Physics is PHUN!!!

1. Physics is PHUN!!! http://fairway.ecn.purdue.edu/~step/class_material

2. Friction • What have we learned from graphing our ideal velocity against our actual velocity from the ramp exercise? • We lose a lot of energy due to outside forces, such as drag, friction, the coaster wobbling, heat loss, etc. • To account for this, we’ve created equations of loss to use for your modeling. These equations were created by doing many trials and mathematical calculations and have been simplified as much as possible for your use.

3. Loss Equations • The upcoming loss equations are set up as follows: • Find the ideal velocity at the location, as you have been. • Subtract the loss from the ideal velocity, and you will have the actual velocity. • You must separate your coaster’s track into sections where it is decelerating, accelerating, and traveling on a horizontal surface. • In the equations, ∆xis the distance from the start of your section (where your coaster first started decelerating or accelerating) to your current location, in feet. • L is the velocity loss of the previous section (from where it started to decelerate or accelerate to where your current section started)

4. Frictional Loss • To find the frictional velocity loss on any section of your coaster, use the following equation: Ln = videal,n – vactual,n where… L : frictional velocity loss on current track section videal : ideal velocity, according to Energy Conservation vactual : actual velocity, calculated with formulas in the next three slides • Note: The very first loss for your coaster, L0, will be equal to 0.

5. Friction - Accelerating When your car is accelerating (i.e. moving downhill), use the following are loss equations: • If the angle is greater than 45º (from horizontal): vactual,curr= videal,curr– (∆x + Lprev) • If the angle is less than or equal to 45º (from horizontal): vactual,curr= videal,curr– (0.75*∆x + Lprev) • If the length of the piece of track is less than 1.5 ft, no matter the angle: vactual,curr= videal,curr– (0.3 + Lprev) where… ∆x : length of current section L : loss from previous section

6. Friction - Decelerating If your coaster is decelerating (i.e. moving uphill): • If your angle is greater than 45º (from horizontal): vactual,curr= videal,curr– (-∆x + Lprev) • If your angle is less than or equal to 45º (from horizontal): vactual,curr= videal,curr– (-0.75*∆x + Lprev) where… ∆x : length of current section L : loss from previous section

7. Friction - Horizontal When you coaster is moving on a horizontal surface, use the following equation for loss: vactual,curr = videal,curr – (1.32*∆x + Lprev) where… ∆x : length of current section L : loss from previous section

8. Velocity Loss Example h1 ∆x1 h3 ∆x2 θ1 θ2 h2 Knowns:Unkowns: ∆x1 = 5 θ1 = 75⁰ h1, h3, ∆x2 = 2 θ2 = 15⁰ Ideal and Actual Velocities V0 = 0 for each section Lprevious for each section

9. Section 1 – Downward Ramp h1 h1 = ∆x1* sin(θ1) = 5 * sin(75) = 4.83 ∆x1 mgh1 + .5mV02 = mgh2 + .5mV2ideal = __________ Videal = √ (2*32*4.83) Videal= ______ft/sec θ1 h2 L1 = Videal - Vactual Vactual = Videal – (∆x1 + Lprev ) L1 = 17.58 – 12.58 L1 = ___ ft/sec Vactual= 17.58 – (5 + 0) Vactual= ______ ft/sec

10. Section 2 – Upward Ramp h3 ∆x2 h3 = ∆x2 * sin(θ2) = 2 * sin(15) = .52 θ2 h2 mgh2 + .5mV02 = mgh3 + .5mV2ideal V0 = Vactual,1 • Vactual= Videal– (-0.75*∆x + Lprev) • Lprev= L1 _____________________ Videal= √ (12.582 - (2 * 32 * .52) Videal= _____ ft/sec • Vactual= 11.18 – (-0.75*2 + 5) • Vactual= ___ ft/sec L2 = Videal- Vactual L2 = 11.18 – 7.68 L2 = ____ft/sec

11. Overall • So, in the example we modeled before, the sections would break down as follows: • Ramp… • Accelerating the entire time • Downward Curve… • Accelerating the entire time • Turn… • Flat surface • Loop… • Decelerating until 90 degrees (from horizontal!) • Accelerating after 90 degrees (from horizontal!) • IF you use a banked turn… • Decelerating for the first half of the turn • Accelerating for the second half of the turn

12. Team Modeling Exercise(That means as a TEAM!) • As a team, calculate the actual velocities at all of the locations in your Excel spreadsheet.

13. Actual Velocities

14. Time • The time at any location is simply the distance traveled since the velocity was last calculated divided by the velocity at that point. • As a team, take 5 minutes and find the elapsed time for every velocity on your Excel spreadsheet.

15. Time Should have 1ft intervals (whole numbers) in here

16. Is this realistic? • Since the loss equations we are using are specific to our materials, we have to transfer our numbers to the size of an actual coaster. This means that the scale factor becomes 1 in. = 1 ft., so from the previous exercises, 40 ft. = 1 ft. • This scale factor will not be used to determine velocity, but it should be used to determine elapsed time in your cost workbook.

17. Model Your Own Coaster! • Now you get to model your own coaster!!! • Your coaster should have the actual velocity, g’s, running distance (1 ft intervals), and running time at every increment. • You should measure the track distance at the beginning and end of every section (broken down into track pieces (i.e. loops, ramps, etc.) and accelerating or decelerating) and at one foot intervals between those increments. For loops or curves use simple angle divisions (45 or 90 degrees) and not one foot intervals. • Your sheet should also have a graph of the g’s vs. running distance and your estimated thrill factor and a graph of the velocity vs. distance. • You may include any other graphs that will make your spreadsheet easier to understand. Don’t forget to format everything!

18. Analyzing Your Data • Getting weird numbers??? • Need to make any changes??? • What kind of changes??? • Have you documented the changes???