What Goes Up
PETER PALIWODA
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?
Position
Sensor
Time
How can we Use Math to Explain this?
Position
Sensor
Time
Parabola
yx
2
y  x  1
2
Parabola Equation
Free Rolling (m)
y
x
Time (sec)
Parabola Standard Form
y  ax  bx  c
2
y = free rolling distance on the ramp in meters
x= 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 
b
a
c
a
Let’s Model A Simple Rollback!
 Experiment:
 Inclined Ramp
 Cart
 Vernier Software: Motion Detector , Lab Pro, and Logger Pro
 Procedure
 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
Product of x-intercepts =(c/a)
Parameter
a
b
c
Second x-intercept
Sum of x-intercepts = -(b/a)
Value
Equation
y  ax  bx  c
2
y   x   x  
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 presented
 Formula for deriving parabola explained
 To build safe roller coasters one needs to understand
the math that is behind it
 The equation derived was close to real behavior of
the cart on the ramp