Ch3_LeibowitzG

= = toc = = = = =**__ Lesson 1- __****// Vectors (A + B) //**=


 * **Vectors and Direction**
 * **Vector Addition**

**__ Vectors and Direction __** A vector quantity is a quantity that is __fully described by both magnitude and direction__ vs. a scalar quantity that is fully described by its magnitude Ex: displacement, velocity, acceleration, force

* Vector quantities are often represented by scaled vector diagrams that depict a vector by use of an arrow drawn to scale in a specific direction. These are known as free body diagrams

This vector diagram depicts a displacement vector
 * A scale is clearly listed
 * 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

* Vectors can be directed due East, due West, due South, and due North BUT some are northeast etc. Therefore…
 * 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
 * The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its “tail” from due Eas t


 * The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale.



Two vectors can be added together to determine the result (or resultant)
 * __ Vector Addition __**

 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

__ Pythagorean Theorem: __ The hypotenuse of the right triangle can be used to determine the resultant



__ Trigonometry: __ SOH CAH TOA - sine, cosine, and tangent functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle
 * 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
 * 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
 * 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



__ Head-to-Tail Method __
 * 1) Choose a scale and indicate it on a sheet of paper
 * 2) 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
 * 3) 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
 * 4) Repeat steps 2 and 3 for all vectors that are to be added
 * 5) Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as R
 * 6) Measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m)
 * 7) Measure the direction of the resultant using the counterclockwise convention



= **__ Lesson 1- __****// Vectors (C + D) //** =

**__Resultants__** The **resultant** is the vector sum of two or more vectors. It is //the result// of adding two or more vectors together. If displacement vectors A, B, and C are added together, the result will be vector R. Vector R can be determined by the use of a scaled, vector addition diagram: gjkdfghjjg **A + B + C = R**  When displacement vectors are added, the result is a **resultant displacement.** But any two vectors can be added as long as they are the same vector quantity. If two or more velocity vectors are added, then the result is a **resultant velocity.** If two or more force vectors are added, then the result is a **resultant force.** If two or more momentum vectors are added, etc.  **__Vector Components__** A vector is a quantity that has both magnitude and direction In situations in which vectors are directed at angles to the coordinate axes, one will //transform// the vector into two parts with each part being directed along the coordinate axes. Ex 1: A vector that is directed northwest can be thought of as having two parts - a northward part and a westward part. Ex 2: A vector that is directed upward and rightward can be thought of as having two parts - an upward part and a rightward part.
 * **Resultants**
 * **Vector Components**

Any vector directed in two dimensions can be thought of as having an influence in two different directions. Each part of a two-dimensional vector is known as a **component**.

= **__ Lesson 1- __****// Vectors (E) //** =


 * **Vector Resolution**

**__Vector Resolution__** The process of determining the magnitude of a vector is known as **vector resolution**. The two methods of vector resolution:
 * The parallelogram method
 * The trigonometric method

__Parallelogram Method of Vector Resolution__
 * 1) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Select a scale and accurately draw the vector to scale in the indicated direction.
 * 2) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Sketch a parallelogram around the vector
 * 3) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Draw the components (sides) of the vector
 * 4) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Label the components of the vectors
 * 5) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Measure the length of the sides of the parallelogram and use the scale to determine the magnitude of the components in real units

<span style="font-family: 'Times New Roman',Times,serif; font-size: medium;">Ex:

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">__Trigonometric Method of Vector Resolution__ <span style="font-family: 'Times New Roman',Times,serif; font-size: medium;">Ex:
 * 1) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Construct a //rough// sketch of the vector in the indicated direction
 * 2) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Draw a rectangle about the vector such that the vector is the diagonal of the rectangle
 * 3) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Draw the components of the vector
 * 4) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Label the components of the vectors
 * 5) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">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. Use algebra to solve the equation for the length of the side opposite the indicated angle
 * 6) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle

= **__ Lesson 1- __****// Vectors (G + H) //** =

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **__Relative Velocity and Riverboat Problems__** Objects can move within a medium that is moving with respect to an observer 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 <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">* Motion is relative to the observer <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Ex 1: a plane flying among a **tailwind** (a wind that approaches the plane from behind, therefore, 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
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Relative Velocity and Riverboat Problems**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Independence of Perpendicular Components of Motion**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> Ex 2: a plane encounters a **headwind** (a wind that approaches the plane from the front, therefore, decreasing the plane's resulting velocity) The velocity of the plane would be relative to an observer on the ground

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> Ex 3: a plane encounters a **sidewind** The resulting velocity of the plane is the vector sum of the two individual velocities

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">* The resultant velocity of a motorboat can be determined in the same manner as was done for the plane

<span style="font-family: 'Times New Roman',Times,serif;"> **<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">__Independence of Perpendicular Components of Motion__ ** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> A force vector that is directed upward and rightward has two parts - an upward part and a rightward part If you pull upon an object in an upward and rightward direction, then you are exerting an influence upon the object in two separate directions - an upward direction and a rightward direction.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> Two parts of the two-dimensional vector are referred to as **components**, which describe the affect of a single vector in a given direction. The vector sum of these two components is always equal to the force at the given angle Any vector 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="display: block; font-family: 'Times New Roman',Times,serif; text-align: center;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">* The two perpendicular parts or components of a vector are independent of each other. A change in one component does not affect the other component

=**__ Activity- __****// Orienteering //**= <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Gabby Leibowitz, Maddie Margulies, and Max Llewellyn **<span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">PART ONE: ** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Our group's starting position was the set of doors on the right on the front of the building side of the courtyard. We started measuring in the middle of the doors on the ground.
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Legs || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Distance (m) || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Direction ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">0 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">0 m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">E ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">1 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">7.05 m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">E ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">2 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">5.72 m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">S ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">3 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">17.34 m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">E ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">4 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">5.68 m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">S ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">5 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">2.95 m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">W ||

<span style="font-family: 'Times New Roman'; font-size: 14pt; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">  <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">Our group got a percent error, between the actual and the measured, of 4.86%, and a percent error, between the actual and the calculated, of 5.81%. Although this is not too high of a percent error, a reason behind it could be from a common error during the "actual" part of the activity. While outside, we could have been slightly off on our measurements, therefore, leading to a difference between the theoretical and the experimental. In addition, the graphical, or measured, aspect of the activity involves drawing and then measuring the lengths of the vectors and angles, which would result in many sources of measurement error. Finally, the calculated, or analytical, aspect of the activity involves rounding, also contributing to a higher precent error. **<span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">PART TWO: ** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Start: 20 yard line, West (football field)
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Leg || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Distance(m) || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Direction ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">1 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">9.22m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">N ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">2 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">24.52m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">E ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">3 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">18.31m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">N ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">4 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">24.23m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">E ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">5 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">18.33m || <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">S ||

<span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Our percent error was much lower for this part of the activity compared to the first part. Our percent error between the actual and the measured was 1.00% and the percent error between the actual and the calculated was 0.605%. This lower percent error could merely be a result of the fact that we were more precise with either our measuring outside or our rounding/drawing for the analytical or graphical parts. However, since for this part of the experiment we had to follow instructions from a previous group, the slight percent error we got could be a result of miscommunication between the two groups and unclear directions, for we could have started at a slightly different starting point than they had anticipated.

= **__ Lesson 2- __****// Projectile Motion (A + B) //** =

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **__What is a Projectile__**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**What is a Projectile?**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Characteristics of a Projectile's Trajectory**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **1. What is a projectile?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> A projectile is an object upon which the only force acting is gravity.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **2. What are examples of projectiles?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> There are many examples of projectiles. An object dropped from rest is a projectile. An object that is thrown vertically upward is also a projectile. An object which is thrown upward at an angle to the horizontal is a projectile, as well.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **3. What are specific properties of a projectile?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> A projectile has a single force that acts upon it, which is the force of gravity. If there were any other force acting upon an object, then that object would not be a projectile. Therefore, there is a specific free-body diagram of a projectile, which would show a single force acting downwards and labeled force of gravity (Fgrav). <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: 0px; overflow-x: hidden; overflow-y: hidden;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **4. What do projectiles have to do with what we are learning about?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> This relates to what we were doing in the previous chapter with free falling objects. The same misconceptions were formed. The only force acting upon an upward moving projectile is gravity. It is false to think that an object moving upward must be affected by an upward force, or an object moving rightward must be affected by a rightward force. That is because this misconception goes against Newton's laws.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **5. How can you apply projectiles to an application?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> A cannonball is shot horizontally from a very high cliff at a high speed. The cannonball is projected horizontally from the top of the cliff. Gravity will act downwards upon the cannonball to affect its vertical motion. Gravity causes a vertical acceleration and the ball will drop vertically below its inertial path.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> *** Main Idea:** The only force acting upon a projectile is gravity.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **__Characteristics of a Projectile's Trajectory__**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **1. What are horizontal components of a projectile's motion?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> If an object is projected horizontally, it will continue its horizontal motion at a constant velocity in the absence of gravity. This is consistent with the law of inertia. If an object is projected in the presence of gravity horizontally, the object would maintain the same horizontal motion as before- a constant horizontal velocity. In addition, there must be a horizontal force to cause a horizontal acceleration.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **2. What are vertical components of a projectile's motion?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> If an object is aimed upward, the projectile would travel along a straight-line, or the inertial path, in the absence of gravity. In the presence of gravity, the object would free-fall below the inertial path. This projectile would travel with parabolic trajectory. The downward force of gravity would act upon the object to cause a downward acceleration.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **3. What is the difference of the effect of gravity on horizontal or vertical motion?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> The presence of gravity does not affect the horizontal motion of the projectile, only vertical motion. A projectile maintains a constant horizontal velocity since there are no horizontal forces acting upon it.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **4. Why do projectiles travel with parabolic trajectory?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> Projectiles travel with a parabolic trajectory due to the fact that the downward force of gravity accelerates them downward from their otherwise straight-line, gravity-free trajectory. This downward force and acceleration results in a downward displacement from the position that the object would be if there were no gravity.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **5. What does projectile trajectory have to do with what we have previously learned?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> If an object is dropped from rest in the presence of gravity, it will accelerate downward, gaining speed at a rate of 9.8m/s. This is consistent with the conception of free-falling objects accelerating at the rate known as the acceleration of gravity, which we previously learned.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> *** Main Idea:** The force of gravity does not affect the horizontal component of motion; a projectile maintains a constant horizontal velocity since there are no horizontal forces acting upon it.

= **__ Lesson 2- __****// Projectile Motion (C) //** =

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**1. What is the significance of a vector diagram?** A vector diagram shows the velocity of an object at intervals of time. It is used to represent how the x- and y- components of the velocity of an object is changing with time by using x- and y- velocity vectors. The lengths of the vector arrows are representative of the magnitude of that quantity. The diagram illustrates that the horizontal velocity remains constant during the course of the trajectory and the vertical velocity changes by 9.8 m/s every second.
 * <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">**Describing Projectiles with Numbers**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**2. What do the numerical values of the x and y components show?** Numerical values, regardless of the diagram, show that a projectile has a vertical acceleration of 9.8 m/s/s, downward and no horizontal acceleration. Therefore, the vertical velocity changes by 9.8 m/s each second and the horizontal velocity never changes. It shows that there is a vertical force acting upon a projectile but no horizontal force. It is the vertical force that causes a vertical acceleration, an acceleration of 9.8m/s/s.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**3. How do the horizontal and vertical components of the velocity vector change with time?** As the projectile rises towards its peak, it is slowing down and as it falls from its peak, it is speeding up. The symmetrical nature of the projectile's motion shows that the vertical speed one second before reaching its peak is the same as the vertical speed one second after falling from its peak. For non-horizontally launched projectiles, the direction of the velocity vector is sometimes considered + on the way up and - on the way down, yet the magnitude of the vertical velocity is the same on either side of its peak. At the peak itself, the vertical velocity is 0 m/s and the velocity vector is entirely horizontal at this point in the trajectory.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**4. How do the horizontal and vertical components of a projectile's displacement change with time?** The vertical distance fallen from rest during each consecutive second is increasing, showing that there is vertical acceleration. Since there is no horizontal acceleration, the horizontal distance traveled by the projectile each second is a constant value- the projectile travels a horizontal distance each second. Displacement values for a projectile launched at an angle to the horizontal is different and would rise a vertical distance equivalent to the time multiplied by the vertical component of the initial velocity.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**5. What equation is used to describe the horizontal and vertical displacement of a projectile?** Vertical displacement: y=1/2gt^2 Horizontal displacement: x= vi (of x) t

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">*** Main Idea:** The horizontal and vertical components of the velocity vector change with time during the course of projectile's trajectory.

= **__ Activity- __****// Ball in Cup //** = <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">Gabby Leibowitz, Maddie Margulies, and Max Llewellyn <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;"> **Procedure:** <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;"> media type="file" key="Ball+in+Cup+Activity.mov" width="300" height="300" <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;"> **PART TWO:**
 * PART ONE:**
 * Conclusion:**  <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">There was only a 1.18% error for our activity. This could be as a result of the fact that the distance from the launcher wasn't 100% accurate and we had to "eye-ball" where to move the cup after launching the ball a few times. However, after this we were able to get the ball in the cup the majority of the time. There existed a few outliers as a result of the inconsistency of the launcher to be completely precise every time. Overall, we maintained accurate results.

= **__ Project- __****// Gourd-O-Rama Contest //** = <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">Gabby Leibowitz and Maxx Grunfeld = =

<span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">**Design:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">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 bottom. We cut an opening in the bottle and gently placed the pumpkin on top of it.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Calculations:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">d = 18.25 m <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">t = 11.35 s <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**vi = 3.22 m/s** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**a = -.284 m/s2**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Conclusion:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">The results of our project turned out extremely well and achieved the assignment's objective. Our prototype was a similar model to our final project, however, the wheels were not securely placed on the bottom of the bottle and the hole in the bottle was cut big enough for the pumpkin to be held within the bottle. However, after this prototype we adjusted our project, gluing instead of taping the wheels onto the popsicle sticks and cutting a slightly smaller hole in the bottle so the pumpkin rested on top. Our first success was the fact that each time we would release 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, illustrating that our project met the expectations of the objective- to travel the farthest distance in the longest amount of time with the least acceleration. In order to improve our experiment and maintain even better results, we could have even further secured the popsicle sticks better to the chassis for more support, and this security may have allowed the pumpkin to travel even farther. We also should 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 caused inaccurate results. Finally, we could have chosen lighter wheels in order to reduce the weight of our project. Regardless, we were pleased with the outcome of our project and the low acceleration we calculated!

=**__ Lab- __****// Shoot Your Grade //**= <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Gabby Leibowitz, Maddie Margulies, and Max Llewellyn <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">**Purpose with Rationale:**Using a launcher and a plastic ball, you must be able to successfully calculate the x and y distances (using knowledge of projectiles) of five different rings and a plastic cup, calculate the trajectory of the ball, set the rings accordingly, and successfully launch your ball into all five of the rings, ultimately landing in the cup. This models an "off-a-cliff" projectile problem. Therefore, I hypothesize that the ball will travel in a curved path from our measured angle of 21 degrees, allowing the velocity, time, and distances to be calculated through previously learned methods (x/y tables). <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;"> **Materials and Method:** <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">Materials: <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">The main materials used included 5 rings of tape rolls strategically hung from the ceiling tiles by the means of string and tape. Their distances were measured using measuring tape and a yard stick. Projectile launchers were used to launch a plastic ball through these rings, held down on the counter by clamps and loaded by a black launcher "loader." Before the actual experiment, carbon paper and paper were put on the ground to mark where the ball hit and allow the plastic cup to be placed at an average distance. Our calculations were all done by knowledge of projectiles and use of x/y tables and the common formulas. <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">Method: <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">First, we had to find the distance of where to place the plastic cup that our balls would eventually be launched into after passing through five rings. To do this, we placed paper with carbon paper on top on the ground and shot the ball multiple times to get an average of where it could most accurately be predicted to land. We took the distances of where the black marks the carbon paper left from where the ball hit and then averaged them together to get an exact distance of where to place the cup in regards to the launcher. We then calculated the initial velocity by using the average x distance and the height of the shooter to the ground plus the height of the cup. We solved for this using the formula D=vit+1/2at^2. We then measured the x distance of rings planted by the previous class. Using this number, the initial velocity, and constant acceleration values, we were able to find the time, and were ultimately able to find a value which, once added to the height of the ground (1.175m), gave us the height or y distance the rings needed to be placed at. After hanging the rings at these specific x and y distances, using string and tape to attach them to the ceiling tiles and making sure the center of each ring was at each height, we experimented by launching the ball and then adjusting our measurements as necessary. We were able to successfully shoot the ball through 4 rings since the trajectory of the ball varied slightly each launch. <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: 0px; overflow-x: hidden; overflow-y: hidden; text-align: left;">  <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;"> Picture of the p lacement of four of our rings  <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;">Video posted below **<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Observations and Data from Initial Velocity ** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Our data shows a table of the distances where the ball landed after being shot a total of 6 times. We took the average of these 6 distances in order to get a relatively accurate distance that would allow the ball to be launched into the cup a majority of the time. Using this data and the average distance as the range, we were able to find velocity using the constant acceleration values and the height of the launcher from the ground.

media type="file" key="Shoot+Your+Grade.m4v" width="300" height="300" <span style="display: block; font-family: 'Times New Roman',Times,serif; font-size: 120%; text-align: left;"> Our best launch made it through 4 of the rings as seen above. We tried changing the contrast and slowing the video down in order to see the ball more clearly. <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Our data above shows results from performance day with a sample calculation to show how we found where to place each ring (x and y distances) in relation to the time. Although our ball only made it through four rings, we still found the placement for the 5th ring and cup. **<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Error Analysis ** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Actual positions of rings measured:
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Observations and Dara from Performance **
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Ring || <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Actual height (m) ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">1 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">1.32m ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">2 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">1.41m ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">3 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">1.34m ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">4 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">1.19m ||
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">5 || <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Did not achieve ||

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">In the end, our hypothesis proved to be correct and the ball traveled in a curved path from our measured angle of 21 degrees, allowing the velocity, time, and distances to be calculated through previously learned methods. However, the problem was that the launcher wasn't consistent and therefore, it was difficult to succeed in getting the ball into all five rings and the cup since the rings had to be slightly shifted in position as we did each trial. Regardless, you can see our hypothesis was accurate. While the ball may not have gone through each of the rings every time, it never missed by too large of a gap, and this could be blamed on the sources of error present in the actual experiment, not the calculations. The percent error calculations illustrates the sources of error and obstacles our group faced. Since we got 0% for our x percent error, it is evident that there was no movement in the x-distance of our rings. However, the height or y percent error was higher, at 3.65%. This is a result of the fact that our angle would alter slightly each trial, making our results alter as well. From all the contact made with the launcher it was human error that the angle wouldn't be the exact degree as it was when we first fired it. We tried to reduce this error by constantly checking and tightening it. Also, the height proved inconsistent because of the other groups working with our launcher and ceiling tiles, changing the height of the rings to fit their calculations. It was difficult to rearrange the rings to mirror the same height we had previously, and therefore, made achieving consistent results a bit more challenging. Both these sources of error could be reduced by changing the materials and setting of the experiment. While the shifting of the angle is merely human error and bound to happen, an even tighter and stronger clamp could have limited the extent of movement of the angle. In addition, if this experiment was performed in a larger room, we would have been the only group working with our rings, limiting any movement from other groups and keeping the y distances relatively the same. This would have also given us more time to perfect our distances since we wouldn't have had to fix our heights at the beginning of every lab. However, this is not realistic, for this experiment was performed in a physics classroom and a larger room is not accessible. Therefore, we could have limited the shift in y-distances //during// our lab period that occurred accidentally from either ourselves or surrounding groups. This could have been achieved through finding a different means to attach the rings to the ceiling tiles, possibly by the use of binder clips in addition to the tape. The concept this experiment demonstrated is similar to the real-life application of playing a variety of sports, such as football, basketball, golf, and baseball. For example, while a football player most likely doesn't take time during a game to calculate the initial velocity, range, or height of the projectile of a football, it is important that these players have a sense of understanding of where the football will land after throwing it at a certain speed. It is this knowledge which provides the player with a keen sense of where and how fast to throw the ball in order to get it as close to a specific target as possible, or in his case, in the hands of a specific teammate. Most sports players use knowledge of projectiles without even knowing it!
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Conclusion: **