1 Welcome to Hersheypark®, a Proud Supporter of STEM Education

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Welcome to Hersheypark®, a proud supporter of STEM education initiatives. Hersheypark offers laboratory manuals focusing on Math and Physics. Both of the manuals are designed to give students the opportunity to apply their content knowledge in a hands-on environment while enjoying the day at one of the country’s favorite amusement parks.

Hersheypark provides this Math Lab Manual that correlates directly to the Common Core State Standards while continuing to adhere to the Pennsylvania State Standards and the NCTM Standards. Students may want to download a graphing calculator App.

Our educational guides are designed to be a resource as you and your students plan your Hersheypark adventure. Feel free to choose the activities that best fit your students and reproduce the worksheets as needed. You will find activities that are appropriate for a variety of ages, grade levels, and curriculums.

Hersheypark would like to recognize several individuals who have contributed their time and talents to updating the current Math Lab Manual. These individuals are continuing the tradition of educational enrichment within Hersheypark.

• Mike Long Ed.D., Shippensburg University of Pennsylvania • Robert Newara, Cumberland Valley SD • Amanda Paveglio, Lancaster-Lebanon Intermediate Unit 13 • Becky Piscitella, Richland School District • Timothy Seiber, Cumberland Valley SD

Hersheypark would like to extend a thank you to the following individuals:

• Don Scheuer, Consultant • Amy Lena, Cumberland Valley SD • Carolyn Marchetti, Berks County IU14

Hersheypark will gladly accept any additional problems, experiments, or corrections to our educational guides. Please feel free to email Hersheypark Group Sales, with your comments or questions to [email protected].

1 Tips for a Successful Math Day • Identify the labs you want the students to complete. • Use the Day Planner to coordinate your day. • Consider making a time and place to meet during the day. • Students may want to download the following Apps to their smart phones ahead of time: • Graphing Calculator App • Accelerometer App • Smart phones allow students to carry fewer items; however, we highly recommend wearing clothing that can secure your phone on rides. Pockets that zipper will help prevent your phone from being lost, broken, or stolen. • Please note that measuring devices, phones and cameras are not permitted on the following rides: Skyrush, Storm Runner, Fahrenheit, Laff Trakk and the Hershey Triple Towers. Hersheypark has the right to add additional rides to this list at any time. All riders must follow the directions of the ride operators at all times. • Wear comfortable shoes that are strapped to your feet. • Check the weather ahead of time and dress appropriately. • Dress in layers to accommodate temperature changes throughout the day. • Don’t forget the sunscreen. • List any other items you want to suggest to your students :

______

3 Math Day Planner

Time Place or Notes Activity

4 Lab Name: Bigger Coaster = More Riders?

Ride(s) involved: Coasters throughout the Park

Key Question:

Student Tech Requirements: Stop Watch

Activity Directions:

Using the following tables, you are going to collect data about the coasters at Hersheypark®. As a train leaves the station or passes a particular point of reference, start the stopwatch. Record the cumulative time in seconds (do not stop the watch) as the next six trains dispatch or pass the particular point of reference. The number of trains will be the independent variable. The dependent variable will be the cumulative number of passengers that have left the station. So in addition to keeping the time, you will need to count the number of people riding in each train to keep a running total of the number of people that have ridden the coaster.

Comet sooperdooperLooper® Great Bear®

Total Time Cumulative # Total Time Cumulative # of Total Time Cumulative # of of Passengers Passengers Passengers

5 Trailblazer Storm Runner® Fahrenheit®

Total Time Cumulative # Total Time Cumulative # of Total Time Cumulative # of of Passengers Passengers Passengers

Sidewinder The Wildcat Lightning Racer®

Total Time Cumulative # Total Time Cumulative # of Total Time Cumulative # of of Passengers Passengers Passengers

6 1. Now that you have collected some data, make scatterplots or one scatterplot of your data. You may choose to use some form of technology for this or you may choose do it by hand.

2. Next, determine the line of best fit for each of the individual data sets.

3. What does the rate of change of each of the lines of best fit tell you?

4. Use the line of best fit to predict the number of people that would ride the coaster in one hour.

5. Compare the predicted riders per hour to the theoretical by making of a ratio.

7 Lab Name: Carrousel Ride(s) involved: Carrousel Key Question: Does every rider on the carrousel experience the same ride? Student Tech Requirements: Smart phone or stopwatch and calculator

Activity Directions: Part 1: Observe the ride and answer the following questions:

1. How many horses are there on the carrousel? ______

2. How many of the horses move up and down? ______

3. If you got on a horse without looking up to see if it could move, what is the probability that you would be on a moving horse? ______

4. What are the odds that you would be on a moving horse? ______

Part 2: Complete the chart according to the list of directions and then answer the questions that follow. Use the data in the back of the book or scan the QR code at the ride entrance.

1. List the radius of each horse’s circular path.

2. Calculate the circumference of each horse’s path. C=2 r

3. If the ride time is 2 minutes and a single rotation is 11 seconds, how many revolutions will each horse make during one complete ride?

4. Calculate the distance each horse travels during one complete ride. Distance = circumference × number of revolutions

5. Now calculate the average speed for each horse. Average speed = d / t

Circular Radius of Circumference Revolutions Distance Average Path each path of each path per ride per ride speed Inner Horse

Middle Horse Outer Horse

6. Are all the paths on the carrousel the same distance?

7. If you decided to ride a horse, which path would you choose and why?

8. If an inner horse rider rides 4 times, about how many times will an outer horse rider need to ride to cover approximately the same distance?

9. Does every rider on the carrousel experience the same ride? Defend your response.

Ride data /elements: CARROUSEL ENGLISH METRIC

Radii 25’ 7.62 m

Inner horse 17.5” 5.31 m

Middle horse 20’5” 6.22 m

Outer 23’5” 7.16 m

Total ride time 2 minutes 2 minutes

Single rotation time 11 seconds 11 seconds

Lab Name: Carrousel 2

Ride(s) involved: Carrousel

Key Question: What is tangential velocity?

Student Tech Requirements: Smart phone, stopwatch

Activity Directions:

Use your smart phone to retrieve the following information:

Radius of outer ring: ______Ride radius: ______Height of platform: ______

1. Measure the period of rotation.

T = ______

2. Calculate the tangential speed vt for the outer ring.

θ ds Useful formulas: s r= , vt = dt

3. Would the tangential velocity increase or decrease for riders riding on the inner ring? Explain.

4. Suppose an outer ring rider’s shoe flew off their foot. How far will the shoe fly before reaching the ground? Assume a reasonable value for the height above the platform of the rider’s foot.

5. What is the distance from the platform that the shoe is now located?

Ride data /elements: CARROUSEL ENGLISH METRIC

Ride Radius 25’ 7.62 m

Radius of outer ring 23’5” 7.16 m

Height of platform 1’ .3048 m

10 Lab Name: Carrousel vs. Great Bear

Ride(s) involved: Carrousel, Great Bear

Key Question:

Student Tech Requirements:

Activity Directions:

Problem 1: The diameter of the carrousel varies depending on whether you are riding a horse on the inner row, 17.5 feet, the middle row, 20.5 feet, and the outer row 23.5 feet. Determine the total distance that a rider travels during one revolution. Then use this to find the total distance that a rider travels during one complete ride. Make your science teacher happy by remembering to use units!

Inner Row: Middle Row: Outer Row:

Problem 2: Determine the average linear speed inner row, middle row, and outer row of the horses on the carrousel during one complete ride. Use appropriate units.

time x speed = distance

Inner Row: Middle Row: Outer Row:

Problem 3: Determine the number of “degrees” that a rider in each row travels during one complete ride on the carrousel. Then determine the angular speed for each row.

Inner Row: Middle Row: Outer Row:

Problem 4: Determine the area of the circle made by the horses of the carrousel as the ride rotates. Again…remember the units.

Inner Row: Middle Row: Outer Row:

Problem 5: If the length of the Great Bear is 2,800 feet, how many revolutions does the carrousel have to make so that a ride travels the same distance as a rider on the Great Bear?

Inner Row: Middle Row: Outer Row:

Problem 6: In the space below, sketch a distance versus time graph for each row of the carrousel. Lab Name: Chocolate Probability

Ride(s) involved: Fahrenheit, SkyrushSM

Key Question: What is the probability?

Student Tech Requirements: Smart phone or calculator

Activity Directions:

1. Review the ride data of the following rides then answer the following questions.

nPr = ___n!____ nCr = ____n!_____ (n – r)! (n – r)! r!

2. If a group of 12 friends wants to ride Fahrenheit, how many different groups can be seated on one ride?

3. If the same group of 12 friends wants to ride Fahrenheit together, how many times will they need to ride to make sure everyone sits in a different seat each time?

4. If you were seated at random on Fahrenheit, what is the probability that you would be seated in the first car? What are the odds in favor of this?

5. If you were seated at random on Skyrush, what is the probability that you would be seated in the first car? What are the odds in favor of this?

6. If you were seated at random on Skyrush two times, what is the probability that you would be seated in the same seat both times?

7. If you were seated at random on Fahrenheit two times, what is the probability that you would be seated in the same seat both times?

8. If you put your hand in a bag that contains 5 Hershey’s® Milk Chocolate Bars with Almonds, 7 Hershey’s Milk Chocolate Bars, 13 Reese’s® Peanut Butter Cups, and 10 Twizzlers®, and randomly picked out 2 items to eat, what is the probability that both items would be chocolate?

Ride data /elements: Fahrenheit 4 seats per car, 3 cars, 12 total riders

Skyrush 4 seats per car, 8 cars, 32 total riders

Lab Name: Circular Function Machines

Ride(s) involved: Entire Park

Key Question:

Student Tech Excel Requirements: 13

Activity Directions: This activity requires that you make some observations at different circular rides as you roam Hersheypark. As you approach each of the rides listed, you need to count the number of arms, which are holding seats, and then determine the number of degrees between each of the arms or the measure of the angle between arms which are side by side. The rides which you are to observe (or ride) are listed:

The Howler Number of Arms:___ Degrees Between Each Arm:_____

Scrambler Number of Large Arms:___ Degrees Between Each Arm:_____

Dinosaurs Number of Arms:___ Degrees Between Each Arm:_____

Music Express Number of Arms:___ Degrees Between Each Arm:_____

Flying Falcon Number of Small Arms:___ Degrees Between Each Arm:_____

Frontier Flier Number of Arms:___ Degrees Between Each Arm:_____

Starship America Number of Arms:___ Degrees Between Each Arm:_____

Granny Bugs Number of Arms:___ Degrees Between Each Arm:_____

After you finish collecting the information, make a display of your data using a graphing calculator or spreadsheet. (You may choose to use graph paper.) The number of arms for the rides should be on the independent variable and the corresponding number of degrees between the arms on the dependent variable. Be sure to use a scale on both axes so that the graph will fit in the window (or paper). Put the ride names that you collected data for beside the number of arms.

• Now that you have collected some data, make scatterplots or one scatterplot of your data. You may choose to use some form of technology for this or you may choose do it by hand.

• Next, determine the line of best fit for each of the individual data sets. Again, you may choose to use some form of technology for this.

• What can you say about the number of degrees between the arms as the number of arms increases?

• CHALLENGE: What would the rate of change of the function be and what would it tell you?

Lab Name: Coaster Hypothesis Tests

Ride(s) involved: Entire Park

Key Question:

Student Tech Requirements: Excel

Activity Directions:

This activity requires that you make some observations at different coasters as you roam Hersheypark. As you approach each of the coasters listed, you need to count the number of empty seats for at least five dispatches of the train. Tables for the data you are to collect are provided below.

15 Comet (wood) The Wildcat (wood) Lightning Racer Red (wood)

Train # of Empty Train Number # of Empty Train # of Empty Number Seats Seats Number Seats

1 1 1

2 2 2

3 3 3

4 4 4

5 5 5

Lightning Racer Green Great Bear sooperdooperLooper (wood)

Train # of Empty Train Number # of Empty Train # of Empty Number Seats Seats Number Seats

1 1 1

2 2 2

3 3 3

4 4 4

5 5 5

16 Trailblazer Storm Runner Sid ewinder

Train # of Empty Train Number # of Empty Train # of Empty Number Seats Seats Number Seats

1 1 1

2 2 2

3 3 3

4 4 4

5 5 5

After you finish collecting the information:

1. Choose two of the wood coasters. Determine an interval where the difference between the population mean number of empty seats for the two coasters lies and interpret the results.

2. Choose two of the wood coasters. Test a claim that the population mean number of empty seats for the one coaster you chose is different from the second.

3. Choose three of the wood coasters. Test a claim that the population mean number of empty seats for at least one of the wood coasters you chose is different.

4. Choose any two coasters. Determine an interval where the difference between the population proportion of empty seats for the two coasters lies and interpret the results.

5. Choose any two coasters. Test a claim that the population proportion of empty seats for the one coaster you chose is different from the second.

Lab Name: Conquering Starship America!

Ride(s) involved: Starship America in Founder’s Circle

Key Question:

Student Tech Requirements: Pasco or Vernier data collection device with and an altimeter

Activity Directions:

Conquering Starship America

Can you ride the Starship America (located in Founder’s Circle), while wearing a data collection device, and create a height versus time (altitude) graph to match the one below. Figure out what you need to do with a partner and then ride.

Please describe: the graph shown above

18 Lab Name: Correlating at Hersheypark

Ride(s) involved: Multiple coasters

Key Question: Is there a correlation between Hersheypark visitors’ ages and number of times they ride on roller coasters in a day?

Student Tech Requirements: None for data collection at the park. For data analysis: software to graph data points, OR graph paper

Activity Directions:

1. In the middle of the afternoon or as you leave the Park, ask at least 30 people how many times they rode on roller coasters today. Try to ask a variety of age groups.

a. Why might it be important to ask at least 30 people?

b. Why might it be important to do this activity in the afternoon?

c. Is there a difference between asking “how many coasters did you ride today?” and “how many times did you ride on roller coasters today?”

2. Ask each person their age and the number of coaster rides they had. If they rode the same coaster more than once today, count each ride separately. Record your data in the table below.

Age Number of Age Number of Age Number of Rides Rides Rides

3. Make a scatterplot with your results. Plot age on the x-axis and number of coaster rides on the y-axis. (You may want to scale the graph by 2, 4, 6, 8, etc.)

20 4. Is there a positive, negative, or no correlation represented? What does that mean?

5. If there is a correlation, would you consider it a strong or weak correlation?

6. Do you think that your results would be similar or different if you had asked fewer people? What about more people?

7. Partner with a classmate who also did this activity. Combine your results together and make a new scatter plot. Is your “combined” graph different than your first graph? Was it what you expected?

Post-Activity: Look in a newspaper or magazine for an article about a poll or a scientific study. Does the article talk about correlation (or the relationship between two variables)? What question was the poll/study trying to answer? What are the variables? Does the article describe how big the sample size was? Does it describe how the sample was chosen? What questions do you have about the poll/study’s results?

Lab Name: Fast… Faster… Fastest! Ride(s) involved: Multiple coasters Key Question: Is the fastest coaster your favorite?

Student Tech Requirements: Smart phone or stopwatch and calculator

Activity Directions:

1. From the chart, obtain the ride rating, track length, and maximum speed.

2. Time each coaster and record the data below.

3. Use the formula R = d / t to find the average speed of each ride in ft/s.

4. Convert the average speed to mph. (1 mile = 5280 ft) .

Coaster Ride Track Time of Average Average Maximum Rating length in Ride Speed Speed Speed feet (ft/s) (mph) mph

Comet 4 3360 50 Fahrenheit 5 2700 58 Great Bear 5 2800 61 Lightning Racer 4 3393 51 Cocoa Cruiser 3 279 16 Sidewinder 5 935 47 Skyrush 5 3720 75 sooperdooperLooper 5 2614 45 Storm Runner 5 2600 72 Trailblazer 3 1600 45 Wildcat 5 3183 50 Wild Mouse 5 1213 28

5. Compare the average speeds.

a. Which coaster has the fastest average speed?

Coaster______Ride Rating______

b. Which coaster has the fastest maximum speed?

Coaster______Ride Rating______

c. Which coaster has the slowest average speed?

Coaster______Ride Rating______

d. Which coaster has the slowest maximum speed?

Coaster______Ride Rating______

e. Which is your favorite coaster? Why?

Lab Name: Filling the Carrousel

Ride(s) involved: Carrousel

Key Question: What is the volume of the carrousel?

What other ways can we think about volume?

Student Tech Requirements: Calculator

Pre-Activity:

You may need any of the following formulas to complete this activity:

2 Volume of a cylinder: V = π • r • h Volume of a rectangular prism: V = l •w •h

2 π•r •h 3 Volume of a cone: V = Volume of a sphere: V = 4 • π • r 3

Area of a circle: A = 2π • r

Use your smart phone to scan the Carrousel QR code and get the dimensions of the Carrousel.

Activity Directions:

1) The Carrousel has a complex shape. Can you break it up into simpler shapes? Draw a sketch of the carrousel. What type of geometric figure does the tent sit on? What type of geometric figure is the tent?

2) Use the given dimensions to find the volume of the carrousel. a) Find the volume of the bottom section of the carrousel. b) Find the volume of the tent of the carrousel. c) Find the total volume of the carrousel.

3) Imagine that all of the horses, seats, etc. are taken out of the Carrousel and riders must stand if they want to ride. For safety, Hersheypark puts a railing inside the Carrousel 3 feet away from the edge of the rotating deck. If each rider stands on the deck in a circle with a 1-foot radius, approximately how many people can ride the “standing room only Carrousel” if it is full?

4) Image that you want to fill the entire Carrousel – both the bottom section and the tent section – with 1-scoop ice cream cones. If a regular-size ice cream scoop has a radius of 2 inches, and a regular-size cone has a radius of 1 inch and a height of 4 inches, what is the volume of a regular-size ice cream cone? How many ice cream cones can fit inside the Carrousel? (Don’t forget about unit conversion!)

Ride data / elements: CARROUSEL ENGLISH METRIC

Circumference of Carrousel 157’ 47.85 m

Radius 25’ 7.62 m

Distance from ground to ceiling riding area 12’ 3.66 m

Distance from ground to peak of the tent 50’ 15.24 m

Mini-Geocaching inside Hersheypark

You will need a longitude/latitude app or a GPS reader.

Due to the variety of apps, all waypoints have been recorded both as degrees (°) minutes (’) seconds (”), and degrees (°) minutes (’) with decimals. See the end of this activity if you wish to learn how to convert seconds into decimal minutes.

Note: mini-geocaching requires precise measurements with little change in the degree and minute units of measure.

Our first stop is to ride a horse while following the path of a conic section. Record your waypoint location:

Record your location: N ______

W ______

Now it is off to a little game of cat and mouse.

Travel NE to: N 40 ° 17’ 34” ( 40° 17.567’ )

W 76° 39’ 18” ( 76° 39.300’ )

Look due EAST and upward. What two water rides do see listed on the same sign?

______

It’s not far to the next stop. Go east to the far entrance of the “Rube Goldberg” water machine. Record your waypoint location:

Record your location: N ______

W ______

Back up the hill for a restful ride. Your ride over the park will take you outside the park to the eastern most point of the ride. Be attentive, the ride does not stop at this location. There was a time, many years ago, when visitors to this building were treated to the smell of chocolate mixing in very large vats.

Record your location: N ______

W ______

26 Over the river and through the valley to our next stop we go. Here live the live residents of Hersheypark.

Travel to: N 40° 17’ 16” ( 40° 17.267’ )

W 76° 39’ 04” ( 76° 39.067’ )

How do you measure up? Take a photo with the sign.

Our final stop leads us to the entrance of an attraction that Chicken Little may never ride. Is the sky really falling or does it just feel that way on the ride?

Record your location: N ______

W ______

Name the ride.

Well done! We hope your trip around the Park was enjoyable.

------

Degrees, minutes and seconds are similar to hours, minutes and seconds. A minute is one sixtieth of a degree. A second is one sixtieth of a minute. For this activity, to convert a second to a decimal part of a minute divide the seconds by 60. To convert the decimal part of a minute to a second multiply by 60.

27 Lab Name: Geometry Scavenger Hunt

Ride(s) involved: Entire Park

Key Question:

Student Tech Requirements: Camera, photo enhancement software (MS Paint or other), printer, or publishing software

Activity Directions:

This activity requires that you take some photos as you roam Hersheypark. Look for the geometric structures in the list below. Take a photo of the structure. When you return to class, upload the pictures into Microsoft Paint and highlight the structure using the drawing features. Also, label the structure. Then, at the direction of your teacher, print out the photos in an appropriate format or create a MixBook of them.

• Segment • Scalene Triangle • Ray • Trapezoid • Right Angle • Parallelogram • Acute Angle • Rectangle

• Obtuse Angle • Rhombus • Complementary Angles • Square • Supplementary Angles • Hexagon • Vertical Angles • Octagon • Parallel Lines • Pentagon • Perpendicular Lines • Congruent Angles • Acute Triangle • Similar triangles • Right Triangle • Mirror symmetry • Obtuse Triangle • Rotational symmetry • Equilateral Triangle • Ellipse • Isosceles Triangle • Hyperbola

Lab Name: Graphical Scavenger Hunt

Ride(s) involved: Entire Park

Key Question:

Student Tech Requirements:

Activity Directions:

Can you identify the ride that has the height vs. time graph shown?

Please write the associated ride name under each graph.

Lab Name: Hersheypark Number Hunt Ride(s) involved: Entire Park Key Question: Student Tech Requirements:

34 Activity Directions:

As you roam Hersheypark, write a mathematical expression for the situation and then find the answers to the mathematical expressions.

1. There are 4 wheels on each of the antique cars on the Turnpike and the wheels each have the same number of spokes. How many total spokes are there on every car?

2. If 1 train on the Trailblazer leaves and every other seat, starting with the first, has 2 people in it and the remaining seats have 1 person, how many people are riding in the train?

3. If every seat on The Howler has 3 people in it, how many people will be riding on The Howler?

4. What percent of the horses on the Carrousel are occupied if only the horses on the outside row are occupied?

5. On Lightning Racer, the odd seats in the trains have 1 person and the evens have 2 (i.e. the first seat has 1 person, the second has 2, the third has 1 person, etc.). How many people are riding the Lightning Racer?

6. If 80% of the seats on the Music Express are filled and all have 3 riders each, how many riders are on the Music Express?

7. If each seat on the Pirate can hold 5 people, what is the maximum number of people that can ride the Pirate at 1 time with 1 seat completely empty?

8. If only the outside seats of each cluster on The Claw are occupied, what fraction of the seats are occupied?

9. If 7/8 of the seats on the sooperdooperLooper are full when it dispatches, how many riders are there in that train?

10. The seats on The Claw are in groups of 4. If one of the middle seats of every cluster is empty, what percent of the seats are occupied?

11. The seats on The Whip seat up to 3 people. If 44% of the booths have 3 people and the remaining have 2 people, how many people are riding?

Lab Name: Scrambler

Ride(s) involved: The Scrambler

Key Question: How do you calculate arc length?

Student Tech Requirements: Smart phone, stopwatch

Activity Directions: Use your smart phone to retrieve the following information:

Radius of gondola arm: ______Rate of gondola arm: ______

Radius of center arm: ______Rate of center arm: ______

1. Measure the time of the operation of the ride. t = ______

2. Estimate the distance a rider travels during the operation of the ride using circumference.

3. Estimate the distance a rider travels during the operation of the ride using arc length. You may want to use a graphing calculator.

4. Which estimation do you think is more accurate? Explain your reasoning.

Ride data / elements: The Scrambler ENGLISH METRIC

Length of center arm 14’6” 4.42 m

Rotation rate of center arm 11.4 rpm 11.4 rpm

Length of gondola arm 11’2” 3.4 m

Rotation rate of gondola arm 14.3 rpm 14.3 rpm

Alternate Activity Directions:

Use your smart phone to retrieve the following information:

Radius of gondola arm:______Rate of gondola arm:______

Radius of center arm:______Rate of center arm:______

1. Measure the time of the operation of the ride. t = _____

2. Estimate the distance a rider travels during the operation of the ride using circumference.

3. Estimate the distance a rider travels during the operation of the ride using arc length.

a) Parameterize the path swept out by both the center and gondola arms and write each as a vector-valued function. Keep in mind that the rates at which the arms rotate are not equal so you will need a coefficient for the angles for one of the paths. You can experiment with ratios of the rates for this coefficient. Useful formulas: x = rcost , y = rsint , r t( ) = x t i( ) + y t j( ) 37

b) Add the paths together to write as a single vector-valued function.

c) Use your calculator to determine the period of the function. Estimate in terms of π.

d) Use to calculator to find the arc length of one period.

e) Estimate how many times the path is traced during the operation of the ride.

f) Estimate the distance a rider travels during the operation of the ride.

4. Which estimation to you think is more accurate? Explain your reasoning.

39 Lab Name: sooperdooperLooper

Ride(s) involved: sooperdooperLooper

Key Question: What is the relationship between position and velocity?

Student Tech Requirements: Smart phone, stopwatch

Pre-Activity:

Discuss how to determine the speed at the top, about half-way down, and at the bottom.

Activity Directions:

Use your smart phone to retrieve the following information:

Length of train:______Mass of loaded train:______Height of first hill:______

1. Determine the speed at the top, about half-way down, and the bottom of the first hill. To do this, pick a place in space that the train will pass through. As the train enters that space, start the stopwatch; as the train leaves that space, stop the stopwatch. Simply d take the length of the train and divide by the time to get a value for speed (r = ). t Time for the train to pass through a place in space at the top = ______

Time for the train to pass through a place in space half-way down = ______

Time for the train to pass through a place in space at the bottom = ______

40 2. Measure the time it takes for the train to get half-way down and all the way down the first hill.

ttop = __0 s___

thalf −way = ______tbottom

= ______

3. Use the above speeds and their respective times to create an exponential equation that models the train’s velocity. This can be achieved through the STAT menu of a TI graphing calculator.

4. Use your model to find the instantaneous velocity of the train when it is 8 m down the first hill.

Ride data / elements: sooperdooperLooper ENGLISH METRIC

Length of train 42’6” 12.95 m

Mass of the loaded train 9400 lbs. 4264 kg

Height of the first hill 81’0” 24.69 m

Lab Name: Storm Runner

Ride(s) involved: Storm Runner

Key Question: What is the relationship between velocity and acceleration?

Student Tech Requirements: Smart phone, stopwatch

Pre-Activity:

Since the concept of work is not directly a part of the AP curriculum, it is an application of integration. It would be good to talk about work in order to get a feel for what it is and if the answer to the question below makes sense. How much is a Joule? How much work is required to lift a book from the floor to a desk? How much work is required to lift a loaded elevator up 2 stories?

Activity Directions:

Use your smart phone to retrieve the following information:

Mass of train:______Number of passengers:______

1. Measure the time for the train to accelerate from rest to the point just prior to its ascent (when the train achieves its maximum velocity).

m t = _____ 2. At t = 0, the train has an acceleration of 3.5 g’s (34.3 2 ). At the time you

found above,

m the train is moving at 1.5 g’s (14.7 2 ). Use this information to create an exponential s equation that models the train’s acceleration. This can be achieved through the STAT menu of a TI graphing calculator.

3. Use your acceleration equation to create a velocity equation. Maximum velocity = 32.187 m s .

u 1 u Useful formula: ∫a du = a +C . lna

4. Use your velocity equation to create a position equation. Distance during the timed acceleration = 46 m.

5. During the acceleration, how much work is done on the train? Assume the average mass of the passengers to be 86 kg. Useful formulas: F ma= , W =∫ F x dx( ) .

Ride data / elements: Storm Runner ENGLISH METRIC

Mass of loaded train 23,501 lbs 10,659 kg

Number of passengers 20 20

Lab Name: Turnpike Ride(s) involved: Twin Turnpike – Sunoco Speedway & Sunoco Classic Cars Key Question: How do we calculate velocity and use it to solve problems? Student Tech Requirements: stopwatch

Activity Directions:

1. Find out how much time it takes to go around the Classic Cars track and the Speedway track. a. Either ride both tracks yourself or work with a friend (each of you ride one track). b. Use a stopwatch to time your rides. Start the stopwatch as soon as the car starts to move, and stop the stopwatch when the car comes to a complete stop. Record the time, in seconds, for each ride in the table below.

Track Time around / lap time (seconds)

Classic Cars

Speedway

43 2. “Velocity” is another word for “speed.” When you calculate velocity, you are answering the question “How far did I go, and how long did it take me to get there?”

The formula to calculate velocity is

distance d V = or Velocity =

t time

Write your lap times in the table below. The distance for each track is provided. Use this information to calculate your velocity for each track.

Track Time around / lap time Length of Track Velocity (feet/second)

Classic Cars 1,167 feet

Speedway 1,076 feet

3. For cars in the United States, velocity is usually described in miles/hour (or mph). For the Classic Cars track and Speedway track, convert your feet/second velocity to miles/hour on your own or following these steps below.

Classic Cars Velocity (ft/sec)

feet 60 seconds 60 minutes 1 mile miles 1 second x 1 minute x 1 hour x feet = hour 5,280

44 Speedway Velocity (ft/sec)

1 mile 1 second 5,286 x x x = 1 minute 1 hour feet hour feet 60 seconds 60 minutes miles

4. How many miles is it to Hersheypark from your school?

5. If you drove as fast as the Speedway car, how many hours would it take you to get back to school? How many minutes?

6. Would it take more, less, or about the same amount of time if you drove as fast as the classic car?

7. If you had to get back to your school in 30 minutes, how fast (in miles/hour) would you have to drive?

Post-Activity:

Graph your velocity information from the Speedway track. Time (in seconds) on x axis, distance (in feet) on y axis.

Do you think this graph tells a “true story” of your trip on the speedway? Why or why not?

**Hersheypark graph of the speedway’s velocity (will be provided in next update to this guide) Ride data / elements: actual graph of speed vs. time for 1 lap of each of the Classic and Speedway tracks

Discuss the similarities and differences you see. What reasons might exist for the differences? 45

Real-Life Racing:

The final race of the 2011 NASCAR Sprint Cup Series was run at Homestead-Miami Speedway on Nov. 20, 2011. In that race, the fastest lap was run by Carl Edwards, with a speed of 171.838 mph.

Convert this speed from miles/hour to feet/second.

If one lap of the Speedway is 1.5 miles long, how many seconds did it take Carl to complete the fastest lap of the race?

NASCAR teams and officials measure lap times to the 1000th of a second, and speeds to the 1000th of mph. Why do you think this might be?

Lab Name: Wave Swinger

Ride(s) involved: Wave Swinger

Key Question:

Student Tech Requirements: Smart phone, stopwatch, altimeter

Pre-Activity:

Discuss how to use an altimeter.

Activity Directions:

Use your smart phone to retrieve the following information:

Radius of the outer ring:______

1. Measure the period of rotation.

T = ______46

2. Calculate the tangential speed vt for the outer ring. ds

Useful formulas: s r= θ, vt = dt

3. Use an altimeter to measure the angle from the ground to one of the seats on the outer ring.

θ = ______

4. Calculate the height of the seat using the above angle. Estimate both the horizontal distance on the ground from the seat to your position and the height at which you are holding the altimeter.

5. Suppose an outer ring rider’s shoe fell off their foot. Find the velocity with which the shoe hits the ground.

Ride data / elements: Wave Swinger ENGLISH METRIC

Radius of outer ring 30’6” 9.3 m

Lab Name: When Will It Happen?

Ride(s) involved: Entire Park

Key Question:

Student Tech Requirements:

Activity Directions:

Determine the number of people described in each situation…

Problem 1: If each seat on the Tidal Force® seats 4 people and each seat is half full every time a boat leaves. How many boats will have dispatched before boat with the 48th person “feels the force?”

Problem 2: If you are standing in line for Fahrenheit and there are 42 people in front of you. How many FULL trains will leave BEFORE the train with you in it leaves?

Problem 3: You enter the line for the Wave Swinger just as a group of people are exiting. You notice there are 100 people in front of you. How many times will the ride start, and be completely full, BEFORE you get on the ride?

Problem 4: You notice that the line for the Whip is not too long, only 24 people. You do not get on the ride because all of the seats are occupied, but none by more than 2 people. Determine how many of the seats are filled with 2 people and how many with just 1 person.

48 Lab Name: ZooAmerica – Animals in their Habitats

Ride(s) involved: ZooAmerica – various habitats

Key Question:

Student Tech Requirements:

Activity Directions:

There are 5 habitats in ZooAmerica: Northlands, Big Sky Country, Eastern Woodlands, The Great Southwest, and Southern Swamps.

1) Visit at least 3 habitats, and count the number of different types of animals you see in each of the following categories. Then convert each count into a percent of that habitat’s total.

Northlands Big Sky Country

Animal # of different % of total Animal # of different % of total Category types Category types

Birds Birds

Reptiles Reptiles

Mammals Mammals

Fish Fish

Other Other

TOTAL TOTAL

49 Eastern Woodlands

Animal # of different % of total Category types

Birds

Reptiles

Mammals

Fish

Other

TOTAL

The Great Southwest Southern Swamps

Animal # of different % of total Animal # of different % of total Category types Category types

Birds Birds

Reptiles Reptiles

Mammals Mammals

Fish Fish

Other Other

TOTAL TOTAL

2) Using the percentages you calculated, make a pie chart for each habitat.

3) Which habitat has the highest percentage of reptiles?

Which habitat has the highest percentage of birds?

Which habitat has the highest percentage of mammals?

4) Is the habitat with the highest number of different types of reptiles also the habitat with the highest percentage of reptiles? If not, explain what causes the difference.

5) Which habitat(s) have no reptiles at all? Why do you think that might be?