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This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Calculus Illustrated -- Projects

## 1 Instructions

It's no secret that a vast majority of students will never use what they have learned. Poor career choices aside, a former calculus student might have trouble recognizing mathematics that surrounds him...

The student will be given a short, one or two sentences, description of a problem that uses no calculus terminology.

• Stage 1: The student's first task is to ask the right questions in order to recast the problem in the language of the appropriate mathematics. It should also include a critique of old submissions on the topic if any.
• Stage 2: Then he is to explore the problem numerically and graphically with a spreadsheet (or other software) and make an approximation of what the answer might be and ask further questions.
• Stage 3: Finally, the problem is solved analytically, if possible, with the calculus tools and the answer is confirmed.

These are modelling problems and they can't be solved by manipulating formulas.

All the steps, the explanations of the methods, the data, the illustrations, and the computations are presented in writing. The methods should be based entirely on the material of the course (and common knowledge). The intended solutions aren't the kind that you can find on the internet. The project should contain your solution and not a discussion of a solution of somebody else.

The actual project statements are written as if by a person with no knowledge of calculus. That is why the problems may be poorly formulated, incomplete, and have many possible answers or none at all. As a result, the structure of the project might be much more complex than the three stages above.

Some projects have links to the actual students projects on the topic. If you pick one of those, you will have two choices: to use or not to use the previous work. In either case, why you accept or reject it is to be addressed in your essay. In the former case, you build your project on top of the old and in the latter you build from scratch. In either case, you should include a critique.

Project statements are below in the order of increasing complexity. All submissions are linked even the failed ones.

## 2 Bouncing ball

Project statement: (a) How do I throw a ball down a staircase so that it bounces off each step? -- (fall 2016)

Project statement: (b) Can I run from one table to another without touching the floor by stepping on top of a bouncing ball?

## 3 Free fall

Project statement: How should you throw a ball from the top of a $100$ story building so that it hits the ground at $100$ feet per second? -- (spring 2017, fall 2016) CLOSED

## 4 Ballistics

Project statement: (a) I would like to use a cannon with a muzzle velocity of $100$ feet per second to bombard the inside of a fortification $300$ feet away with walls $20$ feet high. What if there is side wind? -- (spring 2017, fall 2016, fall 17 Excel)

Project statement: (b) How do I shoot a toy cannon from a table to hit a spot on the floor $10$ feet away from the table? What if the projectile goes through a sheet of paper half-way and loses half of its velocity? -- (spring 2017, fall 2016)

Project statement: (c) How fast do I have to move my hand while spinning a sling in order to throw the rock $100$ feet away? -- (spring 2017, fall 2016)

## 5 Complex motion

Project statement: (a) How hard do I have to push a toy truck from the floor up a $30$ degree incline to make it reach the top of the table at zero speed? -- (spring 2017, fall 2016, (spring 2018) CLOSED

Project statement: (b) Can a sailing boat go in the direction opposite to the direction of the wind? (fall 2017, excel)

Project statement: (c) How fast does the shadow of a falling ball on a sliding ladder move? -- (spring 2017, fall 2017 excel)

Project statement: (d) Where so you end up if you keep moving north-east?

Project statement: How much does moving in or out of the passing lane affect the reading of a radar gun? -- (spring 2017, fall 2016)

## 7 Compounded forces

Project statement: (a) What would the gravitation be like if the Earth was shaped as: a cannonball, two cannonballs, a plane, a coin, a cube, a sphere?

Project statement: (b) How would an object move if it is attached to two-three-four springs? What if it is attached to a hole in the middle of a rubber sheet?

## 8 Reflections

Project statement: (a) Several building with curved walls (in LA, Vegas, London) have been accused of causing sunburns and other damage by focusing the sunlight; is it possible? (fall 2017)

Project statement: (b) Archimedes was told to have set Roman ships on fire by re-directing the sun light with reflective shields; is it possible? (fall 2017, excel)

Project statement: (c) Is possible that a flying object such as the Stealth bomber can be invisible to the radar? (fall 2017)

## 9 Optimization

Project statement: (a) Where should I stand in order to be best illuminated by two light bulbs on the ceiling if the amount of light is inversely proportional to the square of the distance? -- (spring 2017) (fall 2017) CLOSED

Project statement: (b) What is the shape of a slide made of straight parts that allows the fastest time down? (spring 2017, fall 2017)

Project statement: (c) If the speed of my forward progress is proportional the steepness of the terrain, and the steepness changes one-two-three times, how should I plan to get from point A to point B? (fall 2017)

Project statement: (d) Does it really make sense to “aim for the center of mass”? (spring 2018)

Project statement: (e) Where should I put an object in order for it to be best illuminated by two, three, four,... light sources of different power if the amount of light is inversely proportional to the square of the distance?

Project statement: (f) With $100$ yards of fencing material, what is the largest enclosure adjacent to a river I can build? What is the river bends? (spring 2018)

Project statement: (g) Design a sphere which with a refractive index depending on the distance to the center so that all entering light would spiral in to the center, never escaping.

## 10 Dynamical aiming

Project statement: (a) How would two tanks with broken turrets battle if they try to shoot at each other at all times? (spring 2018)

Project statement: (b) How would two tanks -- one with a broken turret and the other with broken tracks -- battle if they try to shoot at each other at all times? (spring 2017)

## 11 Penetrating through a medium

Project statement: (a) What shape of sword is best for cutting? (prototype project)

Project statement: (b) How does one evaluate shapes of swords for cutting? (Fall 2017)

Project statement: (c) How shallower is the depth of penetration of a hollow point bullet in comparison to a regular bullet?

## 12 Predator and prey

Project statement: (a) How would predator and prey populations interact when their dynamics is affected by seasons? (spring 2017)

Project statement: (b) How would predator and prey populations interact when their dynamics is affected by random fluctuation?

Project statement: (c) How would predator and prey populations interact if they are seen as (indivisible) individuals?

## 13 Pursuit

Project statement: (a) How would a hound's pursuit of a rabbit go -- one step at a time -- if it attempted to aim with a lead? (spring 2017)

Project statement: (b) How does a heat-seeking missile pursue an aircraft if it periodically evaluates the difference of the temperature on its wings? (spring 2017)

Project statement: (c) You are standing in the center of an equilateral triangle with a side of $20$ meters. At two corner stand two velociraptors with a speed of $25$ meter per second and at the third a wounded one with a speed of just $10$ meters per second, while your speed is only $6$ meters per second. What do you do? (spring 2018)

## 14 Moving around the solar system

Project statement: (a) What is the trajectory of the Moon around the Sun? (spring 2017)

Project statement: (b) How do I bring a craft from the Earth's orbit to the Moon's?

Project statement: (c) Is it possible to place a satellite (with an appropriate initial velocity) on the orbit in such a way that it will be moving in unison with the Moon?

## 15 Spreading through a medium

Project statement: (a) How does a communicable disease spread -- one person at a time -- through a population: some of the susceptible individuals are infected and then removed from the susceptible population? What if the rates vary with seasons?

Project statement: (b) How does communicable disease spread -- one block at a time -- through a city?

Project statement: (c) How does fire spread -- one patch at a time -- in the prairie?

Project statement: (d) How does heat propagate -- one little piece at a time -- through an object? What if the object also expands when heated?

Project statement: (e) What will happen to the prices of wheat if every farmer adjusts his price according to the average price of his neighbors?

## 16 Complex motion under gravity

Project statement: (a) If the Earth falls apart, what paths will the pieces follow?

Project statement: (b) How would an object dropped down a mine shaft that goes all the way through the Earth behave in the presence of air-resistance?

Project statement: (c) What would Moon's orbit be like if the Earth was shaped as a plane, a coin, a cube?

Project statement: (d) How would a plant grow if the pot is put at the edge of a turn-table?

## 17 Rocket motion

Project statement: What does the trajectory of a satellite become when its rocket engine is turned on? (fall 2017)

## 18 Composite objects

Project statement: What shape a rubber band would take if we attach weights to its locations?

## 19 Image analysis

Project statement: How can you determine if a digital image contains one or two objects?