Ch3_MarguliesM

toc Chapter 3

Vectors Lesson 1: parts a and b (method 1)
Vectors and Direction distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. vector quantity: a quantity that is fully described by both magnitude and direction scalar quantity: a quantity that is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces that occur in two dimensions.Vector quantities are not fully described unless both magnitude and direction are listed. Vector quantities are often represented by scaled vector diagrams. Such diagrams are commonly called as free-body diagrams. An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram.
 * 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. 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: >
 * 1) 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.
 * 2) The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its " tail " from due East. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit.

**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. For example, the diagram at the right shows a vector with a magnitude of 20 miles.

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.

Vector Addition One operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). The //net force// experienced by an object was determined by computing the vector sum of all the individual forces acting upon that object.

A vector directed up and to the right will be added to a vector directed up and to the left. The //vector sum// will be determined for the more complicated cases shown in the diagrams below.There are a variety of methods for determining the magnitude and direction of the result of adding two or more vectors.

**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.

**Using Trigonometry to Determine a Vector's Direction** The direction of a //resultant// vector can often be determined by use of trigonometric functions. Once the measure of the angle is determined, the direction of the vector can be found. In this case the vector makes an angle of 45 degrees with due East. Thus, the direction of this vector is written as 45 degrees. The measure of an angle as determined through use of SOH CAH TOA is not always the direction of the vector.

**Use of Scaled Vector Diagrams to Determine a Resultant** **head-to-tail method** is employed to determine the vector sum or resultant. A common Physics lab involves a //vector walk//. Either using centimeter-sized displacements upon a map or meter-sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position.

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

Vectors Lesson 1: parts c and d (method 1)
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. That is why it can be said that **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//.

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.

Vector Components

In situations in which vectors are directed at angles to the customary coordinate axes, 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. 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.

Vectors Lesson 1: part e (Method 1)
Vector Resolution 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. A step-by-step procedure for using the parallelogram method of vector resolution is:
 * 1) Select a scale and accurately draw the vector to scale in the indicated direction.
 * 2) 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).
 * 3) 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).
 * 4) 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.
 * 5) 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.

A velocity vector with a magnitude of 50 m/s and a direction of 60 degrees above the horizontal may be resolved into two components:

**Trigonometric Method of Vector Resolution** 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:
 * 1) 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.
 * 2) 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.
 * 3) 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).
 * 4) 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.
 * 5) 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.
 * 6) Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

The 60-Newton tension force acts upward and rightward on Fido at an angle of 40 degrees, the components of this force can be determined using trigonometric functions.

The amount of influence in a given direction can be determined using methods of vector resolution. Two methods of vector resolution: a graphical method (parallelogram method) and a trigonometric method.

Vectors Lesson 1: Part f (Method 1)
**Component Method of Vector Addition** This Pythagorean approach is a useful approach for adding any two vectors that are directed at right angles to one another. After rearranging the order in which the three vectors are added, the resultant vector is now the hypotenuse of a right triangle. (Adding North + South and West + East) The size of the resultant was not affected by this change in order. This illustrates an important point about adding vectors: the resultant is independent by the order in which they are added. The above discussion explains the method for determining the magnitude of the resultant for three or more perpendicular vectors.The mnemonic SOH CAH TOA is a helpful way of remembering which function to use. The problem is not over once the value of theta (Θ) has been calculated. This angle measure must now be used to state the direction.

In summary, the direction of a vector can be determined in the same way that it is always determined - by finding the angle of rotation counter-clockwise from due east.

Now we will consider situations in which the two (or more) vectors that are being added are not at right angles to each other. The Pythagorean theorem is not applicable to such situations since it applies only to right triangles.

So whenever we think of a northwest vector, we can think instead of two vectors - a north and a west vector. The two components Ax + Ay can be substituted in for the single vector A in the problem. Vector C is clearly thenasty vector. Its direction is neither due south nor due west. The solution involves resolving this vector into its components.The process involves using the magnitude and the sine and cosine functions to determine the x- and y-components of the vector. Trigonometric functions - sine, cosine and tangent - are then used to determine the magnitude of the horizontal and vertical component of each vector. The work is shown and organized in the table below.

A northward and a southward component would also add together as a positive and a negative. Once the bottom row is accurately determined, the magnitude of the resultant can be determined using Pythagorean theorem.

Vectors Lesson 1: part g and h (Method 1)
Relative Velocity and Riverboat Problems To illustrate this principle, consider a plane flying amidst a **tailwind**. A tailwind is merely a wind that approaches the plane from behind, thus increasing its resulting velocity. If the plane is traveling at a velocity of 100 km/hr with respect to the air, and if the wind velocity is 25 km/hr, then what is the velocity of the plane relative to an observer on the ground below? The resultant velocity of the plane (that is, the result of the wind velocity contributing to the velocity due to the plane's motor) is the vector sum of the velocity of the plane and the velocity of the wind. This resultant velocity is quite easily determined if the wind approaches the plane directly from behind. If the plane encounters a headwind, the resulting velocity will be less than 100 km/hr. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Suppose a plane traveling with a velocity of 100 km/hr with respect to the air meets a headwind with a velocity of 25 km/hr. In this case, the resultant velocity would be 75 km/hr; this is the velocity of the plane relative to an observer on the ground. This is depicted in the diagram below. The resulting velocity of the plane is the vector sum of the two individual velocities. To determine the resultant velocity, the plane velocity (relative to the air) must be added to the wind velocity. This is the same procedure that was used above for the headwind and the tailwind situations; only now, the resultant is not as easily computed. Since the two vectors to be added - the southward plane velocity and the westward wind velocity

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 motorboat may be moving with a velocity of 4 m/s directly across the river, yet the resultant velocity of the boat will be greater than 4 m/s and at an angle in the downstream direction. While the speedometer of the boat may read 4 m/s, its speed with respect to an observer on the shore will be greater than 4 m/s. The resultant velocity of the motorboat can be determined in the same manner as was done for the plane. The resultant velocity of the boat is the vector sum of the boat velocity and the river velocity. Since the boat heads straight across the river and since the current is always directed straight downstream, the two vectors are at right angles to each other. 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.
 * Analysis of a Riverboat's Motion**

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.

Motorboat problems such as these are typically accompanied by three separate questions: 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** The time to cross this 80-meter wide river can be determined by rearranging and substituting into the average speed equation. **time = distance /(ave. speed)** If one knew the **distance C** in the diagram below, then the **average speed C** could be used to calculate the time to reach the opposite shore. Similarly, if one knew the **distance B** in the diagram below, then the **average speed B** could be used to calculate the time to reach the opposite shore. And finally, if one knew the **distance A** in the diagram below, then the **average speed A** could be used to calculate the time to reach the opposite shore.
 * 1) What is the resultant velocity (both magnitude and direction) of the boat?
 * 2) 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) What distance downstream does the boat reach the opposite shore?

The mathematics of the above problem is no more difficult than dividing or multiplying two numerical quantities by each other. The difficulty of the problem is conceptual in nature; the difficulty lies in deciding which numbers to use in the equations. The motion of the riverboat can be divided into two simultaneous parts - a motion in the direction straight across the river and a motion in the downstream direction.

Independence of Perpendicular Components of Motion

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. 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. 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. The resulting motion of a plane flying in the presence of a crosswind is the combination (or sum) of two simultaneous velocity vectors that are perpendicular to each other. Suppose that a plane is attempting to fly northward from Chicago to the Canada border by simply directing the plane due northward. If the plane encounters a crosswind directed towards the west, then the resulting velocity of the plane would be northwest. The northwest velocity vector consists of two components - a north component resulting from the plane's motor (the //plane velocity//) and a westward component resulting from the crosswind (the //wind velocity//). These two components are independent of each other. An alteration in one of the components will not affect the other component.

The resulting motion of the boat is the combination (i.e., the vector sum) of these two simultaneous and independent velocity vectors - the boat velocity plus the river velocity. In the diagram at the right, the boat is depicted as moving eastward across the river while the river flows southward. The boat starts at Point A and heads itself towards Point B. But because of the flow of the river southward, the boat reaches the opposite bank of the river at Point C. The time required for the boat to cross the river from one side to the other side is dependent upon the boat velocity and the width of the river. Only an eastward component of motion could affect the time to move eastward across a river. Suppose that the boat velocity is 4 m/s and the river velocity is 3 m/s. The magnitude of the resultant velocity could be determined to be 5 m/s using the Pythagorean Theorem. The time required for the boat to cross a 60-meter wide river would be dependent upon the boat velocity of 4 m/s. It would require 15 seconds to cross the 60-meter wide river. **d = v • t** So **t = d / v** (60 m) / (4 m/s) **15 seconds** 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.

Lab: Orienteering
Lab Partners: Gabby and Max

Our 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. Measured displacement:


 * Legs || Distance (m) || Direction ||
 * 0 || 0 || E ||
 * 1 || 7.05 || E ||
 * 2 || 5.72 || S ||
 * 3 || 17.34 || E ||
 * 4 || 5.68 || S ||
 * 5 || 2.95 || W ||

Analysis: Graphical Method: Analytical Method: 7.05 m east + 17.34 m east + 2.95 m west= 21.44 m east 5.72 m south + 5.68 m south = 11.40 south

Percent Error: = = Part 2 (Orienteering): Start: 20 yard line, West (football field) Analysis: Graphical Method: Resultant: 49.8 Scale: 1/2 cm : 1 m
 * Leg || Distance(m) || Direction ||
 * 1 || 9.22 || N ||
 * 2 || 24.52 || E ||
 * 3 || 18.31 || N ||
 * 4 || 24.23 || E ||
 * 5 || 18.33 || S ||

Analytical Method: Percent Error:

Conclusion: In conclusion, our results were pretty accurate compared to the original resultant. Some sources of error in this lab were when we were measuring the resultant on the field the tape measure was not long enough. This could have altered the resultant by a few centimeters, so our results could have been off. To prevent this, we could have gotten a longer tape measure. Also, our directions were not particular on which 20 yard line we should have started on, so we had to change the last direction to South. This is so we could have fit our map on the field. In the end, our results were pretty accurate because our percent error was less than ten percent for both the analytical and graphical methods. ==

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Vectors Lesson 2: parts a and b (Method 3)
Section A Questions: 1. What is a projectile? A projectile is an object where only force acting is gravity. An object that is projected in motion by its own inertia and influenced by gravity. 2. What are some examples of projectiles? Some examples of projectiles are objects dropped at rest, objects thrown vertically upward, and an object that is throw upward at an angle to the horizontal. 3. What does a Free body diagram of a Projectile look like? 4.Is a force required to keep an object in motion? A force is not requred to keep an object in motion. It is only required to maintain acceleration 5.What influences the vertical motion of a projectile? Gravity acts to influence the vertical motion of the projectile. Due to absence of horizontal forces, a projectile will remain in motion with constant velocity.

Section B Questions: 1. How to projectiles travel? They travel with parabolic trajectory because the downward force of gravity accelerations them downward 2. What does the downward force and acceleration result in? The downward displacement from the position that the object would be if there were no gravity 3. Does the force of gravity affect the horizontal compent of motion? No, because a projectile maintains a constant horizontal velocity because there are no horizontal forces acting on it. 4. What is the equation for horizontal displacement of a projectile? The equation is: x= Vix * t 5. What is vertical acceleration? Vertical acceleration is -9.8 m/s, down.
 * <  ||< **Horizontal** **Motion** ||
 * < Forces (Present? - Yes or No)(If present, what dir'n?) ||< No ||< Vertical MotionYesThe force of gravity acts downward ||
 * < Acceleration (Present? - Yes or No)(If present, what dir'n?) ||< No ||< Yes"g" is downward at 9.8 m/s/s ||
 * < **Velocity** (Constant or Changing?) ||< Constant ||< Changing(by 9.8 m/s each second) ||

Vectors Lesson 2: part c (Method 3)
1. What are some important conceptual notions about projectiles? Some important notions are that they are objects upon which the only force is gravity, they have a parabolic trajectory, and the there are no horizontal forces upon them. 2. What happens to the horizontal velocity if a cannon shoots a ball straight and horizontally? The horizontal velocity will stay constant during its course of trajectory and the vertical velocity will change 9.8 meters per second every second. 3. What happens to the velocity's of projectiles? Vertical velocity changes by 9.8 m/s each second and the horizontal velocity never changes. 4. What happens to the directions of the velocity vectors of non horizontal projectiles? On the way up it is considered positive and on the way down it is considered negative. The magnitudes is the same equal interval of time on both sides of the maximum height. 5. What is the equation when there is no gravity and when there is gravity. What is it when these two influences upon the vertical displacement? y= viy * 0.5 * g *t^2

Activity: Ball in Cup
Maddie, Max, and Gabby

Procedure media type="file" key="Ball in Cup Activity.mov" width="300" height="300"

Part 1: We found the average of these distances:

1. What is the Initial velocity of our launcher?

Part two: 2. After changing its height, What distance away from to launcher should we place a cup on the floor?

Percent Error: Conclusion: For our experiment we calculated a 3.5% error. This is because our distance from the launcher to the cup at first was a little off. We then moved it a little forward and we were able to get the ball into the cup three times in a row. We had some outliers because the launchers were not always persistent and accurate. Overall, the ball went into the cup the majority of the times.

Gourd-o-rama Contest
Maddie Margulies and Nicole Tomasofsky



Materials
 * Poster board
 * Stickers
 * Pumpkin
 * Duct Tape
 * Plastic Wheels (take off a toy monster car)

Design Our car was two pieces of poster board wrapped in duct tape, then we glued on the four wheels onto the cart. Out pumpkin had a smaller hole in the front and larger one in the back to make it more aero dynamic. We attached our pumpkin to the cart with duct tape.

Calculations

Conclusion Our final results were that our car went 8.0 meters. Our biggest problem was that our car kept turning. This must have happened because our axels were not straight. This could have been fixed by using using straight angles as our guide to put on the wheels. Our car stayed together as it went down the ramp, which was good because a lot of them fell apart. Also, we cut a whole small who in the front of our pumpkin and a larger one in the back of it. We thought that this made it more aero dynamic, so it could go farther in the long run. However, I thing if we did not cut a whole in the pumpkin it would have not made a difference. Our pumpkin was balanced and our car went pretty far, so I was happy with our results.

Shoot Your Grade
Maddie Margulies, Max Llewellyn, and Gabby Leibowitz


 * Purpose with Rationale**: In this lab you have to use your knowledge about projectiles to launch, using a launcher, a plastic ball through five strategically placed rings and finally into a cup.

For our lab we first found the initial velocity by taking the average of x distance that the shooter had shot and then took the height of the launcher to the ground. With this we found the initial velocty by using the formula D= Vit + 1/2at^2. Then we meaured the distance of the rings from the launcher (the rings were already placed from the previous class) and we took the initial velocity that we had found earlier. We then plugged the x distance and the velocity into the D= Vit + 1/2at^2 equation and found the time. Then we plugged the time into the y components to find the height of the ring. However, this height was only the height from the launcher up, so we had to add the height from the ground (1.175 m) to the new height we found. This is how we found where each ring should be placed exactly.
 * Materials and Methods:**


 * Observations and Data from Initial Velocity**

This data below is a table of the distances that the shooter had shot. We did about 6 tests and then took the average to find the distance from the launcher to where we should place the cup. Then we found the initial velocity using this data.

Video of best performance (four of the rings): media type="file" key="Shoot Your Grade.m4v" width="300" height="300"
 * Observations and Data From Performance and Physics Calculations**

We did the calculations for the 5th ring and the cup, however, we were not able to get the ball through them.

Above is the sample calculation for how we found where to place each ring and a table with the time, x- distances, and y distances for where each ring should have been placed.


 * Error Analysis**


 * Ring || Actual height (m) ||
 * 1 || 1.32 ||
 * 2 || 1.41 ||
 * 3 || 1.34 ||
 * 4 || 1.19 ||
 * 5 || did not achieve ||

Yes, my hypothesis was correct. Our calculations were mostly correct, however, the shooter was not consistent. If the launcher had been accurate, we may have been able to get it through all the rings and into the cup with easeo Also, there was some error in our calculations. For the x distance we had a zero percent error because we did not move them at all. However for the y distance aur average percent error was in the height which was about 3 percent. This was because when we launched the ball our angle which was supposed to be 21 degrees kept getting shifted. We tightened the angle on the shooter, however, it did not seem to stay. Also, when another group lifted up the ceiling tile our rings would get moved or gravity would make the ring shift down even the slighted bit and it would make our results off. This could have been fixed by clamping binder clips to the ceiling and using a pulley- system like other groups did. That seemed to work better and they were able to get more accurate results. Another source of error is that the launchers were inconsistent. If we had used a consistent launcher we could have been able to get all of the rings. Again, if we had used a consistent launcher, the binder clips, and made sure the angle was at 21 degrees every single launch our lab would be more precise and successful. A real- life application to this lab could be playing basket ball. It is important to know your trajectory and where the hoop is when you shoot the ball. Although in a basketball you would not take the time to calculate your initial velocity and the exact distance from your hands to the hoop, you do take these concepts into consideration. This is important to understand for simple day to day activities such as playing a game of basketball.
 * Conclusion**

Class Notes: Projectiles