Ch3_SmithJ

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Lesson 1 a and b
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. 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]. 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 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 East.

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). The [|net force] was the result (or [|resultant] ) of adding up all the force 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 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 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. 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. 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 p osition. 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.

 Lesson 1 c + d

The ** resultant ** is the vector sum of two or more vectors. If displacement vectors A, B, and C are added together, the result will be vector R. 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 ... In all such cases, the resultant vector (whether a displacement vector, force vector, velocity vector, etc.) is the result of adding the individual vectors. It is the same thing as adding A + B + C + ... . "To do A + B + C is the same as to do R." Vectors are sometimes 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. 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 **. 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.

 Lesson 1 e, Method 1
Any vector directed at an angle to the horizontal (or the vertical) can be thought of as having two parts (or components). 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 and the trigonometric method. 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. The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. Trigonometric functions will be used to determine the components of a single vector. 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. 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.

 Lesson 1 g and h
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">On occasion objects move within a medium that is moving with respect to an observer. In such instances, the magnitude of the velocity of the moving object (whether it be a plane or a motorboat) with respect to the observer on land will not be the same as the speedometer reading of the vehicle. Motion is relative to the observer. The 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. 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. 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. Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a ** side wind ** of 25 km/hr, West. 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 - are at right angles to each other, the [|Pythagorean theorem] can be used. 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 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. Like any vector, the resultant's [|direction] is measured as a counterclockwise angle of rotation from due East. The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat. If a motorboat were to head straight across a river, it would not reach the shore directly across from its starting point. The river current influences the motion of the boat and carries it downstream. Motorboat problems such as these are typically accompanied by three separate questions: <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The first of these three questions: 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). <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">** ave. speed = distance/time. **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.The two perpendicular parts or components of a vector are independent of each other. 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.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">What is the resultant velocity (both magnitude and direction) of the boat?
 * 2) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">If the width of the river is //X// meters wide, then how much time does it take the boat to travel shore to shore?
 * 3) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">What distance downstream does the boat reach the opposite shore?

<span style="color: #00ff00; font-family: Arial,Helvetica,sans-serif;">Lesson 2 a and b
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Central idea: A projectile is an object upon which the only force is gravity. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Gravity causes a vertical motion, which causes a vertical acceleration.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">What is a projectile?
 * 2) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">It is an object dropped from rest or thrown vertically upwards at an angle to the horizontal
 * 3) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">It is any object projected or dropped that continues in motion by its own inertia
 * 4) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">What forces affect upon a projectile?
 * 5) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The only force is of downward gravity
 * 6) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">What is inertia?
 * 7) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The resistance an object has to a change in a state of motion
 * 8) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">How do we represent projectiles?
 * 9) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">A free-body diagram is a good representation of projectiles.
 * 10) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">What type of motion will we see from this diagram?
 * 11) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Parabolic Trajectory will be seen, similar to a diagram in free fall.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Central idea: A projectile has two components: a vertical motion and horizontal motion. Vertical motion is effected by the downward force of gravity, and therefore is always changing. However, horizontal motion is unaffected by gravity, and therefore stays at a constant rate.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Can projectiles move in a direction other than vertical motion?
 * 2) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Yes, projectiles can move in a horizontal motion as well. Horizontal and vertical are both COMPONENTS of the projectile.
 * 3) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Does gravity affect the horizontal motion of a projectile?
 * 4) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">No, it does not. This is because gravity acts as a downward force, therefore unable to affect the horizontal motion.
 * 5) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">How does a projectile travel?
 * 6) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">With constant horizontal velocity and a downward vertical acceleration.
 * 7) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Is the velocity of downward vertical motion changing or constant?
 * 8) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">It is always changing, due to the account of gravity's acceleration of -9.8 m/s/s.
 * 9) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Is acceleration present in horizontal motion?
 * 10) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">No. It is always moving at a constant rate because it is not affected by gravity.

<span style="color: #00ff00; font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Lesson 2 c
<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Central idea: A projectile has two components: a vertical motion and horizontal motion. <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Vertical motion is effected by the downward force of gravity, and therefore is always changing. However, horizontal motion is unaffected by gravity, and therefore stays at a constant rate.
 * 1) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Can projectiles move in a direction other than vertical motion?
 * 2) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Yes, projectiles can move in a horizontal motion as well. Horizontal and vertical are both COMPONENTS of the projectile.
 * 3) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Does gravity affect the horizontal motion of a projectile?
 * 4) No, it does not. This is because gravity acts as a downward force, therefore unable to affect the horizontal motion.
 * 5) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">How does a projectile travel?
 * 6) <span style="font-family: arial,helvetica,sans-serif; font-size: 90%;">With constant horizontal velocity and a downward vertical acceleration.
 * 7) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Is the velocity of downward vertical motion changing or constant?
 * 8) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">It is always changing due to gravities acceleration of -9.8 m/s squared
 * 9) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Is acceleration present in horizontal motion?
 * 10) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">No it is always moving at a constant rate because it is not affected by gravity.

<span style="font-family: Arial,Helvetica,sans-serif;">Vectors Activity
<span style="font-family: Arial,Helvetica,sans-serif;">Partners: Jake Aronson, Ali Cantor, Kaila Soloman <span style="font-family: Arial,Helvetica,sans-serif;">Data: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">Graphical Method Solving for Resultant: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">Analytic Method Solving for Resultant: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">Percent Error for Analytical: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">Percent Error for Graphical:

<span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">Our percent errors were both very low which was very good. The analytical percent error was less because the graphical method can be less accurate because the measuring tool can move, or the point of view from the eye can be off a little bit. Overall, the percent errors were very good though.

<span style="font-family: Arial,Helvetica,sans-serif;">Launching the Ball (10/25/11)
<span style="font-family: Arial,Helvetica,sans-serif;">Partners: Kaila Soloman, Jake Aronson, Ali Cantor <span style="font-family: Arial,Helvetica,sans-serif;">**Calculations:** <span style="font-family: Arial,Helvetica,sans-serif;">Part 1: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">**Procedure**: <span style="font-family: Arial,Helvetica,sans-serif;">media type="file" key="New Project - Medium.m4v" width="300" height="300"

<span style="font-family: Arial,Helvetica,sans-serif;">**Calculations**: <span style="font-family: Arial,Helvetica,sans-serif;">Part 2: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">Percent Error: <span style="font-family: Arial,Helvetica,sans-serif;">

Gordorama
Partner: Maddy Weinfeld Pictures of the pumpkin:



Calculations:

Our pumpkin weighed 3.2 kg and it went 6.25 m after it rolled down the ramp. Our velocity was 3.33 m/s and the acceleration is -0.888 m/s/s.

Conclusion: If I were able to redo this project, I would first use faster wheels. Many people who used roller blade wheels had farther distances than we did. I think that the wheels effect the project a lot. I would also want to use axles and ensure that they are straight because our pumpkin kept turning to the side and hitting the lockers. Our cart was also very large and heavy. If I could change this now, I would want to make a smaller structure that weighed a bit less. I would also ideally make it pointed in the front because I feel that that would help its aerodynamics.

<span style="font-family: Arial,Helvetica,sans-serif;">Shoot Your Grade (11/9/11)
<span style="font-family: Arial,Helvetica,sans-serif;">Partners: Ali, Jake, and Kaila <span style="font-family: Arial,Helvetica,sans-serif;">**Purpose:** The purpose of this lab was to launch a ball from a launcher at 10 degrees and make it into 5 hoops and a cup. We were trying to find the height of all of the hoops given that we found the initial velocity from a previous activity which was 6.92 m/s. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">**Procedure and Materials:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">media type="file" key="New Project - Medium.m4v" width="300" height="300"
 * Hypothesis:** If the launcher has an initial velocity of 6.92 m/s then when put at 10 degrees and launched it will go through the hoops that we calculated relative to that velocity.


 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Calculations: **

These calculations were done to find the initial velocity of the launcher at 10 degrees.

I was not there the day the calculations were done to determine where to place the hoops so these calculations were done by Ali Cantor: She used the velocity and divided it by 6 to determine the intervals at which the hoops and eventually the cups would be located.

Error Analysis:



My hypothesis was correct because we were able to get the ball through 4 of the 5 hoops. The hoops were placed at 1.275m, 1.240m, 1.094m, .8335m, and .4910m. The launcher had an initial velocity of 6.92 m/s and this was used to determine the height of the hoops. The percent errors were .06, .21, .14, 0, and 3.4 as seen above. These averaged comes to an average percent error of .762 which is overall very good. A possible source of error is that the launchers angle changed a little bit the more it was being launched. If this is the case, the angle would be slightly higher or lower than 10 degrees and the initial velocity would be different than what we had calculated. Another possible source of error is that we did not measure the exact middle of the rings. To find the actual height we measured from the ground up to the center of the ring that was hanging. If we did not get exactly the middle, the numbers may be off and this would cause a percent error. Lastly, there could be a percent error because there is a range that the ball can get through because the rings are lager than the ball. We were placed right in front of the vent, so frequently the string holding up the rings would move because of the air. It is possible that the air blew the string and because the ball still got through we didn't notice the ring had moved, but this would change the measurements. If I had the opportunity to remove the percent error, I would first want to tape the string onto the ceiling. We were unable to do this because other classes were going to be using the same rings as us, but if they were not, it would eliminate the third error I discussed. Another thing that would be changed would be making more accurate measurements. This could be measuring the ring to find where the center is and then measuring from the ground to that point instead of estimating the middle. This can be used with real life implications with many different things. The thing I think of first is basketball. It is possible that basketball players could use this technology to help their performance because they are shooting at the angle of their arms and it is similar to a projectile.
 * Conclusion:**