Motion+in+2D

=Chapter 3: Motion in 2 Dimensions ([|Syllabus]) Vectors & Projectile Motion=

-general vector info -manipulating vectors __Projectile Motion__- It's shooting things... using numbers -general projectile info. -demos and equations**
 * __Vectors__- A new more annoying form of math
 * -practice problems**

**5 Quantities of Vectors (things that vectors can show):** - Displacement - Acceleration - Velocity - Force - Momentum
Bold Face in Print Arrows(length and direction) Arrow over simple
 * Representing Vectors:**

- Arrows show direction and length of a vector. - **Equal** **Vector:**Two arrows next to each, have the same length, going in the same direction - **Not Equal:** arrows pointed in different directions, and are not the same length - **Not Equal:** arrows the same length but pointed in different direction
 * Equal/Not Equal Vector**

__ Manipulating Vectors __
 - Graphing Methods of vectors use //rulers and protractors// and make up shapes like parallelograms and polygons. - Mathematical methods for vectors include sketching and calculators.

THE BASICS OF ADDING VECTORS media type="custom" key="268057" width="278" height="235"
 * __Adding Vectors__:**

__Polygon Method__: Equation= **R** = **A** + **B** 1.) Take vector **A** and redraw it to scale. 2.) Take vector **B** and place it's tail to vector **A**'s head in its original position. 3.) Create a dashed line from the tail of vector **A** to the head of vector **B**. 4.) Place a S on the dashed line to indicate the measured line and the distance of this line represents **R**. //Make sure added vectors are always connected from tail to head and that **R** runs from the tail of the first to the head of the last.//
 * Graphical Methods (ruler / protractor):**

__Parellelogram Method__: 1.)Draw the two vectors from the same point, with their tails at the same point in space. 2.)Draw lines parallel to each vector through the head of the other vector.This should create a parallelogram. 3.)Draw a new vector from the tails of the other two vectors to the intersection of the lines (creating one of the diagonals of the parallelogram) 4.)Measure, using a ruler and protractor, the length and angle of the newly drawn vector.
 * R** = **A** + **B**

__Triangle Method__: 1.) Sketch the vector addtion using the polygon method. 2.) Solve for magnitude. Right Triangle - Pythagorean ( a2 + b2 = c2) Non Right Triangle - Law of Cosines c2 = a2 + b2 – 2ab(cos(c)) 3.) Solve for direction. Right Triangle - SOHCAHTOA (sin=opposite/hypotenuse,cosine=adjacent/hypotenuse,tangent=opposite/adjacent) Non Right Triangle - Law of Sines: the ratio of a side to the sine of the opposite angle will be a constant value within a triangle, or in useful terms, (a/SinA=b/sinB=c/sinC)
 * Mathematical Methods (sketch / calculator):**
 * Pick 2 parts to solve by cross multiplying

__Component Method__: Components are two vectors which are 90º apart, which, when summed, forms the original vector. Most often, these are x and y, but are often used to determine a quantity in another direction. No matter what, we will label them x and y for ease of explanation. 1.) Break all vectors down into their X and Y components **Especially here, make sure that θ is from the positive x axis.** a.) **A**x = │**A**│cosθ b.) **A**y = │**A**│sinθ 2.) Add all the X components to get **R**x 3.) Add all the Y components to get **R**y 4.) Add **R**x + **R**y to get **R** using the triangle method. (Pythagorean for │**R**│, inverse tangent for θ.)

Notes: Only include direction when given in the problem, otherwise, it's not a vector problem. Remember to always use significant figures!

The opposite vector is simply a vector with equal magnitude but opposite direction, but remember that the opposite direction is rarely the same as -θ, so, when working with degrees, it is actually =(rA,θA) + (rB,θB+180º) by the way, the A's and B's in those equations are supposed to subscripts, but i guess wiki doesn't like them.
 * Subtracting Vectors**
 * A** - **B** = **A** + (-**B**)
 * When subtracting vectors, you're basically just taking the opposite of B. To do this, switch the direction of the arrow and keep the same length
 * Mathematically, this is **A** - **B** = (rA,θA) + -(rB,θB)

Equation: **B***(x), x = Desired multiplier
 * Multiplying Vectors:**
 * **Note that this is only by a scalar quantity. If it were multiplied by another vector, it wouldn't be so simple.**
 * Take the original length of vector and multiply by x, while keeping the same direction.
 * Direction does not get multiplied.
 * (r,θ)*x = (r*x,θ)

media type="custom" key="3133782"
 * Adding Vectors using a Calculator:**
 * - This video demonstrates how to add vectors using a calculator by using the SciTools application on graphing calculators**

__ General Projectile Info. __
 __**Projectile Motion**__: Vectors that are 90º apart from each other are independent, meaning vertical movement and horizontal movement can be solved for using separate equations for each. For now, this includes only initial horizontal velocity and gravity. Once any angle other than zero is used, the initial velocity is slightly more difficult (and more will need to be added here). Because they are independent, the equations used for each component can be used as if the other component is zero. Because one component is ignored, it can be treated as one dimension and all equations used will come from last chapter. I'll let someone with a greater motivation and/or need for extra credit list them here again.

__ Demos and Equations __
 If a bullet is fired from a gun horizontally and the second bullet is dropped from the same height, both will hit the ground at the same time.

The ball and the cart have the same velocity and the horizontal motion is constant velocity and the vertical motion is constant acceleration. The ball will drop back into the cart.
 * __Ballistics Chart Demo__**

Range is also called horizontal distance!


 * __Equations__**

__**Time of Flight Demo**__ Remember: The vectors are 90 degrees apart are independent and all things fall at the same acceleration due to gravity. Range Changes, but the time stays the same.

Projected Horizontally o--->(zero degrees) Vo = Vx (the initial velocity is moving in the x direction) Voy = 0 (initial vertical velocity is zero)

Projected at an Angle Vx = Vo(cosø) Voy = Vo(sinø)

Vx (stays the same becasue of horizontal constant velocity) hypotenuse is Vo.

Practice Problems
 __**Practice Problem**__ Arrow is shot at a 30 degree angle horizontally and the velocity is 49 m/s. Horizontally Vx = 42.4 m/s(49*cos30) x = ? t = ?

Vertically g = -9.8 m/s^2 Voy = 24.5 m/s(49*sin30) Vy = 0 (1/2way) y = ?

Vy^2 = Voy^2 + 2gy 0 = (24.5 m/s)^2 + 2(-9.8m/s)(y) 0 = 600.25 - 19.6(y) -600.25 = -19.6y y = 30.6

Time Vy = Voy + gt 0 = 24.5 - 9.8t -24.5 = -9.8t t = 2.5*2 5 seconds

Range Vx = change in x/time 42.4 = change in x/5 seconds x =212 range/horizontal distance

If you aim the gun at the monkey, you will shoot it because all things fall at the same acceleration due to gravity.
 * __Shoot the Monkey Demo__**



A rowboat crosses a river with a velocity of 3.30mi/h at an angle 62.5 degrees north of west relative to the water. The river is .505 mi wide and carries an eastward current of 1.25mi/h. How far upstream is the boat when it reaches the opposite shore?
 * Boat Crossing a River Problem:**

Time Required for the boat to cross the width of the river is = t:

t =(.505mi) / Vbsy= (.505mi) / (2.93mi/h) = .172h Distance traveled by boat up steam = s = Vbsx * t s =-.274mi/h * .172h= .047mi = .047 * 1609m = 75.8m (One mile = 1609m) 