Ch3_MillerJ

toc = Chapter 3 Wikispace!!! =



Physics Classroom Notes Chapter 3 Lesson 1 Sections A&B 10/12/11:
Topic Sentence: Vectors are very specific to angles, directions, and measurements and can be represented by using diagrams called (vector/scaler graphs).
 * __Section A:__**

Vectors and Direction A study of motion will involve the introduction of a variety of quantities that are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories - vectors and scalars. A vector quantity is a quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude.

Examples of vector quantities that have been previously discussed include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed. 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. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object. Such diagrams are commonly called as free-body diagrams. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram. 1) a scale is clearly listed 2) a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail. 3) the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).

Conventions for Describing Directions of Vectors

Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east. Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is not due East, due West, due South, or due North. There are a variety of conventions for describing the direction of any vector. The two conventions that will be discussed and used in this unit are described below: 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. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction). The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "tail" from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east.

Observe in the second example that the vector is said to have a direction of 240 degrees. This means that the tail of the vector was pinned down and the vector was rotated an angle of 240 degrees in the counterclockwise direction beginning from due east. A rotation of 240 degrees is equivalent to rotating the vector through two quadrants (180 degrees) and then an additional 60 degrees into the third quadrant.

Representing the Magnitude of a Vector 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. In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated (such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction. Directions are described by the use of some convention. The most common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.

Topic Sentence: There are several methods which one can use when adding vectors; Pythagorean theorem, trigonometry, and vector diagrams are the method we use to add vectors.
 * __Section B:__**

Vector Addition A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). This process of adding two or more vectors has already been discussed in an earlier unit. Recall in our discussion of Newton's laws of motion, that the net force experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object. That is the net force was the result (or resultant) of adding up all the force vectors. During that unit, the rules for summing vectors (such as force vectors) were kept relatively simple. Observe the following summations of two force vectors:

These rules for summing vectors were applied to free-body diagrams in order to determine the net force (i.e., the vector sum of all the individual forces). Sample applications are shown in the diagram below.

There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors. The two methods that will be discussed in this lesson and used throughout the entire unit are: the Pythagorean theorem and trigonometric methods the head-to-tail method using a scaled vector diagram

The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

Using Trigonometry to Determine a Vector's Direction The direction of a resultant vector can often be determined by use of trigonometric functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA from their course in trigonometry. SOH CAH TOA is a mnemonic that helps one remember the meaning of the three common trigonometric functions - 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.

In the above problems, the magnitude and direction of the sum of two vectors is determined using the Pythagorean theorem and trigonometric methods (SOH CAH TOA). The procedure is restricted to the addition of two vectors that make right angles to each other. When the two vectors that are to be added do not make right angles to one another, or when there are more than two vectors to add together, we will employ a method known as the head-to-tail vector addition method. This method is described below.

Use of Scaled Vector Diagrams to Determine a Resultant The magnitude and direction of the sum of two or more vectors can also be determined by use of an accurately drawn scaled vector diagram. Using a scaled diagram, the head-to-tail method is employed to determine the vector sum or resultant. The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position. Where the head of this first vector ends, the tail of the second vector begins (thus, head-to-tail method). The process is repeated for all vectors that are being added. Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to real units using the given scale. The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East.

A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below. 1) Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the 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 (e.g., SCALE: 1 cm = 20 m). 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 Resultant or simply R. 6) Using a ruler, 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 discussed earlier in this lesson.

Physics Classroom Notes Chapter 3 Lesson 1 Section C&D 10/13/11:
__Section C:__ Topic Sentence: The resultant is the vector sum of all the individual vectors, and can be determined by adding the individual forces together using vector addition methods.

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. As shown in the diagram, vector R can be determined by the use of an accurately drawn, scaled, vector addition diagram.

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, then the result is resultant resultant momentum. 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.

__Section D:__ __Topic Sentence:__ Vectors can travel north, south, east, and west but they can also travel in multiple dimensions which adds a new level of difficulty to the problem. Vector Components A vector is a quantity that has both magnitude and direction. Displacement, velocity, acceleration, and force are the vector quantities that we have discussed thus far in the Physics Classroom Tutorial. In the first couple of units, all vectors that we discussed were simply directed up, down, left or right. When there was a free-body diagram depicting the forces acting upon an object, each individual force was directed in one dimension - either up or down or left or right. When an object had an acceleration and we described its direction, it was directed in one dimension - either up or down or left or right. Now in this unit, we begin to see examples of vectors that are directed in two dimensions - upward and rightward, northward and westward, eastward and southward, etc. In situations in which vectors are directed at angles to the customary coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes. For example, a vector that is directed northwest can be thought of as having two parts - a northward part and a westward part. A vector that is directed upward and rightward can be thought of as having two parts - an upward part and a rightward part.

Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. 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 combined influence of the two components is equivalent to the influence of the single two-dimensional vector.

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. In the next part of this lesson, we will investigate two methods for determining the magnitude of the components. That is, we will investigate how much influence a vector exerts in a given direction.

Physics Classroom Notes Chapter 3 Lesson 1 Section E 10/17/11:
__Topic Sentence__: Two methods for finding the magnitude of a vector are the the parallelogram method and the trigonometric method. Vector Resolution As mentioned earlier in this lesson, any vector directed at an angle to the horizontal (or the vertical) can be thought of as having two parts (or components). That is, any vector directed in two dimensions can be thought of as having two components. For example, if a chain pulls upward at an angle on the collar of a dog, then there is a tension force directed in two dimensions. This tension force has two components: an upward component and a rightward component. In this unit, we learn two basic methods for determining the magnitudes of the components of a vector directed in two dimensions. 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 the parallelogram method the trigonometric method

Parallelogram Method of Vector Resolution The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. Briefly put, the method involves drawing the vector to scale in the indicated direction, sketching a parallelogram around the vector such that the vector is the diagonal of the parallelogram, and determining the magnitude of the components (the sides of the parallelogram) using the scale. If one desires to determine the components as directed along the traditional x- and y-coordinate axes, then the parallelogram is a rectangle with sides that stretch vertically and horizontally. A step-by-step procedure for using the parallelogram method of vector resolution is: Select a scale and accurately draw the vector to scale in the indicated direction. 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 (a special case of a parallelogram). 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. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right). Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled Fnorth. A rightward velocity component might be labeled vx; etc. Measure the length of the sides of the parallelogram and use the scale to determine the magnitude of the components in real units. Label the magnitude on the diagram.

Trigonometric Method of Vector Resolution The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. Earlier in lesson 1, the use of trigonometric functions to determine the direction of a vector was described. Now in this part of lesson 1, trigonometric functions will be used to determine the components of a single vector. Recall from the earlier discussion that trigonometric functions relate the ratio of the lengths of the sides of a right triangle to the measure of an acute angle within the right triangle. As such, trigonometric functions can be used to determine the length of the sides of a right triangle if an angle measure and the length of one side are known.

The method of employing trigonometric functions to determine the components of a vector are as follows: Construct a rough sketch (no scale needed) of the vector in the indicated direction. Label its magnitude and the angle that it makes with the horizontal. Draw a rectangle about the vector such that the vector is the diagonal of the rectangle. 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. Draw the components of the vector. The components are the sides of the rectangle. The tail of each component begins at the tail of the vector and stretches along the axes to the nearest corner of the rectangle. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right). Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled Fnorth. A rightward force velocity component might be labeled vx; etc. 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 some algebra to solve the equation for the length of the side opposite the indicated angle. Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

In conclusion, a vector directed in two dimensions has two components - that is, an influence in two separate directions. The amount of influence in a given direction can be determined using methods of vector resolution. Two methods of vector resolution have been described here - a graphical method (parallelogram method) and a trigonometric method.

Physics Classroom Notes Chapter 3 Lesson 1 Section G&H 10/18/11:
__Lesson G:__ Topic Sentence: When observing a moving object the speed the object actually moves needs to be observed from the side and measured based on that.

Relative Velocity and Riverboat Problems On occasion objects move within a medium that is moving with respect to an observer. That is to say, the speedometer on the motorboat might read 20 mi/hr; yet the motorboat might be moving relative to the observer on shore at a speed of 25 mi/hr. Motion is relative to the observer. The observer on land, often named (or misnamed) the "stationary observer" would measure the speed to be different than that of the person on the boat. The observed speed of the boat must always be described relative to who the observer is.

In this situation of a side wind, the southward vector can be added to the westward vector using the usual methods of vector addition. The magnitude of the resultant velocity is determined using Pythagorean theorem. The algebraic steps are as follows: (100 km/hr)2 + (25 km/hr)2 = R2 10 000 km2/hr2 + 625 km2/hr2 = R2 10 625 km2/hr2 = R2 SQRT(10 625 km2/hr2) = R 103.1 km/hr = R

The direction of the resulting velocity can be determined using a trigonometric function. Since the plane velocity and the wind velocity form a right triangle when added together in head-to-tail fashion, the angle between the resultant vector and the southward vector can be determined using the sine, cosine, or tangent functions. The tangent function can be used; this is shown below: tan (theta) = (opposite/adjacent) tan (theta) = (25/100) theta = invtan (25/100) theta = 14.0 degrees If the resultant velocity of the plane makes a 14.0 degree angle with the southward direction (theta in the above diagram), then the direction of the resultant is 256 degrees. Like any vector, the resultant's direction is measured as a counterclockwise angle of rotation from due East.

Analysis of a Riverboat's Motion

Thus, the Pythagorean theorem can be used to determine the resultant velocity. Suppose that the river was moving with a velocity of 3 m/s, North and the motorboat was moving with a velocity of 4 m/s, East. What would be the resultant velocity of the motorboat (i.e., the velocity relative to an observer on the shore)? The magnitude of the resultant can be found as follows: (4.0 m/s)2 + (3.0 m/s)2 = R2 16 m2/s2 + 9 m2/s2 = R2 25 m2/s2 = R2 SQRT (25 m2/s2) = R 5.0 m/s = R The direction of the resultant is the counterclockwise angle of rotation that the resultant vector makes with due East. This angle can be determined using a trigonometric function as shown below. tan (theta) = (opposite/adjacent) tan (theta) = (3/4) theta = invtan (3/4) theta = 36.9 degrees Motorboat problems such as these are typically accompanied by three separate questions: What is the resultant velocity (both magnitude and direction) of the boat? If the width of the river is X meters wide, then how much time does it take the boat to travel shore to shore? What distance downstream does the boat reach the opposite shore? The first of these three questions was answered above; the resultant velocity of the boat can be determined using the Pythagorean theorem (magnitude) and a trigonometric function (direction). The second and third of these questions can be answered using the average speed equation (and a lot of logic). ave. speed = distance/time Consider the following example.

__Lesson H:__ Topic Sentence: Some vectors have multiple components, components are the affect of a single vector in a given direction.

Independence of Perpendicular Components of Motion A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. That is to say, 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. These two parts of the two-dimensional vector are referred to as components. A component describes the affect of a single vector in a given direction. Any force vector that is exerted at an angle to the horizontal can be considered as having two parts or components. The vector sum of these two components is always equal to the force at the given angle.

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.

A change in the horizontal component does not affect the vertical component. This is what is meant by the phrase "perpendicular components of vectors are independent of each other." 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.

All vectors can be thought of as having perpendicular components. In fact, any motion that is at an angle to the horizontal or the vertical can be thought of as having two perpendicular motions occurring simultaneously. These perpendicular components of motion occur independently of each other. 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. In Lesson 2, we will apply this principle to the motion of projectiles that typically encounter both horizontal and vertical motion.

Orienteering Activity:
Joe Miller, Bennett Sherman, and Sarah Gordon

The data:

Analytical Method:

Graphical Method:



Percent Error: Our analytical work was slightly better than the work of our graph. The analytical method produced a 2.56% error. The graphical method is the tougher method to solve however having been precise in our drawing we were able to get a low percent error at 5.64% error. The theoretical value (what we measure when doing the activity) was 665cm so that's what we used to determine our percent errors.



Physics Classroom Notes Chapter 3 Lesson 2 Section A&B 10/19/11:
Method 3:

__**Section A:**__ __**Main idea:**__ understand projectile motion __**Questions**__ __What is the newton law of inertia?__ __How does gravity affect projectile motion?__ __What are the requirements for projectile motion to occur?__ __Does air resistance act upon an object in projectile motion?__ __Does anything remain constant with a projectile?__

__**Key Terms:**__ Projectile: an object upon which the only force acting is gravity (no air resistance) the object is than only influenced by gravity No force is required to keep the object in motion, but it is required to establish acceleration Newton's Law of Inertia: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force Gravity is the downward force upon a projectile gravity influences an objects vertical motion (causing vertical acceleration) and is responsible for its parabolic trajectory A projectile has a constant horizontal velocity, due to the tendency of objects in motion to remain in motion at a constant velocity

__What is the newton law of inertia?__ An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force __How does gravity affect projectile motion?__ Gravity is the single force that acts upon a projectile __What are the requirements for projectile motion to occur?__ A projectile is only acted upon by one force (gravity). When an object gets dropped it continues the downward motion by its own inertia and is only affected by gravity not gravity and air resistance. __Does air resistance act upon an object in projectile motion?__ No, air resistance does not affect it __Does anything remain constant with a projectile?__ Yes, a projectile has a constant horizontal velocity

__**Section B:**__ __**Main idea:**__ explore the trajectory of projectile motion __**Questions**__ __Does the acceleration of gravity affect the trajectory of a projectile?__ __Why does the trajectory of projectile motion maintain a parabolic shape?__ __Non Horizontally Launched Projectiles result in what shape?__ __What is the difference between horizontal and non-horizontal projectile launch?__ __How is projectile motion different from free fall?__

__**Key Terms:**__ - Projectile motion consists of both vertical and horizontal motion Horizontally Launched Projectiles: - Free-falling objects will accelerate according to the acceleration of gravity - Gravity accelerates an object downward, but does not have an effect on its horizontal motion Non Horizontally Launched Projectiles: Travels with “parabolic trajectory” Presence of gravity does not influence horizontal motion of projectile Parabolic trajectory is a result of downward force of gravity and acceleration, and the downward displacement in position as opposed to the lack of gravity Constant horizontal velocity remains same (inertia from start), as no other horizontal forces act upon the projectile

__Does the acceleration of gravity affect the trajectory of a projectile?__ The downward force of gravity along with acceleration is responsible for the trajectory of a projectile in motion. __Why does the trajectory of projectile motion maintain a parabolic shape?__ It maintains the parabolic shape because of the downward pushing force exerted by gravity and by acceleration of the projectile. __Non Horizontally Launched Projectiles result in what shape?__ They result in a shape called parabolic. (like in math the parabola) __What is the difference between horizontal and non-horizontal projectile launch?__ Non-horizontal launch is when an object is launched upward and at an angle to the horizontal. __How is projectile motion different from free fall?__ They are generally the same only difference is that projectile motion has two different components were free fall is simply one.

Physics Classroom Notes Chapter 3 Lesson 2 Section C Parts 1&2 10/20/11:
//**Lesson 2 Section C Part 1:**// __**Main Idea:**__ to communicate the numerical factors of projectile trajectory and explain its conceptual aspects. __**Questions:**__ __1. How does horizontal motion affect projectile motion?__ __2. What are some key terms to know when it come to doing projectile motion?__ __3. An object is launched horizontally describe it.__ __4. An object is launched non-horizontally describe it.__ __5. How does the vertical component of the vector change trough its fall?__

__**Key Terms/concepts:**__ - the only force that acts upon a projectile is gravity many projectiles have parabolic trajectory because of the gravity - horizontal velocity is ALWAYS constant - no horizontal forces acts upon the object, no horizontal acceleration - vertical acceleration due to gravity= -9.8m/s/s (Same as free fall) - vertical velocity changes by acceleration of gravity each second horizontal motion independent of vertical motion Horizontal velocity is constant AND vertical velocity changes by 9.8m/s/s every second A picture of Horizontal Launch A picture of Non-Horizontal Launch

__1. How does horizontal motion affect projectile motion?__ Horizontal velocity always remains constant. __2. What are some key terms to know when it come to doing projectile motion?__ A few important concepts to know: Horizontal velocity remains constant, while vertical velocity changes by 9.8m/s/s every second. __3. An object is launched horizontally describe it.__ When an object is launched at a horizontal, or dropped from rest. __4. An object is launched non-horizontally describe it.__ This scenario occurs when a projectile is launched at an angle to the horizon. __5. How does the vertical component of the vector change trough its fall?__ The vertical velocity of the velocity of the vector changes by 9.8m/s/s every second. (same as free fall)

//**Lesson 2 Section C P2:**// __**Main Idea**__: evaluate the displacement of projectile trajectory, using specific equations. __**Question**__ __1. In what way will an initial vertical component (non-horizontal launch) affect the resulting displacement?__ __2. Are projectile trajectories symmetrical or unsymmetrical?__ __3. What is the vertical displacement of a projectile based on?__ __4. What is the equation that we use for non-horizontal projectile motion?__ __5. What is the equation that we use for horizontal projectile motion?__

__**Key Terms/Concepts**__: Δd= vit + ½ at2 Use this equation for the vertical displacement of a horizontally launched projectile (above) Δd= vit + ½ at2 Use this equation for the vertical displacement of a non-horizontally launched projectile (above) The symmetrical nature of projectile trajectory is key to understanding the factors of the concept



__1. In what way will an initial vertical component (non-horizontal launch) affect the resulting displacement?__ It will only change the general trajectory of the projectile. The displacement can be easily found by using the correct formula. __2. Are projectile trajectories symmetrical or unsymmetrical?__ The trajectory of a projectile is symmetrical. __3. What is the vertical displacement of a projectile based on?__ The displacement is dependent upon the initial velocity, time, and acceleration __4. What is the equation that we use for non-horizontal projectile motion?__ Δd= vit + ½ at2 __5. What is the equation that we use for horizontal projectile motion?__ Δd= vit + ½ at2

Ball in Cup Activity 10/24/11:
Joey Miller, Bennett Sherman, and Sarah Gordon

Data:

Part A)

Part B)



Percent Error:

Video: media type="file" key="ballcup.mov" width="242" height="295"

Conclusion: The percent error, 1.44% was a pretty good percent error. There were a few sources of error that we experienced during this lab. One example is that someone could have accidentally shifted the launchers position weather it be from side to side or forward and backward. The launcher itself if supposed to be horizontal, however with lots of use by multiple groups it could have been shifted slightly leading to off results. Another source of error is that we could have measured the distances incorrectly which would have thrown our data off. Additionally it is important to consider that the launcher is not precise and that the ball often strays from its intended target. In order to fix this, we could be more careful, have two people check the distance measured. When taking data it is important to always do multiple tests as to ensure that the data we collect is comparable to what occurred during the laboratory. We also should have verified the location of the launcher each time as to assure that it was at an optimal settings.

Pumpkin Vehicle Project:
Joey Miller & Bennett Sherman

My Vehicle:
 * Distance Traveled**: 11.6m
 * Time Taken**: 6.92s

The Competition: Our vehicle did very well in the race down the ramp. It traveled one of the farthest distances in the class at 11.6m. It was very consistent throughout its multiple launches. I contribute the vehicles success to its design. A few things influenced the success of the vehicle. The first thing would be its lightweight yet strong design. The car is made of lawon board and thin plywood, its wheels are banister. The car was built with an angle at the front of the vehicle as to break through the wind. The pumpkin was almost completely inside the vehicle which allowed for less air resistance. The tires were drill pressed and the axles placed straight which allowed the vehicle to drive in a decently straight line (which allowed it to move further).

What I would have changed: The only thing that I could think of that could have improved the vehicle was to maybe make it a little more lightweight. While the vehicle was not overly heavy it was not the lightest in the class. What we could have done to help cut weight would have been to make the bottom, previously constructed out of plywood out of a lighter weight material.

Calculations:

Shoot Your Grade Lab:
Joe, Ben, & Sarah

We believe that the ball will travel in a parabolic trajectory. We will find these positions and respective times and based on this data we will be able to set up the rings and the cup in a way that will allow for the ball to pass through the rings and hopefully land inside the cup.
 * Hypothesis:**

**Purpose**: Launch a ball at a specific angle and speed so that the ball passes through 5 rings suspended above the ground and lands in a cup placed on the floor. The point of the lab is to calculate the trajectory of a ball (projectile) when launched off a cliff (counter top) at a set velocity and at a specific angle, in this case 15º. In order to test the trajectory (parabolic) of the ball we set up 5 rings and a plastic cup and found distances both away from the launcher (x) and above the launcher (y).


 * Materials and Methods:** The entire lab has to do with the projectile launch, this case set at a 15 degree angle. The projectile launched a small yellow ball through 5 hoops made of tape. The ball at the end should land in a small plastic cup. Using calculations from the average range and hang time, the initial velocity components were used to determine the exact position of the projectile at 6 time intervals. We used these measurements to place the hoops at their specific locations and than place the cup in its designated position. Using multiple test runs and a few adjustments we were able to align the parabolic trajectory of the ball.


 * Observations and Data from Initial Velocity:**



This chart is the distances from the launcher that were found from firing the ball and having it land in clusters on the carbon copy paper. For the five runs, the ball had scattered landing points, but it averaged to 4.7m, which is our range.
 * Data From carbon paper:**
 * Run || Distance(m) ||
 * 1 || 4.76 ||
 * 2 || 4.65 ||
 * 3 || 4.71 ||
 * 4 || 4.68 ||
 * 5 || 4.72 ||
 * Avg: || 4.7 ||

We collected our data by using carbon paper to calculate the average range of the ball when it is shot from a 15 degree angle. We than used the average range that we found out to calculate the hang time and both the y and x components of the initial velocity. With the x and y components we will be able to determine the location of the rings and the cup.

Video Clip of the Ball traveling through all 5 rings!!!! (//but not the cup :(// ) media type="file" key="5 hoops.mov" width="300" height="300"
 * Observations and Data from Performance:**


 * Table of our results:**

__**NOTE: The ball traveled through all five rings but failed to land inside the cup.**__
 * || Time (s) || Horizontal Distance away; //x values// (m) || Height of the rings; //y values// (cm) ||
 * Ring 1: || .165 || 1.10 || 133.23 ||
 * Ring 2: || .259 || 1.73 || 130.40 ||
 * Ring 3: || .339 || 2.26 || 121.23 ||
 * Ring 4: || .407 || 2.71 || 108.48 ||
 * Ring 5: || .483 || 3.22 || 88.86 ||
 * The Cup: || .703 || 4.68 || 11.5 ||


 * Physics Calculations:**


 * Error Analysis:**


 * Theoretical Distances:**
 * Ring: || Horizontal Distance (m) || Vertical Distance (cm) ||
 * Ring 1: || 1.10 || 133.23 ||
 * Ring 2: || 1.73 || 130.40 ||
 * Ring 3: || 2.26 || 121.23 ||
 * Ring 4: || 2.71 || 108.48 ||
 * Ring 5: || 3.22 || 88.86 ||
 * Cup: || 4.70 || 11.50 ||


 * Actual Distances:**
 * Ring: || Horizontal Distance (m) || Vertical Distance (cm) ||
 * Ring 1: || 1.10 || 133.23 ||
 * Ring 2: || 1.76 || 131.26 ||
 * Ring 3: || 2.29 || 124.4 ||
 * Ring 4: || 2.71 || 113.13 ||
 * Ring 5: || 3.22 || 88.86 ||
 * Cup: || NEVER ACHIEVED || NEVER ACHIEVED ||



||
 * Percent Error Chart:**
 * Ring: || Horizontal: || Vertical: ||
 * Ring 1: || 0% || 0%
 * Ring 2: || 1.73% || .659% ||
 * Ring 3: || 1.33% || 2.61% ||
 * Ring 4: || 0% || 4.28% ||
 * Ring 5: || 0% || 0% ||
 * Cup: || We never made it into the cup || We never made it into the cup ||

In the lab shoot you grade we were able to get the ball, our projectile, through all five hoops. However we were unable to get the ball into the cup. Our video shows the ball going through all five of the rings, thus achieving the main goal of the laboratory. Our percent errors were decent ranging between 0% and 4.28% which is good! We were unable to get the ball into the cup so we couldn’t get an accurate percent error, but because we did not get the ball into the cup we can assume it would be higher than 5%. Our hypothesis was correct the only part that we were unable to achieve was the ball landing inside the cup. A source of error could have occurred in the measurement of positions. Another source of error would be the inconsistent shooting of the launcher. The launcher doesn’t shoot the ball the same way every time which would produce different results. Another factor with our launcher was that the lock for the angle mechanism was not functioning correctly and often times would cause the angle to change slightly. We would have to readjust this, another source of error no doubt. Another thing to consider is the difficult build of the cup. The cup is standing straight up so you need to 100% hit it correctly or the ball will bounce out of it. A final source of error was in the placement of the rings. The rings were hung from the ceiling by strings and these strings were taped down. However we found that many times when people walked by and came in contact with the string it cause the ring to shift its position slightly. In changing the lab I would make sure that all of our rings were secure properly so that no contact would change their location. I would also have taken time to secure the angle mechanism of the launcher. This project is very relevant to real life situations. For example in the military mortar teams need to locate and destroy enemy encampments with mortars. Similar to the project they need to make sure that the mortar (ball) lands in the spot they want, an enemy encampment (cup), and not a civilians home all while making sure the mortar doesn't hit any potential obstacles like mountains (rings).
 * Conclusion:**