Ch3_AronskyA

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toc =**Lesson 1 Homework ** = **10/12 (section a-b)**

**Vectors can are described by magnitude and direction. An arrow diagram is used to display vector form, and can be used as a quantitative value, where you can multiply, divide, subtract, and add vectors.**

** Vectors and Direction ** A vector quantity is a quantity that is fully described by both magnitude and direction. A scalar quantity is a quantity that is fully described by its magnitude.

Examples of vector quantities include [|displacement], [|velocity] , [|acceleration] , and [|force]. 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. Characteristics:
 * 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 ** Two conventions: ** Representing the Magnitude of a Vector ** The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow.
 * 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. For example: 40 degrees North of West
 * 2) The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "__ tail __" from due East.

** Vector Addition ** Two vectors can be added together to determine the result (or resultant). The [|net force] was the result of adding up all the force vectors. These rules for summing vectors were applied to [|free-body diagrams] in order to determine the net force, which is computing the vector sum of all the individual forces. ** 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. It 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. To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement. ** Using Trigonometry to Determine a Vector's Direction ** The direction of a //resultant// vector can often be determined by use of trigonometric functions. Sine, cosine, and tangent functions functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. These three trigonometric functions can be applied to the __ hiker problem __ in order to determine the direction of the hiker's overall displacement. 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 ** The ** head-to-tail method ** is employed to determine the vector sum or resultant. -Example: ** 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg. ** ** SCALE: 1 cm = 5 m **   The order in which three vectors are added has no affect upon either the magnitude or the direction of the resultant. The resultant will still have the same magnitude and direction. **10/13 (section c-d)** **Vectors have resultants and components. The resultant is the vector sum of all the individual vectors. Components are the individual parts that make up a two-dimensional vector.** ** Resultants ** The ** resultant ** is the vector sum of two or more vectors. Displacement vector R gives the same //result// as displacement vectors A + B + C à ** 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. **Vector Components** 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. Now, we begin to see examples of vectors that are directed in //two dimensions// - upward and rightward, northward and westward, eastward and southward, etc.



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.  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. **10/17 (section e)** **A vector directed in two dimensions has an influence in two separate directions. The amount of influence in a given direction can be determined by using the parallelogram method or the trigonometric method.** ** Vector Resolution ** The process of determining the magnitude of a vector is known as ** vector resolution **. The two methods:
 * the parallelogram method
 * the trigonometric method

** Parallelogram Method of Vector Resolution ** 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
 * 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. Select a scale and accurately draw the vector to scale in the indicated direction. ** Trigonometric Method of Vector Resolution ** Trignometric functions are used to find the length of the sides of a right triangle if an angle measure and the length of one side are known. The method: **10/18 (section g+h)** **Sometimes objects are affected by winds or currents, thus the velocity of the moving object is different to the observer on land.** **Vectors can be composed of perpendicular components, which are independent of each other.** ** Relative Velocity and Riverboat Problems **   On occasion objects move within a medium that is moving with respect to an observer. In such instances as this, 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. 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. 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. Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a ** side wind ** of 25 km/hr, West (100 km/hr)2 + (25 km/hr)2 = R2 ** 103.1 km/hr = R ** theta = invtan (25/100) ** theta = 14.0 degrees ** ** Analysis of a Riverboat's Motion **    If a motorboat were to head straight across a river (that is, if the boat were to point its bow straight towards the other side), 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. 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. (4.0 m/s)2 + (3.0 m/s)2 = R2 ** 5.0 m/s = R ** ** theta = 36.9 degrees ** Motorboat problems such as these are typically accompanied by three separate questions:
 * 1) Construct a //rough// sketch 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.
 * 3) Draw the components of the vector. The components are the //sides// of the rectangle.
 * 4) Meaningfully label the components of the vectors with symbols to indicate which component represents which side.
 * 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.
 * 6) Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.
 * 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?

** 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. 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.

= **Lesson 2 Homework ** = **10/18 (lesson a+b)**
 * Lesson A:** //A projectile is an object with one force acting upon it: gravity. If there was no gravity, objects would continue motion at constant velocity because forces are only required for acceleration. No horizontal forces are needed to maintain horizontal motion.//

**Lesson B:** //Gravity does not affect the horizontal displacement of a projectile, so it stays at a constant rate. However, vertical motion is affected by gravity, and is therefore always changing.// <span style="display: block; font-family: Tahoma,Geneva,sans-serif; font-size: 140%; text-align: left;">**Lesson C:** <span style="font-family: Tahoma,Geneva,sans-serif; font-size: 110%;">**10/20** <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">//Vertical motion and displacement are affected by gravity and change, while horizontal motion and displacement remain constant the throughout.// <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">1.What is the velocity of horizontal trajectory and a vertical trajectory? <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">2.What is the difference between free fall and a projectile? <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">3.List important equations. <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">4.What happens when the projectile falls below the gravity free path by a distance of y= 1/2gt2 <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">5.If thrown upwards, what is the relationship of velocity?
 * 1) What is a projectile?
 * 2) A projectile is an object that has only gravity acting upon it
 * 3) If there are other forces acting upon it, it is not a projectile.
 * 4) <span style="font-family: Tahoma,Geneva,sans-serif;">How many forces act upon projectiles?
 * 5) <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">One --> gravity
 * 6) <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">What are some types of projectiles?
 * 7) <span style="font-family: Tahoma,Geneva,sans-serif;">An object dropped from rest without influence of air resistance
 * 8) <span style="font-family: Tahoma,Geneva,sans-serif;">An object thrown upward vertically without influence of air resistance
 * 1) <span style="font-family: Tahoma,Geneva,sans-serif;">Is a force required to keep an object in motion?
 * 2) <span style="font-family: Tahoma,Geneva,sans-serif;">No, a force is not required to keep an object in motion. A force is only needed to keep an object accelerating. In the case of a projectile that is moving upward, there is a downward force and a downward acceleration --> object is moving upward and slowing down
 * 3) <span style="font-family: Tahoma,Geneva,sans-serif;">What would happen according to Newton’s first law if there were no gravity?
 * 4) <span style="font-family: Tahoma,Geneva,sans-serif;">The object would move in a straight horizontal line at a constant speed
 * 5) <span style="font-family: Tahoma,Geneva,sans-serif;">With gravity it would increasingly move downward because inertia.
 * 1) <span style="font-family: Tahoma,Geneva,sans-serif;">What are two components of projectile motion?
 * 2) <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">Horizontal and vertical motion
 * 3) <span style="font-family: Tahoma,Geneva,sans-serif;">Does gravity affect the horizontal motion of a projectile?
 * 4) <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">No, gravity acts a downward force, thus it is unable to affect the horizontal motion.
 * 5) <span style="font-family: Tahoma,Geneva,sans-serif;">Must there be a horizontal force for horizontal acceleration?
 * 6) <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">Yes because gravity acts perpendicular to the horizontal motion and will not affect it since perpendicular components of motion are independent. Therefore, an object moves with constant horizontal velocity and a downward acceleration.
 * 7) <span style="font-family: Tahoma,Geneva,sans-serif;">How would an object travel in constant motion at constant speed?
 * 8) <span style="display: block; font-family: Tahoma,Geneva,sans-serif; text-align: left;">If there was no unbalanced force, or absence of gravity. However because of the downward force of gravity, projectiles travel with a parabolic trajectory because the downward force of gravity accelerates them downward from an otherwise straight-line, gravity-free trajectory.
 * 9) <span style="font-family: Tahoma,Geneva,sans-serif;">Is the velocity of downward vertical motion changing?
 * 10) <span style="font-family: Tahoma,Geneva,sans-serif;">Yes, it is always changing because gravity’s acceleration is -9.8m/s/s
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">Horizontal is constant while vertical changes by 9.8 m/s each second.
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">Free fall is an example of a projectile, but in many cases, a projectile will have an initial velocity (as opposed to zero velocity) due to a launch or upward motion.
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">Vertical displacement for a 1D object - y= 1/2gt2
 * 2) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">Horizontal displacement for a horizontally launched projectile - X= Vixt
 * 3) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">Vertical displacement for an angled-launched projectile - viyt+ 1/2gt2
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">The gravity-free path is no longer horizontal because the projectile was not launched horizontally.
 * 1) <span style="font-family: Verdana,Geneva,sans-serif; font-size: 90%;">Velocities are symmetrical around the vertex.

=<span style="font-family: Tahoma,Geneva,sans-serif;">Measuring Angle Notes =





=<span style="font-family: Tahoma,Geneva,sans-serif;">Vector Mapping =

<span style="font-family: Tahoma,Geneva,sans-serif;">Graphical: <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;">In the graphical method we found the resultant to be 3.5 m at 197 degrees.

<span style="font-family: Tahoma,Geneva,sans-serif;">Analytical: <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;">In the analytical method we found the resultant to be 4.25 m at 196.3 degrees.

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

<span style="font-family: Tahoma,Geneva,sans-serif;">- As seen the analytical method had a much less percent error. This is because the graphical method takes into account human error with inaccurate measuring and drawing, thus it is not as precise and correct as calculating the resultant mathematically.

=<span style="font-family: Tahoma,Geneva,sans-serif;">Ball In Cup =
 * //(Part One)//**

<span style="font-family: Tahoma,Geneva,sans-serif;">A) How fast does the launcher shoot the ball at medium range (horiztonal)? <span style="font-family: Tahoma,Geneva,sans-serif;">B) Change initial height, calculation wwhere to place the cup on the floor so the ball lands inside.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Data**: <span style="font-family: Tahoma,Geneva,sans-serif; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">

<span style="font-family: Tahoma,Geneva,sans-serif; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">**Part A:**
 * <span style="font-family: Tahoma,Geneva,sans-serif;">We found the velocity to be 52.51 cm/s.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Part B**
 * <span style="font-family: Tahoma,Geneva,sans-serif;">We calculated that the ball would need to be placed at 227.4 cm in order for the ball to land in the cup.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Video:** media type="file" key="My First Project - Medium.m4v" width="300" height="300"

<span style="font-family: Tahoma,Geneva,sans-serif;">**Percent Error:** <span style="font-family: Tahoma,Geneva,sans-serif;">- As seen above, our percent error was very small. To make it more accurate the measuring could have been more precise.

=<span style="font-family: Tahoma,Geneva,sans-serif; font-size: 110%;">Gourdorama Contest = <span style="font-family: Tahoma,Geneva,sans-serif;">Andrea and Maddi <span style="font-family: Tahoma,Geneva,sans-serif;">

<span style="font-family: Tahoma,Geneva,sans-serif;">If Maddi and I were to change anything, we would make sure that the axles are perfectly lined up to ensure that the cart would travel completely straight. We would have gotten a lower acceleration and would have covered a longer distance. We could have also had less heavy skate board wheels so that the whole cart would have a lower mass.

=<span style="font-family: Tahoma,Geneva,sans-serif; font-size: 120%;">Shoot Your Grade Lab = <span style="font-family: Tahoma,Geneva,sans-serif;">**Partners**: Rachel Knapel and Jake Greenstein

<span style="font-family: Tahoma,Geneva,sans-serif;">**Purpose/Procedure:** Our objective is to lauch a ball from the shooter at 25 degrees so that the ball passes through five rings hanging from the ceiling tiles and lands in a cup on the floor. As we had a new angle, we had to readjust our initial velocity from part one of this lab. We had to calculate the horizontal and vertical distances, as well as time to ensure that the ball would land in the cup at the right distance. We placed each hoop and the cup at specific coordinates based on the calculations we had found. The goal was to test how accurate projectile calculations are and if they can apply to real life situations. <span style="font-family: Tahoma,Geneva,sans-serif;">**Hypothesis:** The ball will go successfully through the 5 hoops and land in the cup if all our calculations were correct. The ball will have a parabolic trajectory.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Methods and Materials:** We were able to calculate the height that the ring should be hung by using the off the cliff method for projectiles. With our calculations were able to set up the rings with clamps and tape from the ceiling tiles. We measured the heights we got from the launcher, both horizontally and vertically, to line up the rings. We set up one ring at a time and tested it. Each required a little bit of changing, but they were all in the correct general area. The materials used in this lab were rings of masking tape, string, a shooter, a yellow ball, measuring tape, right angle clamps, a plumb bomb, and carbon paper.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Initial Velocity Calculations:** <span style="font-family: Tahoma,Geneva,sans-serif;">- We used the five trials using carbon paper to find the average range of the ball. Then we measured the vertical height of the launcher along with the average range to calculate the hang time and the initial velocity.

<span style="font-family: Tahoma,Geneva,sans-serif;">- Above are the calculations used to find the initial velocity of the ball from the shooter at a 25 degree angle. The hang time was 7.3 seconds and the velocity was 47.67 cm/s. This velocity will be used to calculate the positions of the rings.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Physics Calculations:** <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;">- ^ the values above are in relation to the initial height, where the ball was launched from. <span style="font-family: Tahoma,Geneva,sans-serif;">- horizontal distance that the cup is placed away from the initial position of the shooter.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Trial Video:** showing the ball going through 4 rings, however we were able to get it through 5. <span style="font-family: Tahoma,Geneva,sans-serif;"> media type="file" key="New Project - Medium.m4v" width="300" height="300" <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;">**Data:** <span style="font-family: Tahoma,Geneva,sans-serif;">Above shows a simplified way to see our horizontal distances, theoretical and experimental heights, and percent error. The ball did not go into the cup so we did not have an experimental height. The height was calculated by adding the displacement from the initial height to the distance the launch point was from the ground.

<span style="font-family: Tahoma,Geneva,sans-serif;">**Percent Error:** <span style="font-family: Tahoma,Geneva,sans-serif;">- Sample calculation for Ring 1

= = <span style="font-family: Tahoma,Geneva,sans-serif;">**Conclusion:** <span style="font-family: Tahoma,Geneva,sans-serif;">Our hypothesis proved to not be completely correct. We predicted that the rings would start off getting higher and then lower, as projectiles travel with a parabolic trajectory. This part of our hypothesis proved to be true as shown by our vertical distances that first increased then became lower. The negative values found in our vertical displacement for the 4th and 5th rings indicate that the rings were placed below the launcher. Although we felt our calculations were accurate, the actual measurements were slightly off. When we set up the scene so that the ball would launch through all five rings and land in the cup, we had to adjust the heights of the rings. Our experiment allowed us to pass through 5 of the rings, but not the cup. The percent error calculations show why part of our hypothesis was incorrect. From our calculations, we got that the initial velocity would be about 47.64 cm/s. Based on this calculated value, we were able to find the vertical and horizontal distance components of where to place the rings. For example, we found the horizontal displacement for our second ring to be 99.6 cm and the vertical height to be 136.2 cm. Our experimental value shows that hoop had to actually be 137.4 cm above ground. However, the percent error was only 0.88%, proving how using the physics calculations could provide us with a successful prediction of where the ball would pass. If we were provided with more time, we most likely would have been able to get the ball into the cup, but given the circumstances, we proudly got the ball through 5 rings! <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;"> As seen above, our percent error ranges from 0.88 % to 17.21%. Our percent error represents the parabolic trajectory of the ball, being that the vertical acceleration is -9.8 m/s/s, the vertical position of the ball changes more the further away it is from the maximum height. This would mean that our positions towards the opposite end of the trajectory had a greater chance of being inaccurate, in comparison to the maximum height. Our data shows that the percent error around the maximum height was at its lowest, and then increased away from that point. Thus, the percent error for the cup, had we gotten it in, should have been the highest. There are many reasons for error that could cause these miscalculations in the vertical heights. One source of error was because there were many groups that used the same shooter; the strings would get moved on a daily basis or they would fall, causing our group to set up the scene from scratch. We were unable to control this factor due to lack of space so we had to work with different approximations of placement as well as the tape giving a little each time the ball was shot, changing the position of the hoop. Another source of error was the angle of the launcher and the inconsistency of the launcher itself. Often, after several trials, the angle would slightly increase from 25 degrees to around 28 degrees. Even the slightest change would cause a huge difference in our results. The problem with the launcher was that as the spring would heat up, it would become more flexible and not fire the projectile as far when we continued to use it more. This would cause our initial velocity to change and the target to not be accurate. The third major problem was air resistance. Our projectile calculations did not take into account air resistance, only the effects of gravity. However, with the environment of the experiment, there was a breeze from the air vent that caused our projectile to not follow its predicted measurements and caused the hoops to swing. <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;"> One way to address the errors in the experiment would be for every group to have their own shooter so that the strings weren’t altered and in an area without any vent blowing air. Another way to change the lab would be to make sure the angle is exactly at 25 degrees, either by checking it every launch or having a screw that held it at that angle tighter. When loosened even a little, the ball wouldn’t even go through the first ring after going through all 5. We could have also made sure that all the strings hung from the ceiling were completely tight so that they could not move. Each time we shot a ball from the launcher and it hit one of the rings of tape, it would change the position of the hoop and throw off the projectile course. We would need a clamp or a caliper that holds the string even tighter to the ceiling because each centimeter off makes a difference to the path. A cooling system would address the problem with the launcher springs by canceling out the effects of the springs getting heated. Then our velocity would be consistent and the ball wouldn’t travel extremely differently each launch. The final way to decrease error could be by having an instrument that could measure air pressure so that we could take this information into account in our calculations. Projectiles can be applied to many real life situations, from playing sports, such as basketball, golf, tennis, baseball, and football, to shooting a gun or missile. With this concept we could understand where all these objects could land and where to position them to get your desired target. Given certain information, such as initial velocity and launch angle, the position of the object can be determined at different points in time. So many objects in our lives follow a parabolic trajectory that is similar to the ball launched in this lab. <span style="font-family: Tahoma,Geneva,sans-serif;">