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 * Chapter 3 **

**Vectors - Fundamentals and Operations**

=Lesson 1a: Vectors and Direction =

A vector quantity is a quantity that is fully described by both magnitude and direction. Vector quantities are not fully described unless both magnitude and direction are listed. Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. The vector diagram depicts a displacement vector:



- a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a //head// and a //tail//. - the magnitude and direction of the vector is clearly labeled.

The direction of a vector is often expressed as an angle of rotation of the vector about its "tail" from east, west, north, or south.

=Lesson 1b: Vector Addition =

Two vectors can be added together to determine the result (or resultant). There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors: - the Pythagorean theorem and trigonometric methods - the head-to-tail method using a scaled vector diagram

<span style="color: #fa0000; font-family: 'Comic Sans MS',cursive; font-size: 90%;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The direction of a //resultant// vector can often be determined by use of trigonometric functions. <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">SOH CAH TOA:

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The **sine function** relates the measure of an acute angle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;"> The **cosine function** relates the measure of an acute angle to the ratio of the length of the side adjacent the angle to the length of the hypotenuse. <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The **tangent function** relates the measure of an angle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Once the measure of the angle is determined, the direction of the vector can be found; however, the measure of an angle as determined through use of SOH CAH TOA is __not__ always the direction of the vector.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Head-to-Tail Method:


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Choose a scale and indicate it on a sheet of paper
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Pick a starting location and draw the first vector //to scale// in the indicated direction. Label the magnitude and direction of the scale on the diagram
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Starting from where the head of the first vector ends, draw the second vector //to scale// in the indicated direction. Label the magnitude and direction of this vector on the diagram.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Repeat steps 2 and 3 for all vectors that are to be added
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as **resultant (r)**
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Measure the direction of the resultant using the counterclockwise convention

=<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Lesson 1c: Resultants =

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The **resultant** is the vector sum of two or more vectors. If displacement vectors A, B, and C are added together, the result will be vector R. <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">When displacement vectors are added, the result is a //resultant displacement//. If two or more force vectors are added, then the result is a //resultant force//. In all such cases, the resultant vector (whether a displacement vector, force vector, velocity vector, etc.) is the result of adding the individual vectors.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">In summary, the resultant is the vector sum of all the individual vectors. The resultant is the result of combining the individual vectors together. The resultant can be determined by adding the individual forces together using vector addition methods.

=<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Lesson 1d: Vector Components =

A vector that is directed upward and rightward can be thought of as having two parts - an upward part and a rightward part.



Each part of a two-dimensional vector is known as a **component**. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single two-dimensional vector. The single two-dimensional vector could be replaced by the two components.

Any vector directed in two dimensions can be thought of as having two different components. The component of a single vector describes the influence of that vector in a given direction.

=<span style="font-family: 'Comic Sans MS',cursive;">Lesson 1e: Vector Resolution =

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The process of determining the magnitude of a vector is known as vector resolution. The two methods of vector resolution that we will examine are: <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">- the parallelogram method - the trigonometric method

<span style="color: #fa0000; font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Parallelogram Method of Vector Resolution**


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Select a scale and accurately draw the vector to scale in the indicated direction.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Sketch a parallelogram around the vector: beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a rectangle
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Draw the components of the vector. The components are the //sides// of the parallelogram. The tail of the components start at the tail of the vector and stretches along the axes to the nearest corner of the parallelogram
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Label the components of the vectors with symbols to indicate which component represents which side
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Measure the length of the sides of the parallelogram in //real// units. Label the magnitude on the diagram.

<span style="color: #fa0000; font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Trigonometric Method of Vector Resolution**


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Construct a //rough// sketch (no scale needed) of the vector in the indicated direction
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Draw a rectangle about the vector such that the vector is the diagonal of the rectangle
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Draw the components of the vector
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Label the components of the vectors with symbols to indicate which component represents which side.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">To determine the length of the side opposite the indicated angle, use the sine function. Substitute the magnitude of the vector for the length of the hypotenuse.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

=<span style="font-family: 'Comic Sans MS',cursive;">Lesson 1g: Relative Velocity and Riverboat Problems =

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">A motorboat in a river is moving amidst a river current - water that is moving with respect to an observer on dry land. The magnitude of the velocity of the moving object with respect to the observer on land will not be the same as the speedometer reading of the vehicle. Motion is relative to the observer. The observed speed of the boat must always be described relative to who the observer is. A tailwind is a wind that approaches the plane from behind, thus increasing its resulting velocity. The resultant velocity of the plane is the vector sum of the velocity of the plane and the velocity of the wind. A headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Side Wind: To determine the resultant velocity, the plane velocity must be added to the wind velocity. The Pythagorean theorem can be used. The direction of the resulting velocity can be determined using a trigonometric function.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Analysis of a Riverboat's Motion**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat. The river current influences the motion of the boat and carries it downstream. Motorboat problems such as these are typically accompanied by three separate questions:


 * 1) <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">What is the resultant velocity of the boat?
 * 2) <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">If the Width of the river is x meters wide, then how much time does it take the boat to travel shore to shore?
 * 3) <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">What distance downstream does the boat reach the opposite shore?

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The second and third of these questions can be answered using the average speed equation.

=<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Lesson 1h: Independence of Perpendicular Components of Motion =

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Any vector - whether it is a force vector, displacement vector, velocity vector, etc. - directed at an angle can be thought of as being composed of two perpendicular components. These two components can be represented as legs of a right triangle formed by projecting the vector onto the x- and y-axis. <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">The two perpendicular parts or components of a vector are independent of each other. A change in the horizontal component does not affect the vertical component. A change in one component does not affect the other component. Changing a component will affect the motion in that specific direction. While the change in one of the components will alter the magnitude of the resulting force, it does not alter the magnitude of the other component.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Any component of motion occurring strictly in the horizontal direction will have no affect upon the motion in the vertical direction. Any alteration in one set of these components will have no affect on the other set.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">**Projectile Motion**

=<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Lesson 2a: What is a Projectile? =

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Central Ideas:** A projectile is an object that has only the force of gravity acting upon it, and it continues its motion by inertia. No forces other than gravity are required because forces only affect acceleration, not motion.

>
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">What is a projectile? A projectile is any object that once //projected// or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">What are the different types of projectiles? An object dropped from rest, thrown vertically upward, and upward at an angle to the horizontal can all be considered projectiles.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">What are the forces acting on projectiles? By definition, a projectile has a single force that acts upon it - the force of gravity. If there were any other force acting upon an object, then that object would not be a projectile.
 * **<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">How are projectiles represented? **<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Projectiles are represented by the free body diagram, showing that gravity is the only force acting upon them. [[image:u3l2a3.gif width="125" height="129"]]
 * **<span style="font-family: 'Comic Sans MS',cursive; font-size: 12px;">What do Newton's laws have to do with projectiles? **<span style="font-family: 'Comic Sans MS',cursive; font-size: 12px;">Because it is difficult for people to comprehend that gravity, pulling downward, is the only force acting upon projectiles, Newton's laws clarify that forces are only required to cause an acceleration (not a motion).

=<span style="font-family: 'Comic Sans MS',cursive;">Lesson 2b: Characteristic's of a Projectile's Trajectory =

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Central Ideas:** Projectiles travel with a parabolic trajectory due to the downward force of gravity; however, gravity only affects the vertical velocity but leaves the horizontal velocity traveling at a constant rate.


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**When is a projectile's velocity constant?** A projectile's velocity is constant in horizontal motion because the vertical force is gravity, and the vertical and horizontal forces are perpendicular of one another so the horizontal motion is not impacted.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**What are the components of a projectile's motion?** Horizontal and vertical velocity are the two components of a projectile's motion, and they are perpendicular of one another.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Is acceleration always present in projectiles?** No, it is not present in horizontal motion because it is not affected by gravity and therefore moving at a constant rate.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Why do projectiles travel with a parabolic trajectory?** They travel with a parabolic trajectory because gravity pulls it downward, and the acceleration factor causes it to move in a parabolic way.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**How would a projectile travel differently without gravity?** Without gravity, the projectile would travel in a straight line.

=<span style="font-family: 'Comic Sans MS',cursive;">Lesson 2c: Describing Projectiles with Numbers =

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Central Ideas:** In projectile motion, the values of the x and y components of the velocity and displacement change depending on the situation; additionally, vector diagrams can be used to show these different types of situations.


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">****What are some important things to remember about projectiles?**** Some important characteristics are that they have only the force of gravity acting upon them, they have a parabolic trajectory, and the there are no horizontal forces upon them.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**What purpose does a vector diagram have?** A vector diagram is used to represent how the x and the y components of the velocity change with time. The vector arrows determine the numerical value of each component.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**How are the numerical values of displacement differ horizontally and vertically?** The horizontal component changes at a constant rate while the vertical component changes according to a particular equation.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**What is the equation used to describe the horizontal and vertical displacement of a projectile?** [[image:Screen_shot_2011-11-11_at_11.02.44_PM.png]] [[image:Screen_shot_2011-11-11_at_11.03.05_PM.png]]
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**How do the x and y components of the velocity change over time for projectiles? [[image:Screen_shot_2011-10-20_at_5.43.15_PM.png]]**

<span style="font-family: 'Comic Sans MS',cursive; font-size: small;">**Orienteering Activity**
<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Partners: Lauren Kostman, Garrett Almeida**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**__Start Position:__ 0 m E**


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Legs || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Distance || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Direction ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">9.28 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">S ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">16.31 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">S ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">3 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">18.36 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">S ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">4 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">16.45 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">W ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">5 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">27.49 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">S ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">6 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">16.34 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">W ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Resultant || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">78.55 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">SW ||


 * || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">%Error ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Analytical || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.19% ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Graphical || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">3.35% ||

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Sample Calculations:**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive;">Launcher Activity (ball in cup)

 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Partners: Lauren Kostman, Garrett Almeida **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Height: 92 cm
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Trial || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Distance(cm) ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">203.1 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">205.4 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">3 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">206.4 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">4 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">206.8 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">5 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">208.2 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">6 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">208.4 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">7 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">209.2 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">8 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">210.2 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Average || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">207.2 ||


 * || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">x || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">y ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">vi || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">4.79 m/s || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">0 m/s ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">a || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">0 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">-9.8 m/s/s ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">t || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.433 sec || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.433 sec ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">d || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.07 m || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.920 m ||


 * Sample Calculations:**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**__Cup__:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Height= 67 cm <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Time= .369 s <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Distance: 177 cm

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Sample Calculations:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Procedure:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">media type="file" key="Movie on 2011-10-25 at 11.04.mov" width="300" height="300"

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Actual =** 1.69 m = 169 cm

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Percent Error:** 4.5% <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">

<span style="font-family: 'Comic Sans MS',cursive;">**Shoot Your Grade Lab**
<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Partners:** Lauren Kostman, Garrett Almeida

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Hypothesis:** If we hang our rings at the correct vertical heights and horizontal distances, the ball should go through all five and land in the cup.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Procedure:** We began this experiment by placing carbon paper on the floor to obtain a relative idea of where our ball landed when shot at medium range. We measured an average of horizontal distances. Then we were able to find the initial velocity and time, so we set our launcher to 25 degrees and hung the rings from the ceiling according to the heights and distances we calculated. We adjusted our rings slightly until the ball went through as many as possible.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Horizontal Distances:
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Trial || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Distance (m) ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.95 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.94 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">3 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.96 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">4 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.96 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">5 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.97 ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Average || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.96 ||


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Velocity at 25 degree angle: **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Vx = 4.23 m/s <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Vy = 1.97 m/s

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Video:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">media type="file" key="through 5 yay!.mov" width="300" height="300"

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Data:**
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Ring: || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">x-position (m) || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">time (s) || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">y-position (m) || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">y-position (m) || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">% error ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.500 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.118 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.20 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.26 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">5.00% ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.960 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.227 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.24 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.28 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">3.23% ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">3 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.25 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.296 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.18 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.20 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.69% ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">4 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.50 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.355 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.11 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.09 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.80% ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">5 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">1.75 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.414 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.926 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.850 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">8.21% ||
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">cup || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">2.96 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.700 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">.090 || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">N/A || <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">N/A ||

<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">**Sample Calculations:**



<span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">In this lab, we obtained relatively close measurements between our experimental and theoretical values calculated for each ring. Our percent error is only about 5%, which is good for an experiment so complex. We got the ball through five rings, which was a very big success. Some possible sources of error in this lab might include the inconsistency of the launcher, alteration of the launcher's angle, and alteration of the rings' heights or distances. If I were to redo this lab, I would try to use a more consistent launcher and use a more fixed and unmovable method of hanging the rings from the ceiling.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;">Conclusion: **

<span style="font-family: 'Comic Sans MS',cursive; line-height: 18px;">Gourd-O-Rama Project
<span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">**Partner: Gabby Leibowitz**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">**Calculations:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">d = 18.25 m <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">t = 11.35 s <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">**vi = 3.22 m/s** <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">**a = -.284 m/s2**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 12px; line-height: 18px;">**Summary:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 90%;"> The results of our project turned out extremely well, especially in spite of the assignment's objective. For our project, we used a plastic liter of soda, popsicle sticks to make the chassis, and finally attached wheels from a toy car to the vehicle. In our first prototype, we had taped wheels to the popsicle sticks, but after testing we altered our design and glued the wheels instead for more sturdiness. We also cut a smaller opening in the bottle and gently placed the pumpkin on top of it. We did this intentionally so that each time we released the vehicle down the ramp, the pumpkin would fall in the bottle, creating extra momentum. Our vehicle traveled a far distance of 18.25 meters in 11.35 seconds, with a low acceleration of -.284 m/s2, showing that our project completely met the expectations of the objective. For even more positive results, I think we could have better secured the popsicle sticks to the bottle for more support, and this security may have allowed the pumpkin to travel even farther. We also could have made sure our wheels were lined up correctly, because sometimes our vehicle would veer off to the right or left and hit the wall, which we had to exclude from our results. Finally, we could have chosen lighter wheels in order to reduce the weight of our project. Otherwise, we were pleased with the outcome of our project and the low acceleration we calculated.