General Physics
Unit 2 – Speed and Graphing
Vocabulary
Term
Definition
Distance
Displacement
Position
Average Speed
Average Velocity
Instantaneous Speed
Acceleration
Section 1.3 Speed (pages 17-21)
Speed
What is it? (definition)
Measurement
Symbol
Units
Speed
s
Meters/second
Distance
d
Meters
time
t
seconds
Velocity
What is it? (definition)
Measurement
Symbol
Units
velocity
v
Meters/second
displacement
d
Meters
time
t
seconds
EquationGives you . . .If you know . . .
v = d/tSpeedDistance and time
d = v x tDistanceSpeed and time
t = d/vTimeDistance and speed
Acceleration
What is it? (definition)
∆v = vf - vi
If an object is accelerating then it is either
Measurement
Symbol
Units
acceleration
a
meters/second2
Change in velocity
∆v
meters/second
time
t
seconds
Speed & Velocity Examples
1. If you start from the Art Gallery and travel to the Cafe and
back to the Art Gallery in 7200 seconds:
a. What is the distance you travel?
b. What is your displacement?
c. What is your average speed?
Looking For
Given
Relationship
Solution
Average Speed
Distance traveled =
time =
d. What is your average velocity?
Looking For
Given
Relationship
Solution
Average Speed
Distance traveled =
time =
2. If you start from the Art Gallery and travel to the Cafe in
3600 seconds:
a. What is the distance you travel?
b. What is your displacement?
c. What is your average speed?
Looking For
Given
Relationship
Solution
Average Speed
Distance traveled =
time =
d. What is your average velocity?
Looking For
Given
Relationship
Solution
Average Speed
Distance traveled =
time =
3. If you start from the Bakery, travel to the Cafe, and then to
the Art Gallery in 120 seconds,
a. What is the magnitude of your displacement?
b. What distance did you travel?
c. What is your average speed?
Looking For
Given
Relationship
Solution
Average Speed
Distance traveled =
time =
d. What is your average velocity?
Looking For
Given
Relationship
Solution
Average Velocity
displacement =
time =
4. Sketch your own example of a situation where a person/object
travels with the same average speed and average velocity.
5. Sketch your own example of a situation where a person/object
travels in two different directions and has a different
displacement and distance.
Acceleration Examples
1. A skater increases her velocity from 2.0 m/s to 10.0 m/s in
3.0 seconds. What is the skater’s acceleration?
Looking For
Given
Relationship
Solution
Acceleration of the skater
Beginning speed = 2.0 m/s
Final speed = 10.0 m/s
Change in time = 3 seconds
The acceleration of the skater is 2.7 meters per second per
second.
2. A car accelerates at a rate of 3.0 m/s2. If its original
speed is 8.0 m/s, how many seconds will it take the car to reach a
final speed of 25.0 m/s?
Looking For
Given
Relationship
Solution
The time to reach the final speed.
Beginning speed = 8.0 m/s; Final speed = 25.0 m/s
Acceleration = 3.0 m/s2
`
The time for the car to reach its final speed is 5.7
seconds.
1. While traveling along a highway a driver slows from 24 m/s to
15 m/s in 12 seconds. What is the automobile’s acceleration?
(Remember that a negative value indicates a slowing down or
deceleration.)
Looking For
Given
Relationship
Solution
2. A parachute on a racing dragster opens and changes the speed
of the car from 85 m/s to 45 m/s in a period of 4.5 seconds. What
is the acceleration of the dragster?
Looking For
Given
Relationship
Solution
3. A car traveling at a speed of 30.0 m/s encounters an
emergency and comes to a complete stop. How much time will it take
for the car to stop if it decelerates at -4.0 m/s2?
Looking For
Given
Relationship
Solution
4. A helicopter’s speed increases from 25 m/s to 60 m/s in 5
seconds. What is the acceleration of this helicopter?
Looking For
Given
Relationship
Solution
∆v
d
d
a
t
t
t
s
v
Class Work
5. What is the average speed of a cheetah that sprints 100 m
East in 4 s?
Looking For
Given
Relationship
Solution
What is the cheetah’s average velocity?
Looking For
Given
Relationship
Solution
6. A runner makes one lap around a 400 m track in a time of 25.0
s. What was the runner's average speed?
Looking For
Given
Relationship
Solution
What is the runner’s average velocity?
Looking For
Given
Relationship
Solution
7. A soccer field is about 120 m long. If it takes Alex 10
seconds to run its length, what is his average speed?
Looking For
Given
Relationship
Solution
8. Calculate the average velocity of a car that drives 50 meters
North East in 25 seconds.
Looking For
Given
Relationship
Solution
9. How long would it take you to run across the high school
parking lot if the lot is 50 meters long and you run with an
average speed of 5 m/sec?
Looking For
Given
Relationship
Solution
10. Bart ran 5000 meters from the cops and an average velocity
of 6 meters/second West before he got caught. How long did he
run?
Looking For
Given
Relationship
Solution
∆v
d
d
a
t
t
t
s
v
Group Work
11. What is the average speed of a cheetah that travels 112.0
meters South in 4.0 seconds? What is the cheetah’s average
velocity?
Looking For
Given
Relationship
Solution
12. Samantha runs a 400 m lap in 53.5 s. What is her average
speed? What is her average velocity?
Looking For
Given
Relationship
Solution
13. What is the average speed of a car that traveled 300.0
meters North West in 3600 seconds? What is the cars average
velocity?
Looking For
Given
Relationship
Solution
14. Elmer Fudd shoots a bullet from his rifle with an average
speed of 720.0 m/s. What time is required to strike a target 324.0
m away?
Looking For
Given
Relationship
Solution
∆v
d
d
a
t
t
t
s
v
Homework
1. On a baseball diamond, the distance from home plate to the
pitcher’s mound is 18.5 m. If a pitcher is capable of throwing a
ball with an average speed of 38.5 m/s, how much time does it take
a thrown ball to reach home plate?
Looking For
Given
Relationship
Solution
2. A bullet travels with an average velocity of 850 m/s. How
long will it take a bullet to go 1000 m?
Looking For
Given
Relationship
Solution
3. Every summer Mr. Magoo drives to Michigan. It is 3900 m to
get there. If he drives with an average speed 100 m/s, how much
time will he spend driving?
Looking For
Given
Relationship
Solution
4. What is the average speed of a cheetah that travels 112.0
meters South in 4.0 seconds? What is the cheetah’s average
velocity?
Looking For
Given
Relationship
Solution
5. After traveling for 6.0 seconds, a runner reaches a speed of
10 m/s. What is the runner’s acceleration?
Looking For
Given
Relationship
Solution
Challenge Problems
1. It is now 10:29 a.m., but when the bell rings at 10:30 a.m.
Suzette will be late for French class for the third time this week.
She must get from one side of the school to the other by hurrying
down three different hallways. She runs down the first hallway, a
distance of 35.0 m, at a speed of 3.50 m/s. The second hallway is
filled with students, and she covers its 48.0 m length at an
average speed of 1.20 m/s. The final hallway is empty, and Suzette
sprints its 60.0 m length at a speed of 5.00 m/s. Does Suzette make
it to class on time or does she get detention for being late again?
Show all of your work.
Given
Formula
Set-up
Solution
2. During an Apollo moon landing, reflecting panels were placed
on the moon. This allowed earth-based astronomers to shoot laser
beams at the moon's surface to determine its distance. The
reflected laser beam was observed 2.52 s after the laser pulse was
sent. The speed of light is 3.0 108 m/s. What was the distance
between the astronomers and the moon?
Given
Formula
Set-up
Solution
3. For many years, the posted highway speed limit was 88.5 km/hr
(55 mi/hr) but in recent years some rural stretches of highway have
increased their speed limit to 104.6 km/hr (65 mi/hr). In Maine,
the distance from Portland to Bangor is 215 km. How much time can
be saved in making this trip at the new speed limit?
Given
Formula
Set-up
Solution
4. The tortoise and the hare are in a road race to defend the
honor of their breed. The tortoise crawls the entire 1000. m
distance at a speed of 0.2000 m/s while the rabbit runs the first
200.0 m at 2.000 m/s The rabbit then stops to take a nap for 1.300
hr and awakens to finish the last 800.0 m with an average speed of
3.000 m/s. Who wins the race and by how much time?
Given
Formula
Set-up
Solution
5. Two physics professors challenge each other to a 100. m race
across the football field. The loser will grade the winner's
physics labs for one month. Dr. Rice runs the race in 10.40 s. Dr.
De La Paz runs the first 25.0 m with an average speed of 10.0 m/s,
the next 50.0 m with an average speed of 9.50 m/s, and the last
25.0 m with an average speed of 11.1 m/s. Who gets stuck grading
physics labs for the next month?
Activity – Toy Car
General Physics
Research Question
· What is the relationship between position and time for a
moving object?
· What does the slope of a position vs. time graph
represent?
· How can we determine if an object moves with constant
speed?
Constant speed: The same amount of distance is covered each
second.
Hypothesis
As time increases the distance a toy car travels will (increase,
decrease, remain the same).
Procedure
1. Choose a starting point for your car. Mark this point with
masking tape, and label it “starting point.”
2. Start the car, and place it on the starting point. Release
the car (your lab partner should start the stopwatch at the same
time). Let the car move in a straight line for 2.0 s. Repeat for
several trials, until you find the point that the car consistently
crosses after 2.0 s. Mark this point with masking tape, and label
it “0.00 m.” Throughout this experiment, you will start the car at
the original starting point, but you will begin to measure the
distance and time of the car’s motion when the car crosses the 0.00
m mark.
3. Start the car, and place it on the floor at the starting
point. Observe the car as it moves. Be sure to start the stopwatch
as the car crosses the 0.00 m mark.
4. After 10.0 s, mark the position of the car with the masking
tape. Label this mark “10.0 s.”
5. Repeat steps 3 and 4 for 9.0 s, 8.0 s, 7.0 s, 6.0 s, 5.0 s,
4.0 s, 3.0 s, and 2.0 s. Be sure to label each point according to
how much time it took for the car to get to that point from the
0.00 m mark.
6. Use the meter stick to measure the exact position from the
0.00 m mark to each time position mark. (Do not measure the
distance from the starting point.)
Name: ______________________________ Date: _____________ Period:
__________ ID: _________
7. For each position marked with tape record in Table 1 on the
next page
12 | Page
Research Question
The research questions of this activity was to
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Hypothesis
I predicted as time increases the distance a toy car travels
will (increase, decrease, remain the same) because
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Data
Table 1: Position vs. Time
Time (sec)
Position (m)
0
1
2
3
4
5
6
7
8
9
10
Average time (seconds)
Position
(meters)
Position vs. time
Conclusion
Directions: Answer the following questions in complete
sentences.
1) What was the shape of the line on your graph, curved or
straight?
______________________________________________________________________________
______________________________________________________________________________
2) Does the shape of your line agree with your hypothesis?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
3) Did your vehicle move at a constant speed? How do you
know?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
4) What variable did you plot on the y-axis on your graph?
5) What variable did you plot on the x-axis on your graph?
6) Slope refers to the steepness of a line or surface and is
found by dividing the change in your vertical axis (rise) over the
change in your horizontal axis (run).
The slope of your graph is equal to: answer to #4/answer to
#5
Answer to #4
Answer to #5
Graphing Little Dudes
General Physics
8
6
4
2
8
6
4
2
Suppose something is moving. If you collect corresponding clock
reading and position measurements, these numbers form ordered pairs
that can easily be graphed. Consider the various little dudes shown
below. They exist and move along a sidewalk marked in 1 meter
increments. We are given snapshots 2-second time intervals.
Analysis
1. Draw the line of the graph and label the line, “Walking
Dude.”
2. What is the independent variable on the graph?
3. What is the dependent variable on the graph?
4. What does the notation Δ t mean and what is Δ t between 4s
and 6s?
5. What does the notation Δ x mean and what is Δ x on the graph
between 4s and 6s?
6. What relation can you use to find the slope of the graph, in
terms of rise and run?
7. What quantity represents rise on our graph? What represents
run?
8. What equation would you use to determine the slope of a
position vs. clock reading graph? (Do not use any numbers yet,
simply state the equation.) Does this equation look familiar? If
not, it is wrong; if so, where have you seen it before?
9. Apply the equation and determine the slope of Walking Dude’s
position vs. clock reading graph.
10. On the axes on the front, plot position vs. clock reading
for the two other little dudes shown below. Running Dudette starts
at 0m and 0s. Reading Dude starts at 8m and 0s. Don’t forget to
label.
Running Dudette
time
(seconds)
Position
(meters)
Speed
(m/s)
0
2
4
6
8
Reading Dude
time
(seconds)
Position
(meters)
Speed
(m/s)
0
2
4
6
8
Qualitative graphs
(m/s)
t (s)
x
(m)
t (s)
(m/s)
t (s)
x
(m)
t (s)
(m/s)
t (s)
x
(m)
t (s)
Walk This Way
Purpose
To develop an understanding of motion graphs by analyzing graphs
made by walking in front of a motion probe. You will use a motion
probe to see what the graph of your motion looks like when you move
with increasing, decreasing and constant speed. To collect motion
data using an ultrasonic motion detector and analyze plots for
evidence that verifies the rules of kinematics.
Research Question
__________________________________________________________________________________________
Hypothesis
Describe how you would walk to produce a graph of constant
speed.
__________________________________________________________________________________________
__________________________________________________________________________________________
Describe how you would walk to produce a graph of increasing
speed.
__________________________________________________________________________________________
__________________________________________________________________________________________
Describe how you would walk to produce a graph of decreasing
speed.
__________________________________________________________________________________________
__________________________________________________________________________________________
Introduction
A motion probe sends out pulses of ultrasound and receives the
echoes from objects in front of it. The motion probe can be
connected to a CBL and graphing calculator or to a computer with an
interface (some motion probes connect directly to a graphing
calculator and need no CBL). When the object in front of the motion
probe moves, a graph of the motion of the object is displayed on
the screen of the calculator. There are several things you should
keep in mind as you carry out this investigation:
1. The motion probe (or wall) is the origin or reference point
from which distances are measured.
2. The motion probe detects the closest object directly in front
of it (including your arms if you swing them as you walk or
anything close to your path).
3. The motion probe will not correctly measure anything closer
than .5 m. When making your graphs don’t go closer than .5 m from
the motion probe.
4. A good way to make the motion graph is to hold the
calculator, interface, and motion probe and point the motion probe
toward a wall. This will allow you to see the graph on the
calculator screen as you walk.
A plot of velocity versus time gives certain information about
an object’s motion including how fast and in what direction an
object is moving and whether the object is speeding up or slowing
down. It is also possible to determine an object’s displacement
using a velocity vs. time plot by computing the area underneath the
graph during a given interval of time. This property is true for a
velocity vs time curve, regardless of its shape.
You will present your findings to the class.
Materials:
Computer interface and software, printer, masking tape, meter
stick
Procedure:
Part I: Graphing Position vs. Time
1. Start collecting data. At this point, the motion probe should
start ticking and a graph should be drawn on the computer. Try
moving back and forth to see what effect this has on the graph.
2. Your group will be assigned one of the qualitative challenges
below.
3. Have each group member try the challenge. After several
trials show Mr. Beatty. Once he approves your graph you can copy it
on the next page.
4. You will then collaborate with a member from each group to
get their qualitative graph. You each must explain what you did to
produce your graph.
Group
Qualitative Graph
A
Motion: You remain at rest (motionless) at the 2 meter mark from
the detector.
B
Motion: You walk slowly from the 1 meter mark to the 3 meter
mark.
C
Motion: You walk slowly from the 3 meter mark to the 1 meter
mark.
D
Motion: Starting at the 1 meter mark, you walk slowly to the 3
meter mark, then quickly back to the 1 meter mark.
E
Motion: Starting at the 3 meter mark, walk quickly to the 2
meter mark. Wait there for 2-3 seconds, then walk very slowly to
the 0.5 meter mark.
Analysis:
1. Explain the significance of the slope of a distance vs. time
graph. Include a discussion of positive and negative slope.
2. What type of motion is occurring when the slope of a distance
vs. time graph is zero?
3. What type of motion is occurring when the slope of a distance
vs. time graph is constant?
4. What type of motion is occurring when the slope of a distance
vs. time graph is changing?
5. Acceleration is the rate of change of speed. When speed is
constant acceleration is zero (no change means the rate of change
is zero). How does the slope of a distance vs. time graph change
when there is acceleration?
6. Describe how a person would have to move to create a distance
vs. time graph as pictured.
Part II: The Matching Challenge
1. Open the file “Position vs. Time”. Study it to determine the
following:
a. How close should you be to the motion Sensor at the
beginning?
b. How far away should you move?
c. How long should your motion last?
2. Walk the graph trying to duplicate the lines as precisely as
possible.
3. Over the course of the next week you will be quizzed on your
ability to produce the motions we have discussed in this activity.
You will be challenged with a graph similar to the matching
challenge.
Part II: For each graph have a group member walk matching each
graph then describe their motion next to the picture of the graph.
Use word like faster, slower, going toward, going away, increasing
speed, decreasing speed, steeper, less steep.
Graph 1
Score:
Graph 2
Score:
Homework for Lab 1: Introduction to Motion
Position—Time Graphs
Answer the following questions in the spaces provided.
1. What do you do to create a horizontal line on a position—time
graph?
2. How do you walk to create a straight line that slopes up?
3. How do you walk to create a straight line that slopes
down?
4. How do you move so the graph goes up steeply at first, and
then continues up less steeply?
5. How do you walk to create a Ushaped graph?
Answer the following about two objects, A and B, whose motion
produced the following position—time graphs.
6. a. Which object is
moving faster
A or B?
b. Which starts ahead? Define what you
mean by “ahead.”
c. What does the intersection mean?
7. a. Which object is
moving faster?
b. Which object has a
negative velocityaccording to the convention we have
established?
8. a. Which object is
moving faster?
b. Which starts ahead? Explain what you
mean by “ahead.”
Project - The Ford Challenge
General Physics
The Challenge
Design a car that is fast and safe. Your design modifications
are limited to 5 sheets of paper, glue, axles, and wheels.
Research Question
How can you calculate the speed of your car to determine if it
is the fastest in the class? How can you determine if your car
accelerates the fastest?
Design Criteria
1. Your car can be made of only the following materials – 5
sheets of paper, 4 wheels, cut drinking straws for the axle
housings, and glue.
2. The drinking straw may only be used as axle housing. It
cannot be used for any other structural element of the car.
3. The car’s width must be less than 6.5 cm.
4. The car’s mass must be less than 30 grams.
5. There can be no physical contact between the device and the
designer once the vehicle has been released on the track.
6. The design must allow for easy removal and inspection of the
egg. The egg will be provided by your teacher, and will be put into
the car when time has elapsed.
7. Repairs requiring additional materials will not be allowed
once the competition has begun.
8. All vehicles must display the following: vehicle’s name,
builder’s name, vehicle’s width in cm, and vehicle’s mass in grams
(without egg).
9. All vehicles must be decorated.
10. All vehicles must have a theme that is consistent with the
way it is decorated.
11. Refer to the grading sheet. Ask your teacher for a copy. You
will receive points for a unique design, quality of construction,
vehicle performance, accuracy of measurements, and completion of
the lab report.
The Race Part I
1. Time how long it takes the car to travel from the top to the
bottom of the race track. Record the time (t) for this trial (#1)
in Data Table 1.
2. Conduct two more trials. Record your time (t) for each trial
in Data Table 1.
3. Calculate the average time from start to finish. Record the
average time of the run (t) in Data Table 2.
4. Compute the average speed (v) of your car by dividing the
distance traveled by the race time
Data Table 1
Trial #
Time to Travel Down Race Track (seconds)
1
2
3
Average
Given
Formula
Set Up (with units)
Solution (with units)
Grading Criteria
50 points
Grading Categories
Points Earned
Design Plan (15 pts.): Is the design unique and creative? Does
the design resemble another student’s in class?
Quality of construction (20 pts.): Does construction show
evidence of quality time and effort or of a last minute rush job?
Is the car decorated? Does the vehicle have a theme? Do the
decorations match them theme of the car? Does the student use color
to add to the vehicle’s appearance? Does the student use more than
just colored pencils, crayons and markers to decorate their
car?
Car holds the egg and rolls entire track length (10 pts.)
Mass under 30 grams
(1 pt.)
Width under 6.5 cm
(1 pt.)
Data written on car
(1 pt.)
Sample Calculation Provided
(2 pts.)
00.050.10.150.20.250.30.350.40.450.50.550000000000000040.60.650.70.750.80.850.90.9511.051.10000000000000011.14999999999999991.21.251.31.351.41.451.51.551.61.651.71.751.81.851.91.9522.04999999999999982.12.152.20000000000000022.252.29999999999999982.352.42.45000000000000022.52.54999999999999982.62.652.72.752.82.852.92.9533.053.13.153.23.253.33.353.43.453.53.553.63.653.73.753.83.853.93.9544.054.09999999999999964.15000000000000044.24.254.34.34999999999999964.40000000000000044.454.54.554.59999999999999964.65000000000000044.74.754.84.84999999999999964.90000000000000044.9555.055.09999999999999965.155.25.255.35.355.45.455.55.555.65.655.75.755.85.855.95.9566.056.16.156.26.256.36.356.46.456.56.556.66.656.76.756.86.856.96.9577.057.17.157.27.257.37.357.47.457.57.557.67.657.77.757.87.857.97.9588.05000000000000078.18.158.19999999999999938.258.30000000000000078.358.48.44999999999999938.58.55000000000000078.68.658.69999999999999938.758.80000000000000078.858.98.949999999999999399.05000000000000079.19.159.19999999999999939.259.30000000000000079.359.49.44999999999999939.59.55000000000000079.69.659.69999999999999939.759.80000000000000079.859.99.94999999999999931010.05000000000000110.110.1510.19999999999999910.2510.310.3510.410.4510.510.5510.610.6510.710.7510.810.8510.910.951111.0511.111.1511.211.2511.311.3511.411.4511.511.5511.611.6511.711.7511.811.8511.911.951212.0512.112.1512.212.2512.312.3512.412.4512.512.5512.612.6512.712.7512.812.8512.912.951313.0513.113.1513.213.2513.313.3513.413.4513.513.5513.613.6513.713.7513.813.8513.913.951414.0514.114.1514.214.2514.314.3514.414.4514.514.5514.614.6514.714.7514.814.8514.914.951515.0515.115.1515.215.2515.315.3515.415.4515.515.5515.615.6515.715.7515.815.8515.915.951616.0516.10000000000000116.14999999999999916.216.2516.316.35000000000000116.39999999999999916.4516.516.5516.60000000000000116.64999999999999916.716.7516.816.85000000000000116.89999999999999916.951717.0517.10000000000000117.14999999999999917.217.2517.317.35000000000000117.39999999999999917.4517.517.5517.60000000000000117.64999999999999917.717.7517.817.85000000000000117.89999999999999917.951818.0518.10000000000000118.14999999999999918.218.2518.318.35000000000000118.39999999999999918.4518.518.5518.60000000000000118.64999999999999918.718.7518.818.85000000000000118.89999999999999918.951919.0519.10000000000000119.14999999999999919.219.2519.319.35000000000000119.39999999999999919.4519.519.5519.60000000000000119.64999999999999919.719.7519.819.85000000000000119.89999999999999919.95200.50.506249999999999980.512499999999999960.518750000000000040.525000000000000020.531250.537499999999999980.543749999999999960.550000000000000040.556250000000000020.56250.568749999999999980.574999999999999960.581250000000000040.587500000000000020.593750.60.606249999999999960.612500000000000040.618750000000000020.6250.631249999999999980.637499999999999960.643750000000000040.650.656250.662499999999999980.668749999999999960.675000000000000040.681250000000000020.68750.693749999999999980.70.706250000000000040.712500000000000020.718750.724999999999999980.731249999999999960.737500000000000040.743750000000000020.750.756249999999999980.762499999999999960.768750000000000040.775000000000000020.781250.787499999999999980.793749999999999960.80.806250000000000020.81250.818749999999999980.824999999999999960.831250000000000040.837500000000000020.843750.850.856249999999999960.862500000000000040.868750000000000020.8750.881249999999999980.887499999999999960.893750000000000040.90.906250.912499999999999980.918749999999999960.925000000000000040.931250000000000020.93750.943749999999999980.950.956250000000000040.962500000000000020.968750.974999999999999980.981249999999999960.987500000000000040.9937500000000000211.00625000000000011.01251.018751.02499999999999991.031251.03750000000000011.043751.051.05624999999999991.06251.06875000000000011.0751.081251.08749999999999991.093751.10000000000000011.106251.11251.11874999999999991.1251.13125000000000011.13751.143751.14999999999999991.156251.16250000000000011.168751.1751.18124999999999991.18751.19375000000000011.21.206251.21249999999999991.218751.22500000000000011.231251.23751.24374999999999991.251.25625000000000011.26251.268751.27499999999999991.281251.28750000000000011.293751.31.30624999999999991.31251.31875000000000011.3251.331251.33749999999999991.343751.351.356251.36251.36874999999999991.3751.38125000000000011.38751.393751.41.406251.41250000000000011.418751.4251.43124999999999991.43751.44375000000000011.451.456251.46249999999999991.468751.47500000000000011.481251.48751.49374999999999991.51.50625000000000011.51251.518751.52499999999999991.531251.53750000000000011.543751.551.55624999999999991.56251.56875000000000011.5751.581251.58749999999999991.593751.61.606251.61251.61874999999999991.6251.63125000000000011.63751.643751.651.656251.66250000000000011.668751.6751.68124999999999991.68751.69375000000000011.71.706251.71249999999999991.718751.72500000000000011.731251.73751.74374999999999991.751.75625000000000011.76251.768751.77499999999999991.781251.78750000000000011.793751.81.80624999999999991.81251.81875000000000011.8251.831251.83749999999999991.843751.851.856251.86251.86874999999999991.8751.88125000000000011.88751.893751.91.906251.91250000000000011.918751.9251.93124999999999991.93751.94375000000000011.951.956251.96249999999999991.968751.97500000000000011.981251.98751.993749999999999922.00625000000000012.01250000000000022.01874999999999982.02499999999999992.031252.03750000000000012.04375000000000022.04999999999999982.05624999999999992.06252.06875000000000012.07500000000000022.08124999999999982.08749999999999992.093752.12.10625000000000022.11249999999999982.11874999999999992.1252.13125000000000012.13750000000000022.14374999999999982.152.156252.16250000000000012.16875000000000022.17499999999999982.18124999999999992.18752.19375000000000012.20000000000000022.20624999999999982.21249999999999992.218752.22500000000000012.23125000000000022.23749999999999982.24374999999999992.252.25625000000000012.26250000000000022.26874999999999982.27499999999999992.281252.28750000000000012.29375000000000022.29999999999999982.30624999999999992.31252.31875000000000012.32500000000000022.33124999999999982.33749999999999992.343752.352.35625000000000022.36249999999999982.36874999999999992.3752.38125000000000012.38750000000000022.39374999999999982.42.406252.41250000000000012.41875000000000022.42499999999999982.43124999999999992.43752.44375000000000012.45000000000000022.45624999999999982.46249999999999992.468752.47500000000000012.48125000000000022.48749999999999982.49374999999999992.52.50625000000000012.51250000000000022.51874999999999982.52499999999999992.531252.53750000000000012.54375000000000022.54999999999999982.55624999999999992.56252.56875000000000012.57500000000000022.58124999999999982.58749999999999992.593752.62.60625000000000022.61249999999999982.61874999999999992.6252.63125000000000012.63750000000000022.64374999999999982.652.656252.66250000000000012.66875000000000022.67499999999999982.68124999999999992.68752.69375000000000012.72.70624999999999982.71249999999999992.718752.72500000000000012.73125000000000022.73749999999999982.74374999999999992.752.75625000000000012.76250000000000022.76874999999999982.77499999999999992.781252.78750000000000012.79375000000000022.82.80624999999999992.81252.81875000000000012.82500000000000022.83124999999999982.83749999999999992.843752.852.85625000000000022.86249999999999982.86874999999999992.8752.88125000000000012.88750000000000022.89374999999999982.92.906252.91250000000000012.91875000000000022.92499999999999982.93124999999999992.93752.94375000000000012.952.95624999999999982.96249999999999992.968752.97500000000000012.98125000000000022.98749999999999982.99374999999999993
00.050.10.150.20.250.30.350.40.450.50.550000000000000040.60.650.70.750.80.850.90.9511.051.10000000000000011.14999999999999991.21.251.31.351.41.451.51.551.61.651.71.751.81.851.91.9522.04999999999999982.12.152.20000000000000022.252.29999999999999982.352.42.45000000000000022.52.54999999999999982.62.652.72.752.82.852.92.9533.053.13.153.23.253.33.353.43.453.53.553.63.653.73.753.83.853.93.9544.054.09999999999999964.15000000000000044.24.254.34.34999999999999964.40000000000000044.454.54.554.59999999999999964.65000000000000044.74.754.84.84999999999999964.90000000000000044.9555.055.09999999999999965.155.25.255.35.355.45.455.55.555.65.655.75.755.85.855.95.9566.056.16.156.26.256.36.356.46.456.56.556.66.656.76.756.86.856.96.9577.057.17.157.27.257.37.357.47.457.57.557.67.657.77.757.87.857.97.9588.05000000000000078.18.158.19999999999999938.258.30000000000000078.358.48.44999999999999938.58.55000000000000078.68.658.69999999999999938.758.80000000000000078.858.98.949999999999999399.05000000000000079.19.159.19999999999999939.259.30000000000000079.359.49.44999999999999939.59.55000000000000079.69.659.69999999999999939.759.80000000000000079.859.99.94999999999999931010.05000000000000110.110.1510.19999999999999910.2510.310.3510.410.4510.510.5510.610.6510.710.7510.810.8510.910.951111.0511.111.1511.211.2511.311.3511.411.4511.511.5511.611.6511.711.7511.811.8511.911.951212.0512.112.1512.212.2512.312.3512.412.4512.512.5512.612.6512.712.7512.812.8512.912.951313.0513.113.1513.213.2513.313.3513.413.4513.513.5513.613.6513.713.7513.813.8513.913.951414.0514.114.1514.214.2514.314.3514.414.4514.514.5514.614.6514.714.7514.814.8514.914.951515.0515.115.1515.215.2515.315.3515.415.4515.515.5515.615.6515.715.7515.815.8515.915.951616.0516.10000000000000116.14999999999999916.216.2516.316.35000000000000116.39999999999999916.4516.516.5516.60000000000000116.64999999999999916.716.7516.816.85000000000000116.89999999999999916.951717.0517.10000000000000117.14999999999999917.217.2517.317.35000000000000117.39999999999999917.4517.517.5517.60000000000000117.64999999999999917.717.7517.817.85000000000000117.89999999999999917.951818.0518.10000000000000118.14999999999999918.218.2518.318.35000000000000118.39999999999999918.4518.518.5518.60000000000000118.64999999999999918.718.7518.818.85000000000000118.89999999999999918.951919.0519.10000000000000119.14999999999999919.219.2519.319.35000000000000119.39999999999999919.4519.519.5519.60000000000000119.64999999999999919.719.7519.819.85000000000000119.89999999999999919.95200.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.50.537499999999999980.574999999999999960.612500000000000040.650.68750.724999999999999980.762499999999999960.80.837500000000000020.8750.912499999999999980.950.987500000000000041.02499999999999991.06251.10000000000000011.13751.1751.21249999999999991.251.28750000000000011.3251.36251.41.43751.47500000000000011.51251.551.58749999999999991.6251.66250000000000011.71.73751.77499999999999991.81251.851.88751.9251.96249999999999992222222222222222222222222222222222222222222222222222222222222222222222222222222222.01250000000000022.02499999999999992.03750000000000012.04999999999999982.06252.07500000000000022.08749999999999992.12.11249999999999982.1252.13750000000000022.152.16250000000000012.17499999999999982.18752.20000000000000022.21249999999999992.22500000000000012.23749999999999982.252.26250000000000022.27499999999999992.28750000000000012.29999999999999982.31252.32500000000000022.33749999999999992.352.36249999999999982.3752.38750000000000022.42.41250000000000012.42499999999999982.43752.45000000000000022.46249999999999992.47500000000000012.48749999999999982.52.51250000000000022.52499999999999992.53750000000000012.54999999999999982.56252.57500000000000022.58749999999999992.62.61249999999999982.6252.63750000000000022.652.66250000000000012.67499999999999982.68752.72.71249999999999992.72500000000000012.73749999999999982.752.76250000000000022.77499999999999992.78750000000000012.82.81252.82500000000000022.83749999999999992.852.86249999999999982.8752.88750000000000022.92.91250000000000012.92499999999999982.93752.952.96249999999999992.97500000000000012.9874999999999998333333333333333333333333333333333333333333333333333333333333333333333333333333333
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