Investigating linear velocity in circular motion.

Background

Amusement parks are filled with rides for people with all kinds of risk tolerance.  Some rides are simple and classic, attracting all kinds of riders and ages with slow speed and high levels of nostalgia.  Others are flashy and exhilarating, incorporating maximum speed and twists. 

An engineer comes to you with an idea: she thinks if you make a carousel big enough, it could be both a kiddie ride AND a thrill ride.  But she’s not quite sure how big it will need to be to achieve this.  Right now the largest carousel is only 24 m in diameter…surely it can be beat!

Her research has shown that kids like rides that move about 2 meters per second (m/s) (approx 4 miles per hour).  A very popular rollercoaster at the park moves at about 55 m/s (125 miles per hour).  Can both of these speeds be achieved at the same time?  Can we actually make a ride safe for little kids and exhilarating enough for thrill seekers? How much space would it take up in the park?

But… is velocity all that matters?

She then leans in and reminds you about something - if thrills were only about how fast you are going, then spinning around the earth at 1,000 miles per hour would lead to our entire lives as a stomach-lurching thrill ride.  It’s not exciting though, because we don’t feel like we are going fast since everything in our classroom is moving at that same speed and direction. Thrills are both about how fast you perceive you are going and acceleration (increasing or decreasing speed).  On a carousel, you get a sense of excitement as you see yourself moving fast compared to the people standing in line, and feel a pull outwards due to centripetal acceleration. 

In our analysis, we need to look at both velocity and centripetal acceleration.

Dataset

This data set was gathered through a simulator.  You can add your own data by going to a site like PHET’s Ladybug Revolution.  

The rotational speed (how long it takes to spin once) was set by timing a basic carousel like this one, which is known to be safe for children.  The rotational speed of this carousel is 12 seconds/rotation

Variables

Distance from center -  this numerical value describes the distance from the central pivot point which the data was taken. Measured in meters.

Time for one rotation - this numerical value describes the amount of time passed since the start of the rotation. It is the total time.  Measured in seconds. 

Total Distance Traveled - this numerical value describes the total distance around the circle that has been traveled (circumference).  Measured in meters.

Linear Velocity - this numerical value describes the distance traveled around the circle in meters (linear measurement).  Measured in meters per second. 

Centripetal Acceleration - this numerical value describes the change in velocity over time on a curved path. Even if the linear velocity value does not change, the direction changes creating an acceleration.  Measured in meters per second per second.

G-force - this numerical value is how many “Earth accelerations” are present. It creates a feeling of weight or pressure, and it is calculated by taking the centripetal acceleration and dividing by 9.81 m/s/s. When this value is zero, you feel no added weight.  If it is more than zero, you feel a pull.  It has no unit. 

Activity

Goal 1 - Where do we put the little kids?  

Location of kiddie section on carousel

1. Make a graph so you can see how distance from the center affects the velocity. Paste your graph below:

2. What equation fits your data best? Include that below:

3. Replace x and y in your equation with the correct variables and units.

4. The engineer said the parameters for a fun kid’s ride is that it does not go any faster than 2 m/s (4 miles per hour). Use your equation to determine how many rows the ride can have for little kids if each row is 1 m apart and starts at 1 m  

Extension A:  Based on your number of rows, how many horses will you will need for the kiddie section of the carousel?   Each horse takes up approx 2 m of space (for the horse itself and spacing to the next one). 

Goal 2 - Where do we put the thrill seats?  

Location of thrill section on carousel

5. The engineer said the fastest rollercoaster in the park moves in a straight line at 55 m/s (125 miles per hour). Use your equation from #3 to determine the best location for 2 rows of thrill ride on the carousel to match that speed.

Extension B: Based on your number of rows, how many horses will you will need for the thrill section of the carousel? Each horse takes up approx 2 m of space (for the horse itself and spacing to the next one).

Goal 3 - Is it safe? 

Looking at amount of acceleration and G-force

As we said before, thrill rides are about both speed and acceleration.  The human body can enjoy high speeds, but not too much acceleration.
Here are some g-force values we have tested on humans:

6. Our engineer advises to find a g-value around 2.5 - 3.5 g’s, to maintain both thrill and safety (people are only on carousel horses after all, not strapped into roller coaster seats!). Make a graph which will tell how data of location and g-force is related:

7. What equation fits the data? Write it below using the variables (not x and y).

8. Determine both the acceleration at your recommended thrill distance (question #5).  If it does not fit with these new safety guidelines, what is the new safe amount of rows for the thrill ride rows?

Final Findings

9. If an average city bus is 12 meters in length, approximately how many city buses (end to end) of space do we need to create this carousel? Is this practical to build?

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