Physics is PHUN!!!
http://fairway.ecn.purdue.edu/
~step/class_material
Balancing Fun and Safety
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We all want our roller coasters to be a
lot of fun, but this cannot come at the
expense of safety.
All of our cool coaster features (e.g.
drops, loops, spirals, hills, etc.) are
strictly governed by physics, and can be
described by velocity, acceleration,
forces, and energy.
Velocity
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Velocity is the measure of a change in
the location of an object with respect to
time.
As velocity increases, the time to travel
between points becomes smaller, and
vice versa.
Velocity = Distance / Time
Conservation of Energy
(Woo!)
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All energy in the universe is conserved (It can
neither be created nor destroyed). This means
that energy only changes from one form to
another.
Example: If you were to
build a loop, put a car on
the side, and drop the
car, would it complete
the loop? Why or why
not?
Example: Racing Cars
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Let’s say you have two cars on your loop:
you put one halfway up the loop and the other
almost at the top. Which car will attain a
greater final height? Why?
Answer: The one almost at the top.
Why? Because more energy is put into it.
Since energy is always conserved, the more
energy you put into the car, the longer it will
be able to resist the pull of gravity.
Forms of Energy
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In our roller coaster example, Potential
Energy was converted to Kinetic Energy and
back again. But PE and KE aren’t the only
forms of energy! Some examples of other
forms include Rotational, Vibrational,
Chemical, Electrical, Nuclear...
However, for our physics work, we’ll primarily
use PE and KE.
Potential Energy
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Potential Energy, PE, is the energy
associated with the position (height) of an
object. It is the measure of how much energy
an object could potentially have of another
form, like kinetic energy.
Example: If you hold an object up in the air, it
has potential energy because it has the
potential to fall and gain kinetic energy.
Kinetic Energy
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Kinetic Energy, KE, is the energy
associated with the motion of an object.
Example: If the same object from the
previous example is now falling, it has
kinetic energy associated with its
motion.
Potential Energy - Formula!
PE = m*g*h
where...
PE : Potential Energy
m : the mass of the object, in kilograms (kg) or
pounds (lb)
g : acceleration due to gravity (either 9.8
meters/sec/sec or 32 feet/sec/sec)
h : height of the object, in meters (m) or feet
(ft)
Make sure your units are CONSISTENT!!
Acceleration?
So far we’ve discussed velocity, but now
we also need to know what acceleration
is. Acceleration is how fast an object
changes velocity. In other words…
Acceleration = Velocity / Time
or
Acceleration = Distance / Time / Time
Gravity is Awesome!
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Gravity is the force that keeps all of us from
floating away!
On Earth – or at least anywhere where you'd
care to build a rollercoaster – objects
accelerate at the rate of 9.8 m/s2 or 32 ft/s2.
Kinetic Energy - Formula!
KE = (1/2)*m*v^2
where...
KE : Kinetic Energy
m : mass of object, in kg or lb
v : velocity of object, in m/s or ft/s
Conservation of Energy… Again
So, in our ideal world, how are these two
related? The energy of an object, E, is equal
to the sum of all the forms of energy it has.
So…
Etotal = KE + PE (for our purposes)
The Next Step…
Since energy is conserved, the total energies at any
states for a (closed) system should be equal. So, if
you were to drop a ball from some height, the energy
of the ball should (and will) be the same when you're
holding it, while it's falling, and as it hits the ground.
So…
KEi + PEi = KEf + PEf
or
(1/2)*m*vi^2 + m*g*hi = (1/2)*m*vf^2 + m*g*hf
(the subscripts i and f refer to the system's state at different times)
Individual Exercise: Energy
(~ 1 minute)
If you drop a penny off the top of the
Empire State Building (1250 ft), how
fast will it be going when it hits the
ground?
Team Modeling Exercise:
The Empire Strikes Back!
(~ 10 minutes)
• Now, as a team, calculate the final velocity of
the penny if dropped from each story (one
story is 10 ft) to the ground, starting at 0 ft
and going to 1250 ft.
• A not-so-subtle hint: USE EXCEL!
Team Exercise: Ramp
(~ 5 minutes)
• This time you are going to roll your
penny from before down a ramp.
• Ramp specifics:
L
– 240 ft tall (h)
– 30 degree incline (θ)
– 480 ft in length (L)
h
θ
• What’s the velocity at the bottom of the
ramp?
Team Modeling Exercise: Ramp!
(~ 2 minutes)
• As a team, model velocity on a ramp
USING EXCEL from the height of 6 ft
down to 0 ft in ½ ft increments.
• Hint: This is not the same as the final velocity
corresponding to each starting height.
Team Modeling Exercise: Spirals!
(~ 5 minutes)
• How would you model velocity on a
spiral?
• If you think about it, a spiral is really just
a rolled up ramp. So now, how do you
model the velocity of a spiral? Model
velocity on a spiral from heights of 10 ft
down to 0 ft in 1 ft increments.
Team Modeling Exercise: Loops
(~ 7 minutes)
• One last step. Now,
create a model for a
loop for heights from
100 ft down to 0 ft in
5 ft increments.
Assume v = 0 at the
apex (100 ft).
Putting It All Together…
Team Modeling Exercise
(~ 10 minutes)
• Goal: Model the following roller coaster:
1. Ramp:
•
•
initial height = 300 ft; final height = 50 ft
length of track = 400 ft
2. Downward Curve:
•
•
initial height = 50 ft; final height = 0 ft
radius = 50 ft; through 90 degrees (pi/2 radians)
3. Turn:
•
•
initial height = 0 ft; final height = 0 ft
radius = 50 ft; through 180 degrees (pi radians)
4. Loop:
•
•
initial height = 0 ft; apex height = 100 ft; final height = 0 ft
radius = 50 ft
• Set up the spreadsheet with any needed constants, the
titles for the track sections, and the initial, apex (for the
loop), and final heights of each track section.
Spreadsheet
with Heights
Team Modeling Exercise: Velocity
(~ 10 minutes)
• The velocity calculations will be made with
our same super-awesome energy
conservation equations.
• The starting velocity for each section will be
the ending velocity of the previous section.
Velocity Calculations
KEi + PEi = KEf + PEf
initial:
final:
(1/2)*m*vi^2 + m*g*hi
=
(1/2)*m*vf^2 + m*g*hf
vf = sqrt(2*g*(hi - hf) + vi^2)
Modeling
Velocity
Acceleration During a Curve
• While an object is moving along a curve, it must
maintain a certain acceleration to remain on that
curve. The magnitude of that acceleration is given
by…
aC = v^2 / R
where…
aC : centripetal acceleration
v : velocity of object
R : radius of curve
This equation works for both straight and curved paths!
G’s!
• You’ve probably all heard of people experiencing
“G’s” in cars, jets... or roller coasters!
• To calculate the G’s experienced by something, you
do…
G’s = a / g
where…
a : acceleration of object
g : gravitational accel. (9.8 m/s2 or 32 ft/s2)
Calculating G’s Felt
(at the bottom of a loop)
∑Fy = m*a = “Normal Force” – “Weight”
• The G’s felt by the rider are due to the
“Normal Force”, so we must calculate the
“Normal Force”, or N…
∑Fy = m*(v^2 / R) = N – m*g
N = m*(v^2 / R) + m*g
N / m = (v^2 / R) + g
[M∙L/T2]
[M∙L/T2]
[L/T2]
Now, recall that… G's = a / g
So…
[unitless]
G’s Felt = aC / g + 1
(at the bottom of a loop)
Golly G Gosh Darn!
yup…
A heads-up: WE WILL ONLY BE CALCULATING G’S
IN THE VERTICAL DIRECTION!!!
• At the top of a loop:
avertical = v^2 / R - g
G’s Felt = (v^2 / R - g)/g
• At the bottom of a loop:
avertical = v^2 / R + g
G’s Felt = (v^2 / R + g)/g
• Halfway up the side of a loop (at 0° and 180 ° from horizontal):
avertical = 0 + g
G’s Felt = g/g = 1
G’s Everywhere Else
You need to find the vertical component of aC, so…
180º
R
θ
0º
On the top half of a “loop”:
G’s Felt = -1 + (v^2 / R)*sin(θ)/g
On the bottom half of a “loop”:
G’s Felt = 1 - (v^2 / R)*sin(θ)/g
G’s Felt by the Rider
Calculating Height in a Loop
• The height at any point during a loop can be found by
some simple trigonometry.
90º
h
180º
R
θ
On any section of a loop:
0º
h = R*(1 + sin(θ))
270º
Team Modeling Exercise: G’s
( ~ 5 minutes)
As a team, calculate the G’s at all the
locations in your Excel file.
Modeling G’s
Calculating Track Length
• Ramp:
– it’s given… 400 ft
• Downward Curve:
– radius = 50 ft; through 90 degrees
• Turn:
– radius = 50 ft; through 180 degrees
• Loop:
– radius = 50 ft; full 360 degrees
“arc length” = 2*pi()*R*(degrees / 360)
Team Modeling Exercise: Distance
(~ 10 minutes)
• As a team, calculate the distance
traveled (track length) at every location
in your Excel file.
Thrill Factor
• Thrill Factor is a measure used by roller
coaster buffs to find out how exciting a roller
coaster is. You can calculate the Thrill Factor
by graphing your G’s vs. distance traveled.
• Next, draw a line through g = 1.
• Find the absolute value of the areas above
and below g = 1 (by ESTIMATING the area
as a series of triangles and rectangles)
• Use the data handed out to your teams.
Ramp Exercise
• Model your ramp’s ideal velocities at each
height.
• Then, calculate your percent error:
%err = (abs(vactual – videal) / videal)*100%
• You will need your project data from last night
for vactual.
Develop an Equation
Using your flat track data, find the constant k (this is
NOT the coefficient of friction) in the following
equation. Hint: Solve for k five times and take the
average of these:
va = vi – k*d
where…
va : actual velocity
vi : ideal velocity
d : distance traveled
k : constant with units of s-1