I often look at cases where things are falling. We typically call this "free fall" motion because the object is moving only under the influence of the gravitational force. With only the gravitational force, the object has a constant acceleration and the motion is fairly simple to model.
However, objects on the surface of the Earth usually have an air resistance force on them also. When can we ignore this extra force and when is it important?
Modeling Air Resistance ———————–
Let's say I drop a ping pong ball. As it falls, I can draw the following force diagram.
The most common model for the air resistance says that the magnitude of the force depends on:
- The density of air (ρ). This typically has a value around 1.2 kg/m3.
- The cross sectional area of the object (A). A ping pong ball would have a cross sectional area equal to π*r2.
- The drag coefficient (C). This depends on the shape of the object. For a spherical object, a unitless value of 0.47 is typical.
- The magnitude of the velocity squared. The faster you go, the greater the air resistance force.
The direction of the air resistance force is in the opposite direction as the velocity of the object. That's why there is a negative sign in the expression along with the r - hat (which is a unit vector in the direction of the velocity).
But how do you find values for the drag coefficients for different objects? The real answer is that you must measure them experimentally. However, Wikipedia has a nice list of some values. What about a falling human? I often have to model the motion of a falling human, but there isn't a C value listed. There is one trick I can use.
The trick involves terminal velocity. Suppose a human jumps out of a stationary hot air balloon. At first, only the gravitational force acts on the human giving an acceleration of -9.8 m/s2. However, as the human increases in speed, the air resistance force also increases. At some point, the air resistance force will be equal in magnitude to the gravitational force and the human will no longer increase in speed. We call this "terminal velocity".
Now for the trick. It seems to be mostly accepted that the terminal velocity for a skydiver is about 120 mph (53.6 m/s). Of course, this is the terminal velocity for the normal skydiving position with head facing down and arms and legs spread out. If I guess at a human mass of 70 kg, I can set the air resistance and gravitational forces equal. Also, for simplicity I am going to call all the constants in front of the velocity squared just K (since they don't change).
I only need the mass and the terminal velocity and I can build a model for air resistance. Yes, this is just a model. If you go super fast, this model probably isn't valid. For now, it's all I have to work with.
How High is Too High? ———————
If I drop an object from some height, there are two things I could do to obtain a value for the falling time. First, I could just ignore air resistance and use the typical kinematic equation:
Solving for the time is fairly straightforward. But what if I add in air resistance? What then? There is a problem. Air resistance is a force that depends on the velocity. This means that the force (and thus the acceleration) is not constant. That's a big problem.
We can still solve this with a numerical calculations. In short, I can use a computer to model just a tiny time interval for a falling object. During this short time interval, the forces are roughly constant. Here is an older post that gives an introduction to numerical calculations. Also, don't forget that my ebook (Just Enough Physics) has a whole chapter on numerical calculations.
Let's just get to the calculation. Here is a model of a ping pong ball falling from a height of 10 meters. Actually, this is a Glowscript program so you can run it yourself and even edit it. Try it! In this calculation, I have a ping pong ball and a ball without air resistance dropped from the same height. In this plot, you can see that the ping pong ball hits after the no-air resistance ball with a time difference of 0.32
But this doesn't answer the question: how high is too high? Of course, there isn't just one answer to this question. The maximum height depends on how accurate you want your model. Here is the real plot that you want. This shows the falling time difference between an object with air resistance and one without for different starting heights. Actually, since larger starting heights will have larger times, I have plotted the fractional difference in times.
From this, it looks like a human drop height of about 160 meters would give a falling time about 10% different than that without air resistance. If you are just getting a rough estimate (like falling off a building), it would probably be fine to ignore air resistance. If you were dropping a ping pong ball instead, I would assume no air resistance for heights around just 4 meters.
But it's not just about the falling time. Sometimes you care about the final velocity instead of the time. Could you just use the same cut-off heights for velocity that you do for time? I don't think so. Let's take a look at a falling human for example. If this human was falling off a building, near the end of the fall the air resistance would be much greater than it was at the beginning of the fall. However, this increase in speed at the very end might not make a huge difference in the falling time.
Here is a plot for the same objects showing the fractional difference in final velocities for with and without air resistance.
If I go with the same idea of getting just a 10 percent velocity error, the falling height for a human would 60 meters instead of 160 meters.
Ok, there are things left to look at - so I will give them as a homework assignment.
- What about the kinetic energy at the end of a fall? How high would a human have to fall so that there was just a 10 percent error in KE?
- Suppose I want to drop a ping pong ball in a lecture class to show that you need to include the air resistance in order to properly model its motion. How high should I drop this ball?
- I want to make some spheres made of wood (let's say wood has a density of 900 kg/m3). Make a plot of drop height for a 10 percent fall time error vs. ball radius. As the ball gets larger, it's mass to cross sectional area ratio changes. Bigger balls should be able to be dropped from a larger height with less error. How small should a wooded ball be such that if dropped from a height of 2 meters, the fall time is off by 10 percent?
That's your homework.
The Nifty Assignments session at the annual SIGCSE meeting is all about gathering and distributing great assignment ideas and their materials. For each assignment, the web pages linked below describe the assignment and provides materials -- handouts, starter code, and so on.
Applying for Nifty is now done as its own track with a similar deadline to special sessions. The format and content of the .zip you submit is unchanged. See the info page for ideas about what makes a nifty assignment and how to apply for the Nifty session.
Please email any suggestions or comments to Nick Parlante @ cs.stanford.edu with "nifty" in the subject. Nick's Home
|Nifty Assignments 2017|
|Falling Sand - Dave Feinberg||CS1 Very engaging falling simulation|
|2048 in Python - Kunal Mishra||CS1 The fantastic 2048 game works great as a CS assignment|
|Fractal Sound - Josh Hug||CS1 Amazing sound generation and visualization|
|SAT Synonyms - Michael Guerzhoy||CS1-CS2 Fun big data application to the familiar SAT word problems|
|NBody Simulation - Kevin Wayne||CS2 Captivating gravity simulation. Gravity .. it's everywhere!|
|Nifty Assignments 2016|
|Mountain Paths -- Baker Franke||CS1 Neat simple algorithm in 2D arrays|
|Restaurant Recommendations Yelp Maps -- Brian Hou, Marvin Zhang, and John DeNero||CS1 Nifty data visualization of restaurant data|
|Rack-O Game -- Arvind Bhusnurmath, Kristen Gee, and Karen Her||CS1 Play and AI code for an easy game|
|Movie Review Sentiment -- Eric Manley and Timothy Urness||CS1/CS2 Neat word analysis from a surprisingly simple algorithm|
|HugLife -- Josh Hug||CS1/CS2 Grid simulation game that shows off testing|
|Autocomplete-me -- Kevin Wayne||CS2 Neat applied use of word storage and binary search|
|Nifty Assignments 2015|
|Counting Squares -- Mark Sherriff, Luther Tychonievich, and Ryan Layer||CS0/CS1 Neat and easy squares activity|
|Speed Reader -- Peter-Michael Osera||CS1 Nifty Animation|
|GeoLocator -- Stuart Reges||CS1 Fun Geo Data|
|Packet Sniffing -- Suzanne Matthews and David Raymond||CS1 Eye Opening Networking|
|Melody Maker -- Allison Obourn and Marty Stepp||CS1 Fun with Sound|
|Seam Carving -- Josh Hug||CS1/CS2 Amazing Image Resize Trick|
|Nifty Assignments 2013|
|Twitter Trends -- John DeNero and Aditi Muralidharan||CS0-CS1 Neat output with a hip big-data source|
|Collage -- Mark Guzdial||CS0 Novel media output by combining images|
|Authorship Detection -- Michelle Craig||CS1 Surprisingly effective data driven categorization with basic coding|
|Recursive TurtleGraphics -- Eric Roberts||CS1 Get at the essential recursive idea very easily|
|Campus Shuttle -- David Malan||CS1 Stunning graphical tour|
|Estimating Avogadro's Number -- Kevin Wayne||CS1/CS2 Surprisingly easy image processing of lab data to get a real-world result|
|Nifty Assignments 2012|
|Stereo Sound Processing -- Daniel Zingaro||CS1 (early) - Fun and impressive early in the quarter - remove vocals from sound|
|Guitar Heroine -- Kevin Wayne||CS1/CS2, Extremely neat -- math model creates realistic guitar sound|
|Uno -- Stephen Davies||CS1, Strategy AI to play Uno.|
|Image Editor -- Joshua T. Guerin and Debby Keen||CS1/CS2 Code to experiment with images, but requiring only the ability to change text files.|
|Igel Ärgern -- Zachary Kurmas||CS2 Hedgehogs in a Hurry game|
|Binary Bomb -- David O'Hallaron||Post CS2 -- neat assignment puzzle to play with understanding of compiled code and memory as they truly are. On the linked page, see the README, Writeup, Release Notes, Self-Study Handout which all work without a password. To play with the code, email Dave and he'll send you what you need to get the binaries.|
|Nifty Assignments 2011|
|Image Puzzles -- Nick Parlante||CS0 or later, great puzzles using images, tiny code required|
|BMP Puzzles -- David Malan||CS1, More and better image puzzles, looking at bytes of BMP file representation|
|Book Recommendations -- Michelle Craig||CS1, Like the Netflix movie-recommendation system, generate book recommendations. Surprisingly simple algorithms give a neat results.|
|Generic Scrolling Game -- Dave Feinberg||CS1, Project pattern which supports a variety of games. Easily allows students to customize rules, graphics etc. of simple game.|
|Wator World -- Mike Scott||CS1-CS2, Shark/fish simulation using GridWorld type abstraction. Neat simulation/modeling example working from simple rules.|
|Hamming Codes -- Stuart Hansen||CS2, Neat exercise with a real algorithm. Push the students to understand that it's really all bytes.|
|Evil Hangman -- Keith Schwarz||CS2 or late CS1 - Awesome variant of Hangman, where the computer cheats by dodging all the user's guesses|
|Nifty Assignments 2010|
|Picobot -- Zachary Dodds||CS0-CS1, day-1 assignment -- neat environment to get students started, works in the browser|
|Pig -- Todd Neller||CS1, intermediate difficulty game to implement, but students love it and lots of variations|
|Song Generator -- Daniel Zingaro||CS1, implement filters with short bits of code, but it all works in the domain of sound, making in a novel and engaging domain for the students|
|CSI: Computer Science Investigation -- David Malan||The instructor accidentally erases the compact flash card containing their images. Students write code to recover the images, solve the treasure hunt using the images|
|Encryption Chase -- Mark Sherriff||CS2, encryption coding, embedded in a team active-learning campus treasure hunt|
|Chatting Aimlessly (IM) -- Thomas Murtagh||CS1, implement simple instant messaging client in CS1 -- talk about a technology near to the student heart!|
|Nifty Assignments 2009|
|Star Map -- Karen Reid||CS1, neat drawing of the night sky and constellations -- simple file reading and drawing|
|Face Pamphlet -- Mehran Sahami||CS1, simple Facebook application built with just CS1 technology, students love it|
|Secrets In Images -- Brent Heeringa, Thomas Murtagh||CS1, hide secret messages inside images -- neat image manipulation with data as simple arrays|
|Random Art -- Christopher A Stone||CS1, build nifty images with recursive nested random symbolic math expressions (python)|
|Enigma Encryption -- Dave Reed||CS1-CS21, range of easy to complex cryptography projects, using paper/manipulation model to get started|
|DNA Splicing -- Owen Astrachan||CS2, surprisingly easy DNA manipulation, set up for the students to measure/experiment with their code|
|Nifty Assignments 2008|
|Catch Plagiarists -- Baker Franke||CS1-CS2, typical CS2 data structures, difficulty can be adjusted. Search within a set of documents to find pairs with copied content|
|Genetic Algorithm TSP -- Raja Sooriamurthi||CS1-CS2, basic genetic algorithms. Use genetic algorithms to solve the traveling salesman problem|
|Asteroids -- Dan Leyzberg, Art Simon||CS1-CS2, objects, inheritance, abstract classes. An impressive implementation of Asteroids with OOP design and inheritance|
|Huffman Images -- Morgan McGuire, Tom Murtagh||CS1(late) or CS2(early). Labs to explore huffman compression in the context of image bitmap manipulation|
|Maze Solver -- Don Blaheta||CS2, stacks, queues, 2d arrays. Play around with algorithms to solve a maze. Works with gridworld|
|Dice Flip -- Cay Horstmann||CS1-CS2, prolog for advanced CS2, java for late CS1 variant. Explore simple but subtle dice game|
|Nifty Assignments 2007|
|Media Manipulation -- John Cigas||CS0-CS1, spreadsheet use or basic code. Transfer media data to spreadsheet form to make manipulation easy|
|Mindreader -- Raja Sooriamurthi||CS1, CS2. basic logic, map interface (arrays or Hashmaps). Build a surprisingly good computer opponent for a guessing game|
|Solitaire OOP -- Robert Noonan||CS2, OOP and patterns to explore family of solitaire games|
|Sliding Blocks Puzzle -- Mike Clancy||CS2, significant recursion and data structures. Recursive and heuristic work to solve the sliding blocks puzzle.|
|Fire -- Angela Shiflet||CS2, 2-d arrays, simulation. Neat, real-world example simulating spread of fire across a terrain, depending on humidity etc.|
|Nifty Assignments 2006|
|Book Code (ISBN) -- John Motil||CS1, basic logic to play with ISBN numbers. Fun because we are surrounded by these numbers .. use them for basic examples|
|Natural Prestidigitation -- Steve Wolfman||CS1, basic logic, loops, arrays. Appears dull, but has a neat surprise ending.|
|Breakout -- Eric Roberts||CS1, basic logic, loops using ACM graphics early in the term|
|Dancing Turtles -- Chris Nevison||CS1, inheritance with dancing turtles and ACM graphics|
|Solitaire Encryption -- Lester I. McCann||CS2, list manipulations, algorithmic code, file reading. Implements a very novel type of encryption.|
|Anagram Solver -- Stuart Reges||CS2, recursive backtracking ... a very fun application of recursive search|
|RSS Reader -- Jerry Cain||CS2, data structures, networking ... neat to implement a client for a real protocol|
|Nifty Assignments 2005|
|Test Me -- David Levine||CS0-CS1, students write tests to examine black box code -- nifty and no code writing|
|Grid Plotter -- Alyce Brady and Pam Cutter||CS1, neat way to learn and practice loop code|
|Complementary Currency -- Paul Kube||CS1, OOP by creating a currency ... has a community/social aspect among the students|
|Name Surfer -- Nick Parlante||CS1, loops, arrays, files. Graph baby name data for the last 100 years. Nifty because the data is nifty.|
|Photomosaics PPT (PDF version) -- Rich Pattis||CS2, create image made of many little images .. but there is a patent on it|
|Image Lab -- Aaron Gordon||CS2, framework to allow students to write filters on 2-d data and see them applied to images|
- Pong -- Grant Braught -- a neat "objects first" assignment
- Lunar Lander -- Stuart Reges -- another fun "objects first" assignment
- HTML Browser -- Scott Dexter and Chaya Gurwitz -- CS2 assignment to render HTML
- Backtracking -- Stephen Weiss -- CS2 all about backtracking
- Random Writer -- Joe Zachary -- a neat CS2 data structure problem
- Blurbs from the proceedings
- Shall We Play A Game? -- Dan Garcia -- A system where students can play around with game playing AI (CS0)
- Adventure -- John Estell -- Using the classic adventure game as a largish project (CS2)
- Sorting Detective -- David Levine -- A fun variation on the old "sorting algorithms" homework (CS2)
- Boggle -- Julie Zelenski (in cahoots with Owen Astrachan) -- Using the Boggle game to explore recursive algorithms and data structure tradeoffs (CS2)
- Blurbs from the proceedings
- Windows and Regions -- Mike Clancy -- an algorithmic problem using 2-d regions. An excuse to do some linked-list (or ArrayList) type manipulation. Give the students a feel for "window" manager region operations.
- Personality Test -- Stuart Reges -- sort and match the personality data of the class (more fun than it sounds!)
- Quilt -- Julie Zelenski -- a fun, drawing-intensive CS1 project that emphasizes decomp
- Word Ladder -- Owen Astrachan -- a string manipulation puzzle
- Tetris -- Nick Parlante -- a large OOP project, with a tetris board, tetris piece, tetris game, and a pluggable tetris brain . Can be used as a small project where students just write a Tetris brain and plug it in, or can be used as a large CS2 OOP project. The nifty materials include a runnable JTetris.jar sample, and an Instructor's Guide
- Blurbs from the proceedings
- Cat And Mouse -- Mike Clancy -- (CS1) a cute problem which requires non-trivial geometry and algorithms, but can be solved in 100 lines.
- Bagels -- Stuart Reges -- (CS1) a fun game with some algorithmic complexity.
- DNA -- Richard E. Pattis -- (CS1) great first data structures and performance tuning problem.
- Huffman Coding -- Owen Astrachan -- (CS2) decomposition and data structures.
- The Random Sentence Generator -- Julie Zelenski -- (CS2) a fun use of grammars, recursion, and ADTs.
- Darwin's World -- Nick Parlante -- (CS2) a simulator featuring decomposition and a simple interpreter.