161 – Calculus Project Presentations Day 2

(these are my quick notes from Day 2 of presentations)

Limits of Human Speed

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What is the theoretical limit of human speed? Showed world records in 100, 200, 400, 800, 1600, and marathon.
mv = F – D(v).
t-t_0 = m integral of dv/(F_0 – D(v’)) from v_0 to v.
Can find the force of Usain Bolt by knowing his mass and velocity. Acceleration 9.49 m/s^2. 800+ Newtons. Showed graph of his position, velocity, and acceleration. Talked about what the graphs mean. Shows slope of secant lines for average velocity over each 10m interval. Possible reasons why humans have increased their speed (spikes, starting blocks, better tracks, weather patterns, performance enhancing drugs, better timing). Showed his power output throughout the sprint. Peaks at over 2500+ watts (3.51HP). The theoretical limit for the 100m dash is around 9.48 seconds. Talked about how the fast humans can run over distance. For every doubling in distance, multiply the time by 2.3 – 2.1. Showed NYTimes videos on the 100m records over time, and then showed usain bolts world record.

Water Bottle Rocket

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Video – 1 liter of water in a 2L bottle. Goes up HIGH (100ft, 300ft/s, 50-60gs of force) 90 to 120psi. Second launch 100g’s of force. 1/2 liter of water, 60ft high 250ft/sec 87g’s. Second launch 130g’s. Reaches top velocity right after the water has been expelled (no more acceleration). Tried to put more than 1L of water in, but it was far more difficult to put air in.

Knot theory

Describing the basics of Knot Theory. Most basic knot: unknot. How you can “untangle” a knot using the Reidemeister moves. Different types of knots, to classify knots, unknot, trefoil, etc. Applications of Knot Theory, untangling knotted DNA, Protein Strutures, Statistical Mechanics, etc. Moving on to the Borromean Rings (any two of the rings are not connected, but all three are inseparable). Talked about the Human Knot game – how sometimes it’s unsolvable. Students demonstrated the human knot, in this case it was unsolvable.

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Encryption and Hashing

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The basics of encryption. How passwords work, and how passwords are hashed. Why computers use hashing to store passwords. Went into the basics of why computers use passwords. Talked about the connection to LARGE prime numbers and how to verify if a number is prime or not. Hashing is easy to go from a word to the hash, but VERY difficult to go the reverse route. Salting the hash allows each username to have unique hash possibility.

Tilings

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Talked about the different types of tiling. Domino Tilings, related to fibonacci numbers. Aztec Diamond tiling, harder to tile than the checkerboard pattern. Hexagon tiling, must think about the area taken by the tiling. Penrose dart kites and rhombi. Formed by the golden ratio to make the angles. phi = 0.5 *  5^0.5 + 0.5 ! Golden triangle, angles of 36, 72, and 72, and the Golden gnomon 180, 36, and 36 degrees. Arc and Edge matching rules. Penrose pentacles and kosh curve. Applications: flooring, etc. This student’s work has already been featured on this blog.

4 years ago

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