Physics is PHUN!!! http://fairway.ecn.purdue.edu/ ~step/class_material Balancing Fun and Safety 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 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!)
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 Lets 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 In our roller coaster example, Potential Energy was converted to Kinetic Energy and back again. But PE and KE arent the only
forms of energy! Some examples of other forms include Rotational, Vibrational, Chemical, Electrical, Nuclear... However, for our physics work, well primarily use PE and KE. Potential Energy 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
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 weve 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! 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: 240 ft tall (h) 30 degree incline () 480 ft in length (L) L
h Whats 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 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! Gs! Youve probably all heard of people experiencing Gs in cars, jets... or roller coasters! To calculate the Gs experienced by something, you do
Gs = a / g where a : acceleration of object g : gravitational accel. (9.8 m/s2 or 32 ft/s2) Calculating Gs Felt (at the bottom of a loop) Fy = m*a = Normal Force Weight The Gs 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 [ML/T2] [ML/T2] [L/T2] Now, recall that G's = a / g So [unitless]
Gs Felt = aC / g + 1 (at the bottom of a loop) Golly G Gosh Darn! yup A heads-up: WE WILL ONLY BE CALCULATING GS IN THE VERTICAL DIRECTION!!! At the top of a loop: avertical = v^2 / R - g
Gs Felt = (v^2 / R - g)/g At the bottom of a loop: avertical = v^2 / R + g Gs Felt = (v^2 / R + g)/g Halfway up the side of a loop (at 0 and 180 from horizontal): avertical = 0 + g Gs Felt = g/g = 1 Gs Everywhere Else You need to find the vertical component of aC, so 180
R 0 On the top half of a loop: Gs Felt = -1 + (v^2 / R)*sin()/g On the bottom half of a loop: Gs Felt = 1 - (v^2 / R)*sin()/g Gs Felt by the Rider
Calculating Height in a Loop The height at any point during a loop can be found by some simple trigonometry. 90 180 R 270
h On any section of a loop: 0 h = R*(1 + sin()))) Team Modeling Exercise: Gs ( ~ 5 minutes) As a team, calculate the Gs at all the locations in your Excel file.
Modeling Gs Calculating Track Length Ramp: its 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 Gs 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 ramps 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
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