PETER PALIWODA

What Goes Up

First roller coasters -Russian Ice Slides

Roller Coasters got their beginning during the long cold winters in the 1700’s. They were found at amusement parks and they were made entirely out of ice.

It consisted of a steep drop with a few bumps at the end. The rider rode in a sled made of wood or ice. Sand was placed at the end of the ride to slow the riders speed.

It took a lot of skills to ride these slides

Russian Ice Slides

Six Flags Great Adventure: Kingda Ka

• Speed 128 MPH• 45 stories = 456’• Length = 3118’• Ride Duration = 59sec

What is Rollercoaster Rollback?

Occurs when the train of a launched rollercoaster is not launched at a speed fast enough to reach the top of the tower; so it rolls back down the tower and is stopped by brakes on launch track

http://en.wikipedia.org/wiki/Rollback_(roller_coaster)

How can we Use Math to Explain this?

What does a plot of a rollback look like?How can we use algebra to describe the

relationship between position and elapsed time? Sensor

Time

Posi

tion

How can we Use Math to Explain this?

Sensor

Time

Posi

tion

Parabola

2xy 21 xy

Parabola Equation

y

x

Fre

e R

oll

ing

(m

)

Time (sec)

Parabola Standard Form

y cbxaxy 2

y = free rolling distance on the ramp in metersx= time in seconds

c=distance at time = 0

The product of the x-intercepts is equal to the ratio The sum of the x-intercepts is equal to

a

c

a

b

Let’s Model A Simple Rollback!

Experiment: Inclined Ramp Cart Vernier Software: Motion Detector , Lab Pro, and

Logger ProProcedure

Zero reference position Collect data Analyze data based on parabolic portion of graph Find x and y intercepts Compute values for equation parameters Write equation for rollback

Taking Data

y-intercept = c First x-intercept Second x-intercept

Product of x-intercepts =(c/a) Sum of x-intercepts = -(b/a)

Parameter Value

a

b

c

Equation

cbxaxy 2

xxy 2

A Minimum that is a Maximum?

In our experiment, the vertex on the parabolic distance vs. time plot corresponds to a minimum on the graph even though this is the position at which the cart reaches its maximum distance from the starting point along the ramp. Why?

A Minimum that is a Maximum?

The motion detector records distance away from itself.

The detector was at the top of the ramp, so when the cart was at its closest (minimum distance) to the detector when the cart was at its highest point.

What if…?

Let’s say that we repeated our simple rollback, BUT instead of placing the motion detector at the top of the ramp we placed it at the bottom of the ramp.

Sketch your predicted distance vs. time plot.Does changing the location of the motion

detector affect coefficient a?

What if…?

Placing the motion detector at the bottom of the ramp does not affect the distance data. The data will still be parabolic.

However, the parabola will open downward and the sign of coefficient a changes.

Conclusion

Basic math behind roller coasters presentedFormula for deriving parabola explainedTo build safe roller coasters one needs to

understand the math that is behind itThe equation derived was close to real

behavior of the cart on the ramp