STUDENT PACKET MATH IN MOTION Math in Motion – Idlewild & SoakZone Copyright 2000 1 Your visit to Idlewild & SoakZone is an opportunity to not only have fun, but learn about math and the use of technology throughout the park. Use our Outdoor Classroom to broaden your appreciation of Math in Motion.
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STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
1
Your visit to Idlewild & SoakZone is an opportunity to
not only have fun, but learn about math and the use of
technology throughout the park. Use our Outdoor Classroom to broaden your appreciation of
Math in Motion.
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
2
AANN AANNGGLLEE OONN DDIISSTTAANNCCEE
To determine the height of a ride, use a simple “protractor elevation finder.”
Have one student sight through the straw
(using the proper elevation given standing on the designated area) at the top of the ride.
Another student reads the angle on the protractor. The angle read is then subtracted from 90 degrees.
To calculate the height of the ride you will also need the distance (given to the students at the designated elevation sighting locations) between the students and the ride.
Tan =
H = d (tan )
height
distance
distance
(given)
height
Wild Mouse
Protractor
Straw
String
and
Weight
90 degrees
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
3
0
10
20 0
30 0
4 0 0
6 0 0
50 0
9 0 0
80 0
70 0
Protractor Elevation Finder
1. Cut out the protractor including the dashed line section.
2. Trace the protractor part only on a piece of cardboard. (The back of a tablet works
nicely).
3. Glue or staple the cardboard to the back of the paper protractor.
4. Roll the top section around the straw and tape.
5. Punch a hole and tie a 9 inch string of heavy black thread through
the hole. On the other end tie a nut or fish sinker.
6. Follow the directions on the page titled “An Angle on Distance.”
Cardboard
d
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
4
MMOOTTIIOONN
There are two basic types of motion. Motion that is uniform and accelerated motion.
For uniform motion, forces are balanced. There are no net or resulting forces. Under these
conditions calculating the velocity is straight forward.
Velocity = =
This velocity is an average for the trip.
Whenever an unbalanced force acts on an object, an acceleration is produced.
Acceleration = = or F = ma
F= Force, m= Mass, a= Acceleration
As you can see force and acceleration are related. Acceleration is the change in velocity over a
period of time. (How fast something is going faster.)
Acceleration = =
Acceleration occures anytime there is a change in velocity. For objects moving in a curved path,
velocity is changing even through speed may be constant. Velocity is a vector and therefore must
have speed and direction. If your direction is changing, like on the Super Round Up, then there is
an acceleration toward the center of the Super Round Up. This acceleration is called centripetal
acceleration.
Centripetal Acceleration = (velocity)2 ÷ radius V
Ac = Ac = Centripetal Acceleration
V = Velocity Ac
r = radius of the circle
In the case of an object spining in a circle, the size of the velocity (speed) is calculated by
measuring the time for one complete spin and dividing this into the circumfrance of the circle.
V = where (s) = distance = circumference
Distance traveled
Time of travel
s
t
Force
mass
Change in Velocity
Time
Change in V
t
V2
r
s
t
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
5
If there is an acceleration there must be an unbalanced force producing it. The force causing the
circular motion is called centripetal force (Fc ). This force causes the object to change direction
thereby creating the acceleration in the same direction (towards the center).
As stated previously; F = ma
This is Newton’s 2nd
Law of motion and must apply to circular motion.
But note that: Fc = mac
ac = note the equation for ac
If we substitute in for (ac), we find the equation needed to calculate centripetal force.
Fc = mac becomes Fc =
This force is easy to see and understand if you swing a rubber stopper on the end of a string. You
can see your hand is producing the force which is transferred through the string to make the stopper
follow a circular path.
In the Super Round Up, the wall produces the centripetal force. This force keeps you moving in a
circular path by providing an acceleration on you toward the center. You, on the other hand, have
the impression that there is a force throwing you toward the wall. This is very similar to being in an
automobile at rest and the driver pushes the accelerator to the floor. If the car has a lot of
horsepower, you feel like you are being pushed back in the seat. In reality, the seat is accelerating
you forward. So, in the Super Round Up, the force you feel out against the wall, called centrifugal
force is a fictious force. You are reacting to the wall pushing you in!
Think of centripetal force as the action force and centrifugal force as the reaction force.
Remember, centrifugal force is considered to fictitious. It can only be observed in the acceleration
frame of reference.
These forces are also found on many other rides at Idlewild. Any ride which moves in a circular
motion or curved path will produce centripetal and centrifugal forces.
V2
r
V2
r
mv2
r
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
6
Acceleration Finder The accelerometer is a device used to measure the acceleration of any moving object. For this project you will be observing the centripetal acceleration generated by the Carousel. Accelerometer Construction Materials Needed: Clear plastic bottle with lid that is closely the
same diameter of the bottle. Glue String or fishing line Fishing bobber (1”-2”dia.) the bigger your
bottle the bigger your bobber should be. Example: If the bottle is 3½“ in diameter the bobber should be about 1”-1½” in diameter.
Strong box tape Scissors Newspapers Water
Accelerometer Construction Directions: 1) Lay out your newspaper for easy
cleanup. 2) Attach the bobber to one end of the
string and hang it inside the bottle with the bobber just barely touching the bottom of the bottle.
3) Cut the string at the top of the bottle exactly at the same height as the lid. 4) Tape the string in the center of the lid with your box tape. Make sure the string is in the
exact center of your lid.
5) Place the bobber inside the bottle and close the top. The bobber should hang inside the
bottle without touching the bottom of the bottle. Approximately ½ inch from the bottom. 6) Fill the bottle with water approximately ½ inch from the top. Place the bobber inside the
bottle again and close the top tightly and securely. 7) Turn the bottle upside down and check for leaks. If it leaks, turn it back over and let it dry
where it leaked. You will need to seal the lid with glue with the assistance of your teacher.
cut
Bobb
er
String
Water
level
Lid
Bottle upside-down
Air gap
Accelerometer
Bottled
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
7
AAVVEERRAAGGEE && PPEERRCCEENNTTAAGGEE
Average
Averages are obtained by adding the numbers in a set and dividing their sum by the total of the
elements involved.
For example, the average of 3 and 7 is 5
Calculated 3 + 7 = 10; 10 ÷ 2 (the total of elements in the set) is 5.
The average of 96, 89, 13, and 2 is (96 + 89 + 13 + 2) ÷ 4 = 50.
Percentage
The term “percentage” is derived from the Latin word per centum, meaning “per hundred,” this
term essentially represents fractions with the denominator of 100. Therefore, 35 percent (35%)
means the fraction 35/100.
To find the percentage of a number—for example, 20 percent of 40—20 must be changed to a
common fraction (20/100) or to a decimal (.20) and the figure multiplied by the whole (40); see
below
The percentage relationship of one number (5) to another (20), is calculated by dividing the first
number by the latter number then just multiplying this by 100; see below
5 ÷ 20 = .25, .25 x 100 = 25 percent or 25%
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
8
Lengt Gravit Gravity: Acceleration of gravity (g) = 9.8 m / s2 Length: 1 inch = .0254 meters 1 foot = .3048 meters 1 mile = 1609.3 meters
1 meter = 3.28 ft Speed: Speed: 1 m/s = 2.23 mph
RREEFFEERREENNCCEE SSHHEEEETT
Circles: Circumfrence: C = D or 2r; where: = 3.14, D = diameter, r = radius
Area of a Circle = r2 Circumfrance of a circle = 2r
Triangles: c2 = a2 + b2 sin A = a/c cos A = b/c tan A = a/b
r
Angle
(degrees)
Tangent
Value 1 0.174
2 0.035
3 0.052
4 0.070
5 0.087
6 0.105
7 0.123
8 0.140
9 0.158
10 0.176
11 0.194
12 0.212
13 0.230
14 0.249
15 0.268
16 0.287
17 0.306
r
a c
b A
B
C
Angle
(degrees)
Tangent
Value 18 0.325
19 0.344
20 0.364
21 0.384
22 0.404
23 0.425
24 0.445
25 0.466
26 0.488
27 0.510
28 0.532
29 0.554
30 0.577
31 0.601
32 0.625
33 0.649
34 0.674
Angle
(degrees)
Tangent
Value 52 1.280
53 1.327
54 1.380
55 1.428
56 1.483
57 1.540
58 1.600
59 1.664
60 1.732
61 1.804
62 1.881
63 1.963
64 2.050
65 2.144
66 2.246
67 2.356
68 2.475
69 2.605
70 2.747
71 2.904
72 3.078
73 3.271
74 3.487
75 3.732
76 4.011
77 4.331
78 4.705
79 5.145
80 5.671
81 6.314
82 7.115
83 8.144
84 9.514
85 11.430
86 14.301
87 19.081
88 28.636
89 57.290
Angle
(degrees)
Tangent
Value 35 0.700
36 0.727
37 0.754
38 0.781
39 0.810
40 0.839
41 0.869
42 0.900
43 0.933
44 0.966
45 1.000
46 1.040
47 1.072
48 1.111
49 1.150
50 1.192
51 1.235
System o System of Measurment: System Length Mass Time Force Velocity Acceleration Metric (MKS)
meter (m) kg sec newton (N) m/s m/s/s
Metric (CGS)
cm gram sec dyne cm/s cm/s/s
English (FPS)
ft slug sec Pound (lb) ft/s ft/s/s
STUDENT PACKET MATH IN MOTION
Math in Motion – Idlewild & SoakZone Copyright 2000
Carousel (Accelerometer required) Safety Precautions: You MUST stay between the inner and outer horses AT ALL TIMES while the ride is moving. Do not walk around the ride while in motion. Failure to follow this rule could result in personal injury. It will also result in your removal from the ride and possible loss of your rider’s pass for the remainder of the day. Directions:
1. Place the accelerometer lid side down on the deck of the carousel behind an inside horse.
2. Stand beside the accelerometer and look down so that one eye is directly over the bobber. Notice the location of the bobber.
3. As the ride begins to move, notice the motion of the bobber. 4. Take note of where the bobber is in relation to the center of the accelerometer. 5. After observing the accelerometer for one ride move behind an outside horse after
the ride stops. 6. Once again place the accelerometer on the deck of the carousel.
7. Notice where the bobber is in relation to the center of the accelerometer. Continue watching the accelerometer until the carousel has nearly stopped.
8. Once the ride has come to a complete stop you may step off the platform.
1. From the radius, calculate the circumference of the Ferris
Wheel.
(Note: circumference = times 2r
where = 3.14 and r = radius.)
Circumference = _________________
2. Using your watch or stopwatch, measure how long it
takes to make one trip around. (You may find it easier to
time 5 trips and divided by five. Note: some rides take a
minute or two to reach full speed.)
One trip = _________________
3. Calculate the speed of rotation. The speed
will be the distance traveled
(circumference) divided by the time for
that distance (period).
V = =
=
= _________________
4. Graph your results.
distance
time
circumference
period
Radius (r) = 7.62 meters
STUDENT PACKET MATH IN MOTION
Investigation #4 Speed or Velocity – Idlewild Park 2000 33
Carousel
1. From the radius, calculate the
circumference of the Carousel.
(Note: circumference = times 2r
where = 3.14 & r = radius.)
Circumference =
_________________
2. Using your watch or stopwatch,
measure how long it takes to make
one trip around. (You may find it
easier to time 5 trips and divided by five. Note: some rides take a minute or two to reach full speed.)
One trip = _________________
3. Calculate the speed of rotation. The speed
will be the distance traveled (circumference)
divided by the time for that distance (period).
V = =
=
V = ______________
4. Graph your results.
distance
time
circumference
period
Radius (r) = 7.16 meters
STUDENT PACKET MATH IN MOTION
Investigation #4 Speed or Velocity – Idlewild Park 2000 34
Flying Aces
1. From the radius, calculate the circumference of the Flying Aces.
(Note: circumference = times 2r where = 3.14 and r = radius.)
Circumference = _________________
2. Using your watch or stopwatch, measure how long it takes to make one trip around. (You may
find it easier to time 5 trips and divided by five. Note: some rides take a minute or two to reach
full speed.)
One trip = approx.
_________________seconds.
3. Calculate the speed of rotation.
The speed will be the distance
traveled (circumference) divided
by the time for that distance
(period).
V = =
=
V = ______________
4. Graph your results.
distance
time
circumference
period
Radius (r) = 15.85 meters
Timing the Flying Aces: Find a spot along the outside perimeter fence (outside the ride area). Select one of the ten 2-seated cars to observe. When it appears that the ride has reached full speed, start timing when your selected car passes the center of the ride. Stop it once that same car passes the center of the ride again. Good Luck.
STUDENT PACKET MATH IN MOTION
Investigation #4 Speed or Velocity – Idlewild Park 2000 35
Super Round Up
1. From the radius, calculate the
circumference of the Super Round Up.
(Note: circumference = times 2r
where = 3.14 & r = radius.)
Circumference = _________________
2. Using your watch or stopwatch,
measure how long it takes to make one trip around. (You may find it easier to time 5 trips and
divided by five. Note: some rides take a minute or two to reach full speed.)
One trip = approx. ______ seconds
3. Calculate the speed of rotation. The speed
will be the distance traveled (circumference)
divided by the time for that distance (period).
V = =
=
V = ______________
4. Graph your results.
distance
time
circumference
period
Radius (r) = 6.09 meters
STUDENT PACKET MATH IN MOTION
Investigation #4 Speed or Velocity – Idlewild Park 2000 36
Howler (located in Hootin' Holler') The Howler, while in full motion, has a radius (r) of 9.89 meters.
From the radius, calculate the
circumference of the Howler in full motion.
(Note: circumference = times 2r
where = 3.14 and r = radius.)
Circumference = _________________
Using your watch or stopwatch, measure how long it takes to make
one trip around. (You may find it easier to time 5 trips and divide
by five. Note: some rides take several seconds to reach full speed.)
One trip = approx. __________ seconds
Calculate the speed of rotation. The speed
will be the distance traveled
(circumference) divided by the time for
that distance (period).
V = =
=
V = ____________
Graph your results.
distance
time
circumference
period
Radius (r) = 3.35 meters
Timing the Howler To make things easier stand to the left side of the little shed where the ride operator sits facing the ride. (You must be outside the ride area). Select one of the six 4-seated ride cars to observe. When it appears that the ride has reached full speed, start timing when your selected car passes the center of the ride. Stop it once that same car passes the center of the ride again. Good Luck.
The Rollo Coaster was built by the Philadelphia Toboggan Company and opened to Idlewild’s Guests in 1938. It’s two trains carry riders up and down along a wooded hillside, only to turn around in a swooping curve and return back towards the station. The rides last approximately 70 seconds on a track that runs nearly 1400 feet. The Rollo Coaster has been named an American Coaster Enthusiasts Classic Coaster.
Please keep the following in mind while doing your calculation: Some days the Rollo Coaster operates with two coaster trains. If you are visiting on one of these days, please know that one coaster normally runs a little faster than the other. Also, rain tends to make the coasters run a little faster and the heavier or larger group of people in a coaster will cause it to run faster, while cold temperatures cause casters to run slower.
Note: You must be 48 inches tall to ride or be accompanied by an adult.) This investigation will require you to ride the Rollo Coaster. Because of this fact, one individual would have to ride the coaster between 15 and 36 times to obtain the necessary information, which we do not suggest. Therefore, it is recommended that this investigation should be done as a TEAM, in groups of 5 to 10 individuals, sharing the findings.