Math in Motion Idlewild & SoakZone Copyright

STUDENT PACKET MATH IN MOTION 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. Math in Motion Idlewild & SoakZone Copyright 2000 1

STUDENT PACKET MATH IN MOTION AN ANGLE ON DISTANCE 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. Straw Protractor String and Weight 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 = height distance H = d (tan ) height distance (given) Wild Mouse Math in Motion Idlewild & SoakZone Copyright 2000 2

STUDENT PACKET MATH IN MOTION Protractor Elevation Finder 0 10 20 0 30 0 4 0 50 0 6 0 9 0 80 0 70 0 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 Math in Motion Idlewild & SoakZone Copyright 2000 3

STUDENT PACKET MATH IN MOTION MOTION 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. Distance traveled Time of travel Velocity = = This velocity is an average for the trip. Whenever an unbalanced force acts on an object, an acceleration is produced. s t Acceleration = Force mass = 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.) Change in Velocity Time Acceleration = = Change in V t 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 Ac = V 2 r Ac = Centripetal Acceleration V = Velocity r = radius of the circle Ac V 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 s t Math in Motion Idlewild & SoakZone Copyright 2000 4

STUDENT PACKET MATH IN MOTION If there is an acceleration there must be an unbalanced force producing it. The force causing the circular motion is called centripetal force (F c ). 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 2 nd Law of motion and must apply to circular motion. But note that: F c = ma c a c = V 2 r note the equation for a c If we substitute V 2 r in for (a c ), we find the equation needed to calculate centripetal force. F c = ma c becomes F c = mv 2 r 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. Math in Motion Idlewild & SoakZone Copyright 2000 5

STUDENT PACKET MATH IN MOTION cut 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 Bottled Water Bottle upside-down Air gap Bobb String Accelerometer Accelerometer Construction Directions: 1) Lay out your newspaper for easy cleanup. 2) Attach the bobber to one end of the Lid 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. Math in Motion Idlewild & SoakZone Copyright 2000 6

STUDENT PACKET MATH IN MOTION AVERAGE & PERCENTAGE 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% Math in Motion Idlewild & SoakZone Copyright 2000 7

STUDENT PACKET MATH IN MOTION REFERENCE SHEET Circles: Circumfrence: C = D or 2r; where: = 3.14, D = diameter, r = radius Area of a Circle = r2 r Circumfrance of a circle = 2r r Triangles: Angle (degrees) Gravity: Acceleration of gravity (g) = 9.8 m / s 2 Length: Speed: c2 = a2 + b2 sin A = a/c cos A = b/c tan A = a/b 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 A 1 inch =.0254 meters 1 foot =.3048 meters 1 mile = 1609.3 meters 1 meter = 3.28 ft 1 m/s = 2.23 mph Angle (degrees) c b B a C 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 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 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 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 Math in Motion Idlewild & SoakZone Copyright 2000 8

STUDENT PACKET MATH IN MOTION INVESTIGATION # 1 WILD MOUSE COASTE R 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. Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 10

STUDENT PACKET MATH IN MOTION 1. Height of column A from the ground: Column A Sight to the top of column while standing in front of the large marked tree. Height The height of your eye from the = degrees ground meters Distance from Track = 14.63 meters Stand at the red bull s-eye to use the correct distance given. (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 Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 11

STUDENT PACKET MATH IN MOTION 2. Height of column B from the ground: Column B Sight to the top of column while standing in front of the large marked tree. Height The height of your eye from the = degrees ground meters Distance from Track = 11.27 meters Stand at the red bull s-eye to use the correct distance given. (Ground Level) Height of point B = 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 B= + = meters Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 12

STUDENT PACKET MATH IN MOTION 3. Height of column C from the ground: Column C Sight to the top of column while standing in front of the large marked tree. The height of your eye from the = degrees Height ground meters Distance from Track = 9.44 meters Stand at the red bull s-eye to use the correct distance given. (Ground Level) Height of point C = 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 C = + = meters Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 13

STUDENT PACKET MATH IN MOTION 4. Height of column D from the ground: Column D Sight to the top of column while standing in front of the large marked tree. The height of your eye from the = degrees Height ground meters Distance from Track = 10.05 meters Stand at the red bull s-eye to use the correct distance given. (Ground Level) Height of point D = 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 D = + = meters Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 14

STUDENT PACKET MATH IN MOTION 5. Height of column E from the ground: Column E Sight to the top of column while standing in front of the large marked tree. The height of your eye from the ground meters Distance from Track = 12.80 meters Stand at the red bull s-eye to use the correct distance given. = degrees (Ground Level) Height Height of point E = 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 E = + = meters Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 15

STUDENT PACKET MATH IN MOTION 6. Height of column F from the ground: Column F Sight to the top of column while standing in front of the large marked tree. The height of your eye from the = degrees Height ground meters Distance from Track = 16.15 meters Stand at the red bull s-eye to use the correct distance given. (Ground Level) Height of point F = 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 F = + = meters Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 16

STUDENT PACKET MATH IN MOTION 7. Height of column G from the ground: Column G Sight to the top of column while standing in front of the large marked tree. Height The height of your eye from the = degrees ground meters Distance from Track = 20.11 meters Stand at the red bull s-eye to use the correct distance given. (Ground Level) Height of point G = 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 G = + = meters Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 17

STUDENT PACKET MATH IN MOTION 8. Height of column H from the ground: Column H Sight to the top of column while standing in front of the large marked tree. Height The height of your eye from the = degrees ground meters Distance from Track = 24.38 meters Stand at the red bull s-eye to use the correct distance given. (Ground Level) Height of point H = 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 H = + = meters Investigation #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 18

STUDENT PACKET MATH IN MOTION 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 / s 2 (h) = height of the ride V = gh = (9.8 m/s 2 )( ) = = 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 #1 Wild Mouse Coaster Idlewild & SoakZone Copyright 2000 19

INVESTIGATION #2 HIGHEST OF THE HIGH 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: 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. Height = degrees The height of your eye from the ground meters Distance from Ride: 19.81 meters (Ground Level) Stand at the red bull s-eye to use the correct distance given. Height of the ride = distance away ( Tan ) Height = ( ) = meters Don t forget the height of your eye 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 Investigation #2 Highest of the High Idlewild & SoakZone Copyright 2000 20

13. Height of the Rafters Run: Sight to the top of the roof from the red bull s-eye located near the exit of the ride. Height = degrees The height of your eye from the ground meters Distance from Ride: 76.20 meters (Ground Level) Stand at the red bull s-eye to use the correct distance given. Height of the ride = distance away ( Tan ) Height = ( ) = meters Don t forget the height of your eye Height of the ride = height from above + height of your eye. Height of the ride = + = meters Investigation #2 Highest of the High Idlewild & SoakZone Copyright 2000 21

14. Height of the Ferris Wheel: Sight to the top of the ride from the red bull s-eye located near the entrance of the ride. Height = degrees The height of your eye from the ground meters Distance from Ride: 14.02 meters (Ground Level) Stand at the red bull s-eye to use the correct distance given. Height of the ride = distance away ( Tan ) Height = ( ) = meters Don t forget the height of your eye Height of the ride = height from above + height of your eye. Height of the ride = + = meters Investigation #2 Highest of the High Idlewild & SoakZone Copyright 2000 22

15. Height of the Jumpin Jungle Tree House: Sight to the top of the roof from the red bull s-eye located near the entrance of the ride. Height = degrees The height of your eye from the ground meters Distance from Ride: 21.64 meters (Ground Level) Stand at the red bull s-eye to use the correct distance given. Height of the ride = distance away ( Tan ) Height = ( ) = meters Don t forget the height of your eye Height of the ride = height from above + height of your eye. Height of the ride = + = meters Investigation #2 Highest of the High Idlewild & SoakZone Copyright 2000 23

16. Height of the Rollo Coaster: Sight to the top of the railing from the red bull s-eye located near the exit of the ride. Height = degrees The height of your eye from the ground meters Distance from Ride: 16.76 meters (Ground Level) Stand at the red bull s-eye to use the correct distance given. Height of the ride = distance away ( Tan ) Height = ( ) = meters Don t forget the height of your eye Height of the ride = height from above + height of your eye. Height of the ride = + = meters Investigation #2 Highest of the High Idlewild & SoakZone Copyright 2000 24

STUDENT PACKET MATH IN MOTION INVESTIGATION 3 CENTRIPETAL FORCE & CENTRIPETAL ACCELERATION 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. Investigation #3 Centripetal Force & Centripetal Acceleration Idlewild & SoakZone Copyright 2000 26

STUDENT PACKET MATH IN MOTION 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. Investigation #3 Centripetal Force & Centripetal Acceleration Idlewild & SoakZone Copyright 2000 27

STUDENT PACKET MATH IN MOTION 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? Investigation #3 Centripetal Force & Centripetal Acceleration Idlewild & SoakZone Copyright 2000 28