2. When you finish plotting the data, draw a trend line. A trend line is a line that represents the general relationship between data points on a graph. Begin the trend line at the same x-coordinate as the first data 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.

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





5 of 10

y-la

bel

:

x-label:

See Part 5 sample graph.

a. What is the general shape of the data in the graph? What does the graph show you about the motion of the falling marble?

Looking from left to right, in general, the data falls along a linear path. The data shows that the speed of the marble increases at a steady rate over time.

b. Looking at the graph, what is happening to the motion of the marble in free fall?

The graph shows that as the time increases from A to B, the speed of the marble increases at photogate B. This means that the marble is speeding up over time.




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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.

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 indicatethat students used an inconsistent dropping technique.

Developing a modelIn 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 events. For instance, imagine if you wanted to know what the speed of the marble might be if it fell farther 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 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 to compare the measured values to the modeled values of speed, and this comparison may help to identify 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 other data, then it is likely some sort of measurement error has occurred. Based on this, you might change your 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.

Part 5 sample graph

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

Sp

eed

of t

he M

arb

le a

t B

(cm

/s)

Falling Time A to B (s)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350

50

100

150

200

250

300

350

400

450

500

34

B2




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0 0.05 0.10 0.15 0.200

50

100

150

200

250

300

Part 7 sample graph

Sp

eed

(cm

/s)

Time (s)

TrialDistance between

photogates (cm)

Time from A to B

(s)

Speed at B (cm/s)

1 5 0.0422 147.286822

2 10 0.0745 175.925926

3 15 0.1 172.727273

4 20 0.13564 240.536777

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 simplest way to accomplish this.

0 0.05 0.10 0.15 0.200

50

100

150

200

250

300

Part 7 sample graph

Sp

eed

(cm

/s)

Time (s)

0.135 – 0.040 = 0.095

240 – 140 = 100




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Starting with the horizontal dimension of the trend line, we find that the line extends from roughly 0.040 to 0.135. On the graph, these coordinates are rearranged so that the resulting difference is a positive number, and the result is 0.095. This is the horizontal distance, or “run,” value in the slope.

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:

=100 cm/s0.095 s

1052.6 cm/s2

a. What is the value for the slope of the line on your speed vs. time graph? Show your work on your own graph.

See 7a sample graph.

b. The slope of the line on any speed vs. time graph is equal to the object’s _____________________.

Using the graph to check measured values

As mentioned in Part 7, a graph can be used to check the measurement accuracy for a system like the Gravity Drop.

The equation for a line on a graph is: y = mx + b

y variable y interceptslope x variable

If you substitute the specific variables from your speed vs. time graph into the linear equation, the equation will appear like this:

= +v at vB AB A

This equation uses slightly different symbols than are found in the linear equation above. In science, speed and 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.

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 calculate the speed at B. Record your calculated speeds in Table 2.

See Part 8, Trial 1 sample answer.

acceleration

Part 8, Trial 1 sample answer7a sample graph

According to Table 1,vA = 104.40 cm/stAB = 0.0422 sAccording to Part 7,a = 989.40 cm/s2

Solving for vB using the values above,vB = atAB + vAvB = (989.40 cm/s2 )(0.0422 s)+104.40 cm/svB = 146.133 cm/s

Speed vs. time

Sp

eed

of t

he m

arb

le a

t B

(cm

/s)

Falling time A to B (s)

(0.3109, 413.5793)

(0.0422,147.7061)

0 0.05 0.10 0.15 0.20 0.25 0.30 0.350

100

200

300

400

50

150

250

350

450

500

(413.5793 – 147.7061 = 265.8733)

(0.3109 – 0.0422 = 0.2687)

=slope = 265.8733 cm/s0.2687 s

989.48 cm/s2

Gravity Drop 35





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Table 2: Calculating speed

Trial Acceleration (cm/s2)

Time from A to B

(s)

Speed at A (cm/s)

Calculated speed at B (cm/s)

Measured speed at B

1 989.046 0.0422 104.3956 146.133345 147.2868

2 989.046 0.0745 103.8251 177.5090626 175.9259

3 989.046 0.1049 104.9724 208.7232997 211.1111

4 989.046 0.1293 103.8251 231.7087827 240.5063

5 989.046 0.1523 102.7027 253.3344064 253.3333

6 989.046 0.1728 102.7027 273.6098492 271.4286

7 989.046 0.19 102.7027 290.6214401 287.8788

8 989.046 0.2075 103.8251 309.0521788 339.2857

9 989.046 0.2261 101.6043 325.2275756 316.6667

10 989.046 0.2407 101.0638 339.1271987 333.3333

11 989.046 0.2563 102.1505 355.643024 365.3846

12 989.046 0.2713 102.7027 371.0308788 387.7551

13 989.046 0.2843 101.6043 382.790052 372.549

14 989.046 0.2986 102.1505 397.4796692 404.2553

15 989.046 0.3109 101.0638 408.558227 413.0435

1. Plot the times and calculated speeds on your speed vs. time graph. Use a different color to distinguish the calculated speeds from the measured speeds.

See Part 8 sample graph.




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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 line that was drawn through the experimental values.

b. What are some possible explanations for any differences between the measured and the calculated values of velocity?

Answers will vary. Sample answer: The calculated and the acceptedvalue are very close. The acceleration value I calculated is 989.046 cm/s2. This value is higher than the accepted value for the acceleration of gravity, 980.0 cm/s2. There is about a 1% difference between these two values.

Percent error

We said above that gravity on Earth causes objects to accelerate at about 9.8 m/s2. How much error was there between your measurements of gravity using the Gravity Drop and this figure? To answer this question, you 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:

× =Exact value - found valueExact value

100 % error

a. In this case, the acceleration of gravity, 9.8 m/s2 is the exact value because it has been found through many independent experiments. Meanwhile, the value of slope you found experimentally is the “found value.” Calculate the percent error of your findings.

× =

−×

trend line slope = 989.48 cm/sConverting cm/s to m/s :

989.48 cm/s 1m/s100 cm/s

9.8948 m/s

Expressing slope as a percentage of gravity (9.8 m/s ) :

9.8 m/s 9.8948 m/s

9.8 m/s100 = 0.967347%, about 1%

2

2 2

22

22

2

2 2

2

Part 8 sample graph

0.05 0.10 0.15 0.20 0.25 0.30 0.350

50

100

150

200

250

300

350

400

450

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

Sp

eed

of t

he M

arb

le a

t B

(cm

/s)

Time A to B (s)

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

Calculated speed at B

36

B2Notes and Reflections



Name ____________________________________________ Date ________________________

Evaluate INVESTIGATION B2

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.

Distance A to B (m)

Speed at A (m/s)

Speed at B (m/s)

Time A to B (s)

0.10 1.005 1.769 0.0783

0.20 1.000 2.247 0.1305

0.30 1.000 2.653 0.1724

0.40 1.005 3.081 0.2086

0.50 1.000 3.351 0.2409

0.60 1.000 3.604 0.2703

0.70 1.000 3.898 0.2977

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:

TrialDistance between

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.

0.00 0.10 0.20 0.30Time from A to B (s)

Spee

d at

B (m

/s)

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50Speed vs. time

C

D

BA

WRAPPING UP

Have your students reflect on what they learned from the investigation by answering the following questions:

1. What is the difference between an object’s acceleration and an object’s speed?

2. What is the shape of a graph that describes the motion of a falling marble?

3. How can a graph be used to predict the speed of an object in free fall?

Question 4 sample answer

According to Table 1,vA = 104.40 cm/stAB = 0.0422 sAccording to Part 7,a = 989.40 cm/s2

Solving for vB using the values above,vB = atAB + vAvB = (989.40 cm/s2 )(0.0422 s)+104.40 cm/svB = 146.133 cm/s

Gravity Drop 37


Notes and Reflections

38

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