STUDENT PACKET MATH IN MOTION - Idlewild and SoakZone WEB student... · Think of centripetal force...

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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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



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



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



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



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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



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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%



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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



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Investigation #1 Wild Mouse Coaster – Idlewild & SoakZone Copyright 2000 10

IINNVVEESSTTIIGGAATTIIOONN ## 11

WWIILLDD MMOOUUSSEE CCOOAASSTTEE

RR

Hidden among the trees, Idlewild’s Wild Mouse is a modern version of the classic

Wild Mouse roller coasters, featuring mouse-shaped cars carrying four riders each. Once

the cars are hoisted to the top of the lift hill, they must follow the sudden twists, dips, and

hills that will lead them back to the station.

The velocity at the base of the Wild Mouse first drop is represented by the formula:

V= gh (squared)

Using the method discussed in the Student Activity Guidebook “An Angle on Distance”,

find the height of the Wild Mouse at several points of the first drop. (Use the markers on

the ride and the designated sighting location to make your calculations.) The sighting

location is just after entering the entrance line to the ride. Follow the path until you pass

some rocks making up the entrance line. When the rocks end, look to your left. The tree

is located approximately 3 meters to the left marked with a red bull’s eye.



1. Height of column A from the ground:

Height of point A = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters

Don’t forget the height of your eye

Height of the Wild Mouse = height from above + height of your eye.

Height of point A = ____________ + _____________ = ______________ meters

Distance from Track = 14.63 meters

The height of your eye from the

ground _____________ meters

Height

Sight to the top of column while

standing in front of the large

marked tree.

= ___________ degrees

(Ground

Level) Stand at the red bull’s-eye to use the correct distance given.

Column

A



2. Height of column B from the ground:

Height of point B = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters



Height of point B= ____________ + _____________ = ______________ meters




Height



marked tree.

= ___________ degrees

(Ground Level)

Stand at the red bull’s-eye to use the correct distance given.

Column

B



3. Height of column C from the ground:

Height of point C = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters



Height of point C = ____________ + _____________ = ______________ meters




Height



marked tree.

= ___________ degrees

(Ground Level)


Column

C



4. Height of column D from the ground:

Height of point D = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters



Height of point D = ____________ + _____________ = ______________ meters




Height



marked tree.

= ___________ degrees

(Ground Level)


Column

D



5. Height of column E from the ground:

Height of point E = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters



Height of point E = ____________ + _____________ = ______________ meters




Height



marked tree.

= ___________ degrees

(Ground Level)


Column

E



6. Height of column F from the ground:

Height of point F = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters



Height of point F = ____________ + _____________ = ______________ meters




Height



marked tree.

= ___________ degrees

(Ground Level)


Column

F



7. Height of column G from the ground:

Height of point G = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters



Height of point G = ____________ + _____________ = ______________ meters




Height



marked tree.

= ___________ degrees

(Ground Level)


Column

G



8. Height of column H from the ground:

Height of point H = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters



Height of point H = ____________ + _____________ = ______________ meters




Height



marked tree.

= ___________ degrees

(Ground Level)


Column

H



9. Height of the column A from the height of the lowest column_________?:

Therefore the height of the first drop from the bottom of the drop would be the distance

between theses two points.

Height of Point A – height of point ______? = height of the first drop from the bottom of

the drop.

______________ – ______________ = ___________________

10. Velocity of the first drop:

Now that you know the height of the drop at it’s highest point and lowest point, use the

velocity formula at the beginning of the question to calculate the velocity at point B in the

diagram with the help of knowing the speed of gravity.

Speed of Gravity (g) = 9.8 m / s2

(h) = height of the ride

V = gh

= (9.8 m/s2)(_________)

= ________ = ___________

11. On the chart below, graph the heights of each column. Then draw a line representing the track the cars run on. Each column is approximately 5 meters apart from

one another.

Investigation #2 Highest of the High – Idlewild & SoakZone Copyright 2000 20

IINNVVEESSTTIIGGAATTIIOONN ##22

HHIIGGHHEESSTT OOFF TTHHEE HHIIGGHH

Using the method discussed in the student Activity Guidebook “An Angle on

Distance”, find the height of the Wild Mouse at its highest and lowest points of the first

drop. (Use the markers on the ride and the designated sighting location to make your

calculations.)

12. Height of the Wild Mouse:

Height of the ride = distance away ( Tan )

Height = _____________ ( ___________ ) = __________________ meters


Height of the ride = height from above + height of your eye from the bridge + the height of the bridge (1.82 meters).

Height of the ride = ____________ + _____________ = ______________ meters

Distance from Ride: 19.81 meters

The height of your

eye from the ground

_____________ meters

Height

Sight to the top marker from

the red bull’s-eye on the

bridge crossing the creek.

The marker is located on the

wooden railing on the wild

mouse side of the bridge.

= ___________ degrees

(Ground Level)



13. Height of the Rafters Run:


Height = _____________ ( ___________ ) = __________________ meters


Height of the ride = height from above + height of your eye.



The height of your

eye from the ground

_____________ meters

Height

Sight to the top of the roof from

the red bull’s-eye located near the

exit of the ride.

= ___________ degrees

(Ground Level)



14. Height of the Ferris Wheel:


Height = _____________ ( ___________ ) = __________________ meters





The height of your

eye from the ground

_____________ meters

Height

Sight to the top of the ride from

the red bull’s-eye located near

the entrance of the ride.

= ___________ degrees

(Ground Level)



15. Height of the Jumpin’ Jungle Tree House:


Height = _____________ ( ___________ ) = __________________ meters





The height of your

eye from the ground

_____________ meters

Height

Sight to the top of the roof

from the red bull’s-eye

located near the entrance of

the ride.

= ___________ degrees

(Ground Level)



16. Height of the Rollo Coaster:


Height = _____________ ( ___________ ) = __________________ meters





The height of your

eye from the ground

_____________ meters

Height

Sight to the top of the railing

from the red bull’s-eye

located near the exit of the

ride.

= ___________ degrees

(Ground Level)



17. Graph your results.

Draw a line to represent each ride in order of height. Label the ride and it’s height.


Investigation #3 Centripetal Force & Centripetal Acceleration – Idlewild & SoakZone Copyright 2000 26

IINNVVEESSTTIIGGAATTIIOONN 33

CCEENNTTRRIIPPEETTAALL FFOORRCCEE &&

CCEENNTTRRIIPPEETTAALL AACCCCEELLEERRAATTIIOONN

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.



Questions:

A. What happened to the bobber as the carousel began to move?

B. While riding by the inside horse, where was the bobber (in relation to the center of the accelerometer)?

C. While riding by the outside horse, where was the bobber (in relation to the center of the accelerometer)?

D. By which horse, the inner or outer, did the bobber move further away from center?

E. Based upon your observations, where is the centripetal acceleration greatest? Why?

STUDENT PACKET MATH IN MOTION - Idlewild and SoakZone WEB student... · Think of centripetal force...

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