Vectors & One Dimensional Motion

Advanced Placement Physics 1Mr. Kuffer

1

Mr. Küffer’sGuide to Problem Solving…

Just Guesss!

Given- Identify given information

Unknown- Identify unknown information… What is the problem asking you to solve?

Equation- Select the appropriate equation

Substitute - Substitute the given information into the equation

Solve - Solve the problem!

Stupid Step- Put a square around your final answer to separate it from the

rest of your work, double check your units, and ask yourself… Does the answer I got seem reasonable?

Trig Review

2

Geometric Challenge

3

4

5

Vector Practice

6

The Paper River LabMr. Kuffer

Objective:

Determine the sum of two vectors Solve problems dealing with relative velocity Analyze how a boat travels on a river

Procedure / Analysis:

1. Write a brief statement explaining how the boat’s speed can be determined.

2. Determine the time that is needed for your “boat” to cross the river3. Determine your boat’s speed, with the river at rest, run at least

three trials and take the average speed.4. Average speed ______________ .5. Without discussing your ideas… record whether it will take more or

less time for the boat to cross the river when the river is flowing? _______Why?________________________________________________________________________________________________________

6. Have the “hydro-engineer” pull the paper at a constant speed, by pulling the river along the floor.

7. Measure the time it takes the boat to cross the flowing river, run at least three trials.

8. How do these results compare to the time it takes to the boat to cross the non-flowing river? ______________________________________

_________________________________________________________9. How do the two averages compare (river at rest vs moving river)?

______________________________________________________________10. Devise a method to measure the speed of the

river__________________________________________________________________________11. Solve for the resultant velocity of the boat while the river is

flowing!

*Pull the river at a constant speed and collect the necessary data

Vector Addition Worksheet

7

Vector Addition Using Graphical & Component Method

8

Vector Practice Problems

9

Vector Practice Problems Continued…

10

Vector Practice Problems Continued…

11

Vector Practice Problems Continued…

12

13

Answers to Vector Worksheet (Pages 8-14)

14

15

Solution to ‘Trig Review’ (from page 3)

16

Solution to ‘Vector Practice’ (from page 6)

17

Solution to ‘Vector Addition Worksheet’ (from page 8)

18

One Dimensional Motion

Advanced Placement Physics 1Mr. Kuffer

19

Notes on One Dimensional Motion

20

Speed and Velocity

21

Instantaneous Velocity & Speed Continued

22

Instantaneous Velocity & Speed Continued

23

Speed and Velocity Practice Problems

24

Graphical Analysis & Visualizing Data

Advanced Placement Physics 1Mr. Kuffer

25

Just a thought…Suppose you were given 8 wires…identical with the exception of length. Briefly discuss this with your lab partners and make a plan that enables you to predict the mass of a 10th wire, given its length. Once you have devised a plan, present it to Mr. Kuffer… He has a few more questions for you!

Graphical Analysis with Excel(Refer to Visualizing Data Packet as a supplemental)

1) What are the two (main) types of relationships that we will discuss this year in Concepts of Physics?

2) During an experiment, a student measured the mass of 10.0 cm3 of alcohol. The student then measured the mass of 20.0 cm3 of alcohol. In this way the data in the table below was collected. a. Plot a graph of the data, including a best-fit line. b. Describe the resulting curve.c. Use the graph to write a three variable equation for the volume

to the mass of alcohol.d. Find the units of the slope of the graph. What is the name given

to this quantity?e. What would the mass of 35 cm3 of alcohol be? (interpolate, step

17)f. A mass for 60 cm3 of alcohol be? (extrapolate, step 23)

Volume (cm3) Mass (g)10 7.920 15.830 23.740 31.650 39.6

FYI: VOCAB:

1. DEPENDENT VARIABLE – the variable that may change as a result of changes purposely made in independent variable. (Plotted on the vertical axis)

1. INDEPENDENT VARIABLE (manipulated variable) – the variable that is changed on purpose by the experimenter. (Plotted on the horizontal axis)

3) During a class demonstration, Mr. Kuffer placed a 1.0 kg mass on a horizontal table that was nearly frictionless. Mr. Kuffer then applied various horizontal forces to the mass and measured the rate at which it gained speed (was accelerated) from each force applied. The results of the experiment are shown below.a. Plot a graph of the data, including a best-fit line.

26

AP Physics 1Mr. Kuffer

b. Describe the relationship between force and acceleration according to the graph.

c. Use the graph to write an equation relating force and acceleration (Hint: Newton’s 2nd Law… F = ma).

d. Find the units of the slope of the graph.

Acceleration (m/s2) Force (N)4.9 59.8 10

15.2 1520.1 2025.0 2529.9 30

4) This time Mr. Kuffer changes the procedure a bit. The mass was varied while the force was kept constant. The acceleration of each mass was recorded. The results of the experiment are shown below.a. Plot a graph of the data, including a best-fit line for the curve. b. Describe the resulting curve.c. According to the graph, what is the relationship between mass

and the acceleration produced by a constant force?d. Find the units of the slope of the graph.

Mass (kg) Acceleration (m/s2)1 12.02 5.93 4.14 3.05 2.56 2.0

27

Velocity is the slope of a displacement time curve…

Figure a

Figure b

28

Graphing Motion & Graphical Analysis Rules:

What does it all mean?

Identify the graph

Identify the slope (+ / -)

Is the line straight? Must be constant

Is the line curved? Must be changing

Is it increasing or decreasing?

Using the rules above, write a complete description of the motion indicated by each graph below. (SEPARATE SHEET OF PAPER)

A B C D E

F G H

Graph MatchingOne of the most effective methods of describing motion is to plot graphs of position, velocity, and acceleration vs. time. From such a graphical representation, it is possible to determine in what direction an object is going, how fast it is moving, how far it traveled,

29

t t tt t

dd d

d d

d

t

d

t

d

t

and whether it is speeding up or slowing down. In this experiment, you will use a Motion Detector to determine this information by plotting a real time graph of your motion as you move across the classroom.

The Motion Detector measures the time it takes for a high frequency sound pulse to travel from the detector to an object and back. Using this round-trip time and the speed of sound, you can determine the position to the object. Logger Pro will perform this calculation for you. It can then use the change in position to calculate the object’s velocity and acceleration. All of this information can be displayed either as a table or a graph. A qualitative analysis of the graphs of your motion will help you develop an understanding of the concepts of kinematics.

w alk back and forth in front o f

M otion D etector

OBJECTIVES Analyze the motion of a student walking across the room. Predict, DESCRIBE, sketch, and test position vs. time kinematics graphs. Predict, DESCRIBE, sketch, and test velocity vs. time kinematics graphs.

MATERIALScomputer Vernier Motion DetectorVernier computer interface meter stickLogger Pro masking tape

30

PRELIMINARY QUESTIONS1. Use a coordinate system with the origin at far left and positive positions increasing to

the right. Sketch the position vs. time graph for each of the following situations:a) An object at restb) An object moving in the positive direction with a constant speedc) An object moving in the negative direction with a constant speedd) An object that is gradually accelerating in the positive direction, starting from rest

2. Sketch the velocity vs. time graph for each of the situations described above.

PROCEDUREPart l Preliminary Experiments

1. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface.

2. Place the Motion Detector so that it points toward an open space at least 4 m long. Use short strips of masking tape on the floor to mark the 1   m, 2   m, 3 m, and 4   m positions from the Motion Detector.

3. Open the file “01a Graph Matching” from the Physics with Computers folder.

4. Using Logger Pro, produce a graph of your motion when you walk away from the detector with constant velocity. To do this, stand about 1 m from the Motion Detector and have your lab partner click . Walk slowly away from the Motion Detector when you hear it begin to click.

5. Sketch what the position vs. time graph will look like if you walk faster. Check your prediction with the Motion Detector.

6. Try to match the shape of the position vs . time graphs that you sketched in the Preliminary Questions section by walking in front of the Motion Detector.

Part Il Position vs. Time Graph Matching7. Open the experiment file “01b Graph Matching.” A position vs. time graph will

appear.8. Write a complete description of how you would walk to produce this target graph.

Do this in your lab notebook.

9. To test your prediction, choose a starting position and stand at that point. Start data collection by clicking . When you hear the Motion Detector begin to click, walk in such a way that the graph of your motion matches the target graph on the computer screen.

10. If you were not successful, repeat the process until your motion closely matches the graph on the screen. Sketch your graph, including all significant data points.

11. Open the experiment file “01c Graph Matching” and repeat Steps 8 – 10, using a new target graph.

12. Answer the Analysis questions for Part II before proceeding to Part III.

31

Part IIl Velocity vs. Time Graph Matching13. Open the experiment file “01d Graph Matching.” A velocity vs. time graph will appear.

14. Describe how you would walk to produce this target graph. (written explanation should be included in your lab notebook)

15. To test your prediction, choose a starting position and stand at that point. Start by clicking . When you hear the Motion Detector begin to click, walk in such a way that the graph of your motion matches the target graph on the screen. It will be more difficult to match the velocity graph than it was for the position graph.

16. Open the experiment file “01e Graph Matching.” Repeat Steps 14 – 15 to match this graph. (Again sketch your graph)

17. Remove the masking tape strips from the floor.

ANALYSIS – (ANSWER ALL QUESTION IN LAB NOTEBOOK)Part II Position vs. Time Graph Matching

1. Describe how you walked for each of the graphs that you matched. (You should already have this question completed!!!)

2. Explain the significance of the slope of a position vs. time graph. Include a discussion of positive and negative slope.

3. What type of motion is occurring when the slope of a position vs . time graph is zero?

4. What type of motion is occurring when the slope of a position vs . time graph is constant?

5. What type of motion is occurring when the slope of a position vs . time graph is changing? Test your answer to this question using the Motion Detector.

6. Return to the procedure and complete Part III.

Part III Velocity vs. Time Graph Matching

7. Describe how you walked for each of the graphs that you matched.

8. Using the velocity vs . time graphs, sketch the position vs . time graph for each of the graphs that you matched. In Logger Pro, switch to a position vs . time graph to check your answer. Do this by clicking on the y-axis and selecting Position. What does the area under a velocity vs . time graph represent ? (You may have trouble with this question, do the best you can to answer it. We will discuss it further after the lab.) Test your answer to this question using the Motion Detector.

10. What type of motion is occurring when the slope of a velocity vs . time graph is zero?

11. What type of motion is occurring when the slope of a velocity vs . time graph is not zero? Test your answer using the Motion Detector.

32

Graph # 1

0

5

10

15

20

0 10 20 30 40

Time (s)

Dis

tanc

e (m

)

A

BC

DE

Graph # 2

0

5

10

15

20

0 10 20 30 40

Time (s)

Velo

city

(m/s

)

A

B

C

D

EF

AP PHYSICS 1 Name___________________

33

HW

HW

Motion Graphs Period_________

The following questions pertain to Graph #1.

1. Describe what is happening in each segment.A

B

C

D

E

2. What is the velocity in segment B?3. What is the velocity in segment D?4. Where does the object end up with respect to the origin?5.

The following questions pertain to Graph #2.1. Describe what is happening in each segment.

A

B

C

D

E

F

2. Where does the object start out?3. What is the acceleration in segment A?4. How far does the object travel in segment B?5. What is the acceleration in segment C?6. How far did the object travel in segment D?7. What is the acceleration in segment E?8. How far did the object travel in segment F?

The following questions pertain to Graph #3.1. Describe what is happening in each segment.

A

B

C

D

34

HW

E

2. What is the acceleration in segment A?3. What is the acceleration in segment C?4. What is the acceleration in segment E?5. If the object started 5 m behind the origin, where did it end up

after 36 seconds? *******

The following questions pertain to Graph #4.1. Describe what is happening in each segment.

A

B

C

D

E

F

2. How far does the object travel in segment A?3. What was the acceleration in segment B? 4. What was the acceleration in segment D?5. What was the acceleration in segment F?

35

GRAPH # 3

0

5

10

15

20

25

0 10 20 30 40

TIME (S)

VELO

CITY

(M/S

)

A

B

CD

E

GRAPH # 4

02468

101214

0 10 20 30 40

TIME (S)

DIST

ANCE

(M)

A

B C

DE

F

36

HW

Motion Graphs Continued (d-t graphs)…

Part 1: For each of the following graphs state the direction of motion, if the object is speeding up or slowing down, and if there is positive or negative acceleration.

Part 2: Sketch the corresponding v-t graph for each on a separate sheet of paper.

** all curves are parabolic unless noted otherwise **37

Motion Graphs Continued (v-t graphs)…

For each v-t graph, determine the direction of the motion, changes in speed and acceleration, and whether the acceleration is positive or negative.

38

Interpreting v-t Graphs

39

Interpreting v-t Graphs Continued…

For each v-t graph, create the corresponding d-t graph

40

Construct a d-t & a-t Graph

41

Construct an a-t Graph

42

Graphical Analysis of Motion

43

44

45

46

47

48

Solutions

49

Kinematics & Free Fall

Advanced Placement Physics 1Mr. Kuffer

50

Mathematical Models of MotionEquation Sheet

V = Δd / Δt = df – di / tf – ti Eqn 1

df = di + vt Eqn 2

a = Δv / Δt = vf – vi / tf – ti Eqn 3

vf = vi + at Eqn 4

Δd = ½ (vf + vi)t Eqn 5 Δd = vit + ½ at2 Eqn 6

vf2 = vi2 +2aΔd Eqn 7

List the six steps to the GUESSS Method

G - _____________ S - _____________U - _____________ S - _____________E - _____________

51

HW # 1

Use the GUESSS Method to solve all Problems (Show all Work)

1. An dragster releases its parachute at the end of a race, its velocity decreases from 36.0 m/s to 4.0 m/s over a 1.5 s time period . What is the average acceleration of the car?

2. A golf ball rolls up a hill toward a miniature-golf hole. Assign the direction toward the hole as being positive.

a) If the ball starts with a speed of 2 m/s and slows at a constant rate of .5 m/s2, What is its velocity after 2s?

b) If the Constant acceleration continues for 6 seconds, what will be the velocity then?

c) Describe in words and in a motion diagram the motion of the golf ball in part b.

SHORT ANSWER

3. What quantity is represented by the area under the curve of a velocity-time graph? _________________

4. If a velocity-time curve is a straight line parallel to the t-axis, what can you say about the acceleration? ______________________________

5. Sketch a velocity-time graph for an ambulance that goes 25 m/s toward the east for 100s, then 25 m/s toward the west for another 100s.

Mathematical Models of Motion ProblemsLinear Motion

Name:_____________ Mr.Kuffer Date: _____

52

HW # 2

1. Light form the sun reaches the Earth in 8.3 minutes. The velocity of light is 3.0 x 108 m/s. How far is the Earth away from the sun?

2. You and a friend drive 50 km. You travel at 90 km/h; your friend travels at 95 km/h. How long will your friend wait for you at the end of the trip?

3. A cyclist maintains a constant velocity of 5.0 m/s. At time t = 0.0, the cyclist is 250.0 m from point A.

a. Plot a position vs. time graph of the cyclist’s location from point A at 10.0 second intervals for 60.0 s.

b. What is the cyclist’s position from point A at 60.0 s?

c. What is the displacement from the starting point at 60.0 s?

Determining g on an InclineDuring the early part of the seventeenth century, Galileo experimentally examined the concept of acceleration. One of his goals was to learn more about freely falling objects. Unfortunately, his timing devices were not precise enough to allow him to study free fall

53

LAB # 1

directly. Therefore, he decided to limit the acceleration by using fluids, inclined planes, and pendulums. In this lab exercise, you will see how the acceleration of a rolling ball or cart depends on the ramp angle. Then, you will use your data to extrapolate to the acceleration on a vertical “ramp;” that is, the acceleration of a ball in free fall.

If the angle of an incline with the horizontal is small, a ball rolling down the incline moves slowly and can be easily timed. Using time and position data, it is possible to calculate the acceleration of the ball. When the angle of the incline is increased, the acceleration also increases. The acceleration is directly proportional to the sine of the incline angle, (. A graph of acceleration versus sin( can be extrapolated to a point where the value of sin( is 1. When sin is 1, the angle of the incline is 90°. This is

measuring time, as Galileo did, you will use a Motion Detector to determine the acceleration. You will make quantitative measurements of the motion of a ball rolling down inclines of various small angles. From these measurements, you should be able to decide for yourself whether an extrapolation to large angles is valid.

x h

Figure 1

OBJECTIVES* Use a Motion Detector to measure the speed and acceleration of a ball and a

cart rolling down an incline.* Determine the mathematical relationship between the angle of an incline and the acceleration of ball rolling down the ramp.* Determine the value of free fall acceleration, g, by extrapolating the acceleration vs. sine of track angle graph.* Compare the results for a ball with the results for a low-friction dynamics cart.* Determine if an extrapolation of the acceleration vs. sine of track angle is valid.

54

HO

MATERIALScomputer hard ball, approximately 8 cm diameterVernier computer interface rubber ball, similar sizeLogger Pro dynamics cartVernier Motion Detector meter stickramp books

PRELIMINARY QUESTIONS (THOUGHT EXPERIMENTS!!)1. One of the timing devices Galileo used was his pulse. Drop a rubber ball from a

height of about 2 m and try to determine how many pulse beats elapsed before it hits the ground. What was the timing problem that Galileo encountered?

2. Now measure the time it takes for the rubber ball to fall 2 m, using a wrist watch or wall clock. Did the results improve substantially?

3. Roll the hard ball down a ramp that makes an angle of about 10° with the horizontal. First use your pulse and then your wrist watch to measure the time of descent.

4. Do you think that during Galileo’s day it was possible to get useful data for any of these experiments? Why?

5. What should the d vs. t graph look like as the cart rolls down the incline? Sketch it!

6. What should the V vs. t graph look like as the cart rolls down the incline? Sketch it!

PROCEDURE1. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface.

2. Place a single book under one end of a 1 – 3 m long board or track so that it forms a small angle with the horizontal. Adjust the points of contact of the two ends of the incline, so that the distance, x, in Figure 1 is between 1 and 3 m.

3. Place the Motion Detector at the top of an incline. Place it so the ball will never be closer than 0.4 m.

4. Open the file “04 Determining g” from the Physics with Vernier folder.

5. Hold the hard ball on the incline about 0.5 m from the Motion Detector.

6. Click to begin collecting data; release the ball after the Motion Detector starts to click. Get your hand out of the Motion Detector path quickly. You may have to adjust the position and aim of the Motion Detector several times before you get it right. Adjust and repeat this step until you get a good run showing approximately constant slope on the velocity vs. time graph during the rolling of the ball.

7. Logger Pro can fit a straight line to a portion of your data. First indicate which portion is to be used by dragging across the graph to indicate the starting and ending times. Then click on the Linear Fit button, , to perform a linear regression of the selected data. Use this tool to determine the slope of the velocity vs. time graph, using

55

only the portion of the data for times when the ball was freely rolling. From the fitted line, find the acceleration of the ball (a.k.a… The Slope). Record the value in your data table.

8. Repeat Steps 5 – 7 two more times.

9. Measure the length of the incline, x, which is the distance between the two contact points of the ramp. See Figure 1.

10. Measure the height, h, the height of the book(s). These last two measurements will be used to determine the angle of the incline.

11. Raise the incline by placing a second book under the end. Adjust the books so that the distance, x, is the same as the previous reading.

12. Repeat Steps 5 – 10 for the new incline.

13. Repeat Steps 5 – 11 for 3, 4, and 5 books.

14. Repeat Steps 5 – 13 using a low-friction dynamics cart instead of the ball.

DATA TABLE

Data using CART

O H X Acceleration YNumber of

booksHeight of books, h

(m)

Length of incline, x

(m)

sin(

trial 1(m/s2)

trial 2(m/s2)

trial 3(m/s2)

Average acceleration

(m/s2)1

2

3

4

5

ANALYSIS1. Calculate the average acceleration for each height.

2. Using trigonometry and your values of x and h in the data table, calculate the sine of the incline angle for each height. Note that x is the hypotenuse of a right triangle.

3. Plot a graph on Excel of the average acceleration (y axis) vs. sin(. Carry the sin( axis out to 1 (one) to leave room for extrapolation.

4. Use the linear fit feature on Excel, and determine the slope. The slope can be used to determine the acceleration of the ball on an incline of any angle.

56

5. On the graph, carry the fitted line out to sin(°) = 1 on the horizontal axis, and read the value of the acceleration.1

6. How well does the extrapolated value agree with the accepted value of free-fall acceleration (g = 9.8 m/s2)? Solve for % Error!!!! Explain what it means (accuracy).

7. Why do you think the data for the dynamics cart resulted in an extrapolated value of g that was closer to the accepted value than the rolling ball data?

8. Discuss the validity of extrapolating the acceleration value to an angle of °.

EXTENSIONS1. Use the Motion Detector to measure the actual free fall of a ball. Compare the results

of your extrapolation with the measurement for free fall.

2. Compare your results in this experiment with other measurements of g.

3. Investigate how the value of g varies around the world. For example, how does altitude affect the value of g? What other factors cause this acceleration to vary from place to place? How much can g vary at a school in the mountains compared to a school at sea level?

1 Notice that extrapolating to the y value at the x = 1 point is equivalent to using the slope of the fitted line.

57

Kinematics Practice

58

59

Solutions to page 54 Kinematics Problems

60

Free Fall

61

Galileo’s Hypothesis

62

Examples Continued…

63

Picket Fence Free FallWe say an object is in free fall when the only force acting on it is the Earth’s gravitational force. No other forces can be acting; in particular, air resistance must be either absent or so small as to be ignored. When the object in free fall is near the surface of the earth, the gravitational force on it is nearly constant. As a result, an object in free fall accelerates downward at a constant rate. This acceleration is usually represented with the symbol g.

Physics students measure the acceleration due to gravity using a wide variety of timing methods. In this experiment, you will have the advantage of using a very precise timer connected to the computer and a Photogate. The Photogate has a beam of infrared light that travels from one side to the other. It can detect whenever this beam is blocked. You will drop a piece of clear plastic with evenly spaced black bars on it, called a Picket Fence. As the Picket Fence passes through the Photogate, the computer will measure the time from the leading edge of one bar blocking the beam until the leading edge of the next bar blocks the beam. This timing continues as all eight bars pass through the Photogate. From these measured times, the program will calculate the velocities and accelerations for this motion and graphs will be plotted.

Picket fence

Figure 1

OBJECTIVE* Measure the acceleration of a freely falling body (g) to better than 0.5%

precision using a Picket Fence and a Photogate.

64

LAB # 2

MATERIALScomputer Vernier PhotogateVernier computer interface Picket FenceLogger Pro clamp or ring stand to secure Photogate

PRELIMINARY QUESTIONS1. Inspect your Picket Fence. You will be dropping it through a Photogate to measure g.

The distance, measured from one edge of a black band to the same edge of the next band, is 5.0 cm. What additional information will you need to determine the average speed of the Picket Fence as it moves through the Photogate?

2. If an object is moving with constant acceleration, what is the shape of its velocity vs. time graph?

3. Does the initial velocity of an object have anything to do with its acceleration? For example, compared to dropping an object, if you throw it downward would the acceleration be different after you released it?

PROCEDURE1. Fasten the Photogate rigidly to a ring stand so the arms extend horizontally, as shown

in Figure 1. The entire length of the Picket Fence must be able to fall freely through the Photogate. To avoid damaging the Picket Fence, make sure it has a soft surface (such as a carpet) to land on.

2. Connect the Photogate to the DIG/SONIC 1 input of the Vernier computer interface.

3. Open the file “05 Picket Fence” in the Physics with Computers folder.

4. Observe the reading in the status bar of Logger Pro at the top of the screen. Block the Photogate with your hand; note that the GateState is shown as Blocked. Remove your hand and the display should change to Unblocked.

5. Click to prepare the Photogate. Hold the top of the Picket Fence and drop it through the Photogate, releasing it from your grasp completely before it enters the Photogate. Be careful when releasing the Picket Fence. It must not touch the sides of the Photogate as it falls and it needs to remain vertical. Click to end data collection.

6. Examine your graphs. The slope of a velocity vs. time graph is a measure of acceleration. If the velocity graph is approximately a straight line of constant slope, the acceleration is constant. If the acceleration of your Picket Fence appears constant, fit a straight line to your data. To do this, click on the velocity graph once to select it, then click to fit the line y = mt + b to the data. Record the slope in the data table.

7. To establish the reliability of your slope measurement, repeat Steps 5 and 6 five more times. Do not use drops in which the Picket Fence hits or misses the Photogate. Record the slope values in the data table.

65

DATA TABLE

ANALYSIS1. From your six trials, determine the minimum, maximum, and average values for the

acceleration of the Picket Fence. Record them in the data table.

2. Describe in words the shape of the position vs. time graph for the free fall.

3. Describe in words the shape of the velocity vs. time graph. How is this related to the shape of the position vs. time graph?

4. The average acceleration you determined represents a single best value, derived from all your measurements. The minimum and maximum values give an indication of how much the measurements can vary from trial to trial; that is, they indicate the precision of your measurement. One way of stating the precision is to take half of the difference between the minimum and maximum values and use the result as the uncertainty of the measurement. Express your final experimental result as the average value, ± the uncertainty. Round the uncertainty to just one digit and round the average value to the same decimal place.

For example, if your minimum, average and maximum values are 9.12, 9.93, and 10.84 m/s2, express your result as g = 9.9 ± 0.9 m/s2. Record your values in the data table

5. Determine Relative Deviation. Refer to the Paper Football Lab if you have forgotten how to solve for Relative Deviation (Sample calculation… no need to show all)

6. Compare your measurement to the generally accepted value of g (from a textbook or other source). Does the accepted value fall within the range of your values? If so, your experiment agrees with the accepted value. Determine the % Error.

7. Inspect your velocity graph. How would the associated acceleration vs. time graph look? Sketch your prediction on paper.

66

6 times 1 time

Max – Min = Unc. 2

6 times

1 time

Minimum Maximum Average

Acceleration (m/s2)

Acceleration due to gravity, g ±± m/s2

Precision (Relative Deviation) %

Accuracy %

EXTENSIONS – THOUGHT EXPERIMENTS 1. Use the distance vs. time graph to explain the motion in terms of velocity.

2. Would dropping the Picket Fence from higher above the Photogate change any of the parameters you measured? Try it.

3. Would throwing the Picket Fence downward, but letting go before it enters the Photogate, change any of your measurements? How about throwing the Picket Fence upward? Try performing these experiments.

4. How would adding air resistance change the results? Try adding a loop of clear tape to the upper end of the Picket Fence. Drop the modified Picket Fence through the Photogate and compare the results with your original free fall results.

5. Investigate how the value of g varies around the world. For example, how does altitude affect the value of g? What other factors cause this acceleration to vary at different locations? How much can g vary at a location in the mountains compared to a location at sea level?

67

Additional Problems

68

Free Fall Problems

1. A juggler performs in a room with a ceiling 2 meters above hand level.

a. What is the maximum upward speed she can give a ball without letting the ball hit the ceiling?

b. How long is the ball in the air before it is caught?

2. A stone is thrown from the top of a building with an initial velocity of 20 m/s straight upward. The building is 50 meters high, and the stone just misses the edge of the roof on its way down. Determine

a. The time needed for the stone to reach its max height.b. The max height.c. The time needed for the stone to return to the level of the

thrower.d. The velocity of the stone at this instant. e. The velocity and position of the stone at t = 5 seconds.f. The total time of flight.

3. A foul ball is hit into the stands at a baseball game. The ball rises to a height of 40 meters and is caught by a fan at the height of 30 meters as it drops back toward the field. What is its velocity in the vertical direction just before it is caught? Calculate the time in the air.

4. A parachutist descending at a speed of 10 m/s drops a camera from an altitude of 50 meters.

a. How long does it take the camera to reach the ground?b. What is the velocity of the camera just before impact?

5. A rocket moves upward starting from rest with an acceleration of 29.4 m/s2 for 4 seconds. It runs out of fuel at the end of this 4 s, and continues to move upward, acted upon only by gravity. How high does it rise?

69

70

71

72

73

74

75

76

77

78

79

Motion in One Dimension Solutions

80

11)

Unit 1 Practice Test Printed Separately

Good Luck!

81

82

83

84