Student+View+-+Week+3

=Week 3 - Two-dimensional motion=

**1) Introduction**
** Studying one-dimensional projectile motion has been both interesting and informative, but our problem domain has been limited to falling objects - what goes up, must come down. In reality though, most projectiles that we encounter are not constrained to one dimension, but rather move through two and three dimensions.

** Thrown balls, shot bullets, jumping athletes, motorcycle jumpers, and even Super Mario all exhibit similar trajectories in higher dimensions despite being very differently shaped objects. This path in two dimensions resembles the shape of a parabola.

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media type="custom" key="5695253" ** In this session, we will combine our knowledge of vectors with our experience with one-dimensional equations of motion to better ** handle more complex, real-world scenarios where our projectiles not only move up and down due to gravity, but also left and right due to some initial force vector.  **//Thought Questions 3.x//** //- For an object to move in the horizontal as well as the vertical// //planes// ** //, what must we change about the initial launch velocity vector from the previous session?//

We will ** learn why two-dimensional projectiles follow parabolic paths and how initial conditions change its shape. We will also be able to c alculate total air time, flight range, maximum height, and optimal launch angles for our projectiles. To make problem solving easier, we will take advantage of the vector property that allows us to decompose any vector into independent perpendicular components. This allows us to directly extend what we have learned in one-dimensional cases and apply it two-dimensional ones.


 * //Thought Questions 3.x -//** //Does vector decomposition allow us to extend our tools to three, four, or even higher dimensions?//


 * To help develop our two-dimensional projectile flight intuition, we will begin this session by playing a computer game called Scorched Earth.

[ select a version that we want to use ] ** http://www.scorch2000.com http://www.newgrounds.com/portal/view/255611 ** [ select a version that we want to use ]

** Scorched Earth is a simple, but endlessly entertaining strategic tank combat game where players take turns calculating the launch angles and velocities necessary to shoot an explosive projectile over to their competitors. Each turn, players must take into account the range and altitude of the other tanks and factor in wind and environmental conditions as well. Since players alternate turns, the one that can make two-dimensional projectile calculations quicker usually wins the match.


 * Playing the game against the computer is definitely fun, but to play against classmates is even better! In the spirit of fun competition, we are going to have a quick tournament to decide the class champion. Play your classmates according to the bracket and we'll post the results when we have a champion. There can be o ** nly one...

Please find our "Scorched Earth" bracket below:


 * //Review Question 3.x -//** //How do you decide on your attack vector? How many turns did it take you to hit the other tank? Did you overshoot or undershoot your target more often?//

**2) Review**
Last session we each created unique one-dimensional free fall and return to sender projectile examples for inclusion in our chart [link to chart]. A number of interesting sites from which to drop things where chosen. Based on that data, we were able to calculate the air time and impact velocities of our items. Examples included everything from _ and _ __to__ _ __and__ ___.


 * //Thought Question 3.x//** //- Are the impact velocity and launch velocity the same? Why?//

We also had to convert the units of some of our data from feet/s² to m/s² to conform with certain international scientific standards.
 * //Review Question 3.x//** //- What units do you normally think in? Does it matter what units we use in our projectile calculations? Why?//


 * //Thought Question 3.x//** //- What does the fact that we can decompose a vector into two independent one-dimensional components mean to how we approach two-dimensional problems?//

Please remember to review and vote on the best student questions from the previous session for inclusion in our year end course guide. The questions are maintained by UserVoice. See sample below.



Vote, rank, and discuss them each week!

**3) New Content and Assignment**
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scribd paper
 * Reading**

Independent motion of falling objects... media type="custom" key="5692953" @http://www.youtube.com/watch?v=z24_ihikEqQ
 * //Thought Question 3.x -// //How do the forces on the projectile change through its flight? Is this the case in reality?// **

Different launch angles media type="youtube" key="N0H-rv9XFHk" height="385" width="640" **Thought Question 3.x - **How is the range of the projectile affected by the launch angle? Why do you think this is?

Parabolic projectile motion @http://www.physicsclassroom.com/mmedia/vectors/bds.cfm **//Thought Questions 3.x -// //If there were a secondary gravitational force in the horizontal plane, would projectiles still follow parabolic paths?//

Interactive Projectiles** @http://www.physicsclassroom.com/shwave/projectile.cfm **//Thought Questions 3.x//** //- How does the initial launch altitude affect projectile motion? Does that make intuitive sense?// Aiming at falling target example **//Thought Questions 3.x//** //- Why do we lead a player when passing a ball in team sports, but we don't lead falling targets?// Now that we are studying two-dimensional motion, we can go ahead and start varying the launch angle in our simulation.
 * http://www.ngsir.netfirms.com/englishhtm/ThrowABall.htm **
 * Interactive Examples**

**//Thought Question 3.x//** //- Given the same velocity, are there multiple launch angles that allow projectile to reach the same horizontal range?//
 * //Thought Questions 3.x -//** //What does a two-dimensional analysis of the 90º case that we looked at in the one-dimensional case look like? Why did we choose that angle? What other angles have similar properties?//

//**
 * //Teacher Crafted Activity 3.x : Inserted Dynamically by Teacher Assignment Administration Page

**4) Assignment Submission**
Similar to last session, we are going to use the Internet to select interesting locations to build our assignment around. We are going to imagine taking on the postal service with our own company, "Projectile Messaging." Each student needs to select at least three locations in their neighborhood and compute the launch velocities and angles necessary to propel a 5kg projectile message canister from one location to the next and finally return it back to the starting point. Who needs this interweb thing?


 * //Thought Questions 3.x//** //- Why do you think we added the neighborhood restriction on valid locations? What would happen if we picked New York City and Moscow and Sidney?//

In addition to picking appropriate initial velocities and angles, we will need to compute the hangtime, maximum height, and impact velocities for each trajectory.

Here are some examples of distance calculators, but feel free to find and use your own: @http://www.distancefromto.net @http://www.daftlogic.com/projects-google-maps-distance-calculator.htm

Using your knowledge of two-dimensional projectile motion to support your answers, fill in the worksheet with the requested information. All of your launch locations //must// be unique. Early contributors are at an advantage, so get your selections in early! Ask questions on the discussion board.

**//Thought Questions 3.x -//** //What configuration should you use to get it to the target in the shortest amount of time? This is a business after all...//
 * //Review Questions 3.x -//** //What configuration would you use to reach the highest height? Why might this be important?//
 * //Thought Questions 3.x -//** //What configuration would you use to have the projectile in the air the longest?//

If you know three points on a trajectory (path of a projectile), you can find the exact quadratic equation using a system of equations. That is, three points determine a particular parabola that can be expressed as a quadratic equation of the form: math ax^2+bx+c math
 * Finding the Quadratic Equation**

Suppose you have the following three points: (0, 0), (4, 6), and (10, 3) Find a, b, and c in the standard form of the quadratic equation from above by solving this system: math 0=ax(0)^2+b(0)+c 6=ax(4)^2+b(4)+c 3=ax(10)^2+b(10)+c math

Use substitution or linear combination to find a, b, and c. (Hint: solve the first equation first.)

From equation 1, c = 0 plugging in 0 for c in equations 2 and 3, we get 6=a(4)2+b(4), which simplifies to 6=16a+4b 3=a(10)2+b(10), which simplifies to 3=100a+10b

After some simplifying … −5(16a+4b=6) → 80a+20b=30 −2(100a+10b=3) → 200a−20b=−6 −120a=24 a=−0.2

16(−.2)+4b=6 −3.2+4b=6 4b=9.2 b=2.3

We have a=-.2, b=2.3, and c=0, which can be rewritten in standard quadratic form as: math y=-0.2x^2+2.3x math

This equation can be used to find the vertical height (Y) given a horizontal distance (X). Use data from the projectile motion simulator above to calculate the quadratic equation and verify two points you observed in the simulation. If you need a refresher on parabolas, review the section below

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 * Take a quiz to see what you know**

**5) Personal Learning Log Contributions**

 * As we do at the end of each session, we will contribute a summary of what we have learned in our own words, images, and links to our personal course review log. The purpose of which is to evolve a review book for the entire course.

Additionally, each of you will submit three unique questions and answers based on the content learned this session, Make them fun and challenging for your classmates. We will vote on the best ones, so be creative! **