Ch3_GreensteinJ

toc =Honors Physics Chapter 3 (Vectors) =
 * Jake Greenstein**



=Physics Classroom Lesson 1a=

Distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. are all quantities that 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]. 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. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram.
 * Vectors and Direction**
 * 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. >
 * 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, 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) of 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction).
 * 2) 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. A vector with a direction of 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due east. A vector with a direction of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise direction relative to due east.

**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.



**Theme:** Vectors are two dimensional measurements that take into account magnitude and direction. Scalar values only use magnitude. Vectors can be represented with a a scaled vector diagram. A vector arrow is drawn to represent the vector, since it is a force in a specific direction. The angle of the vector is its position in relation to the +x axis.

=Physics Classroom Lesson 1b=

Two vectors can be added together to determine the result (or resultant). That is the [|net force] was the result (or [|resultant]) of adding up all the force vectors.
 * Vector Addition**

 For example, 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 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.  Theme: Two vectors producing a single result, called a resultant. It is a combination (addition) of the two vectors. Vector addition is the method used to find the net force/resultant of the two vectors. Another way to add 2 vectors (provided they meet at a right angle) is to use the Pythagorean theorem. The two vector's resultant is the hypotenuse of the right triangle they create.

=Physics Classroom Lesson 1c=

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. 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**
 * Resultants**

When displacement vectors are added, the result is a //resultant displacement//. 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//.

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].

A resultant is the same as the addition of its vectors. Basically, instead of exerting different forces in different directions, the resultant combines them and has the same net difference of the 3 separate components.
 * Theme:**

=Physics Classroom Lesson 1d=

A [|vector] is a quantity that has both magnitude and direction.
 * Vector Components**

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.


 * Visual examples of vector components:**







This shows, visually, how a vector is made of two parts. A vector is two dimensional, and can be broken up into multiple two dimensional parts. Usually, the two components are x and y. X + Y = resultant.
 * Theme:**

=Physics Classroom Lesson 1e= (Method 1)

Vectors directed at an angle to the horizontal (or the vertical) can be thought of as having two parts (or 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.
 * 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]

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. 
 * Parallelogram Method of Vector Resolution **
 * 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.

** Trigonometric Method of Vector Resolution ** 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.


 * Topic Sentence:**
 * A vector is a two dimensional measurement that can be broken down into its two separate components, x and y. The two components, when combined, produced a vector that can be found using Vector Resolution. A graphical method involves drawing the vector to scale, revealing the magnitude of the vector as a simple measurement (to scale). A trigonometric method involved using a rough drawing (to help). Sine, Cosine, or Tangent can be used to find the different components of the vector (SOH-CAH-TOA). Using an angle and a side, the parts of the vector can be calculated.**

=Physics Classroom Lesson 1g=

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. To illustrate this principle, consider a plane flying amidst a **tailwind**. 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? 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. [|Pythagorean theorem] can be used. This is illustrated in the diagram below.
 * Relative Velocity and Riverboat Problems**

The direction of the resulting velocity can be determined using a [|trigonometric function]. tan (theta) = (opposite/adjacent) tan (theta) = (25/100) theta = invtan (25/100) **theta = 14.0 degrees**

**Analysis of a Riverboat's Motion**

The resultant velocity of the boat is the vector sum of the boat velocity and the river velocity. The [|Pythagorean theorem] can be used to determine the resultant velocity. 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.

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**
 * 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?

**time = distance /(ave. speed)**


 * Topic Sentence:**
 * Motion is relative. While a boat's speedometer may read 30 mph, other forces acting upon the boat make the speed of the boat different to an observer standing on land. When multiple forces are involved, the resultant velocity is a vector. If a plane has a wind pushing it from behind, the addition of the vectors will yield a greater velocity. The actual path an object takes is called the resultant, and it can be broken down into an X and Y component. Once again, using kinematics equations and trigonometry, we can solve for the different parts of the vector.**

=**Physics Classroom Lesson 1h**= A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. 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. This is depicted in the diagram below.
 * 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 one component does not affect the other component. Changing a component will affect the motion in that specific direction.

Perpendicular components of motion do not affect each other. If the wind velocity increased, the air balloon would begin moving faster in the eastward direction, but its downward velocity would not be altered. If the balloon were located 60 meters above the ground and was moving downward at 3 m/s, then it would take a time of 20 seconds to travel this vertical distance. **d = v • t** So **t = d / v** (60 m) / (3 m/s) **20 seconds**


 * Topic Sentence:**
 * This section clarifies what is meant by two separate components of a vector. They are totally independent of one another; a change in one component has no affect on the other. The only thing it would affect is the resultant velocity. However, the resultant velocity is only altered in terms of the given vector you are manipulating.**

=Physics Classroom Lesson 2a=

An object only acted upon by gravity.
 * What is a projectile?**

A constant horizontal component and gravity as the vertical component.
 * What are the components of projectile motion?**

An object launched horizontally or at an angle where the only force acting upon the object is gravity.
 * What types of projectiles are there?**

Inertia is a physics law which states that an object in motion will stay in motion unless acted upon by a force.
 * What is inertia?**

This law explains how, without gravity, a projectile would travel forever in a straight line. Instead, a force acts upon it to stop the motion.
 * How does inertia apply to projectiles?**

A projectile is an object only acted upon by gravity. If not for this, the object would travel in a straight line indefinitely. Gravity causes parabolic trajectory, and is the force on the vertical component. A projectile has no horizontal forces acting on it.
 * Theme:**

=Physics Classroom Lesson 2b=

They travel with a parabolic trajectory affected by only gravity, with a constant horizontal speed and a vertical speed changing at -9.8m/s/s
 * What are the characteristics of projectiles?**

They are calculated similarly, except horizontally launched projectiles start at an angle of 0 while non-horizontally launched projectiles have an angle which affects their velocity.
 * What is the difference between horizontally and non-horizontally launched projectiles?**

Newton proved that an abject does not need a force constantly acting upon it to keep it in motion.
 * What was the misconception that Newton's law proved wrong?**

A diagram which shows a projectile's motion, with gravity labeled as a downward force and a constant horizontal velocity.
 * What is a free-body diagram?**

This passage explained inertia and its relationship to projectiles. Essentially, it explains the rules behind a separate x and y component of a projectile, but with the only force being gravity. If not for this, the object would continue to travel in a straight line indefinitely, when instead gravitational force alters the path of the projectile.
 * Theme:**

=Physics Classroom Lesson 2c=

Horizontal is constant while vertical changes by 9.8 m/s each second.
 * What is the velocity of horizontal trajectory and a vertical trajectory? **

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.
 * What is the difference between free fall and a projectile? **

Yes. It will give the x a constant velocity, and the y an initial velocity to be acted upon by gravity.
 * Does initial velocity affect the x and y components? **

Velocities are symmetrical around the vertex.
 * If the projectile is launched upwards at an angle, what are characteristics of its motion.**

**What are the x and y components in terms of?** Meters (usually) displaced from the starting point.

This section shows how, numerically, the concepts in the previous lesson apply. We saw how the horizontal velocity truly is constant at all points in the trajectory. The sole factor contributing to the parabolic trajectory of a projectile is gravity, which curves the trajectory due to its negative acceleration.
 * Theme:**

=Gourd-o-rama Project=


 * Partners**: Sarah Gordon & Jake Greenstein


 * Procedure**: Design and build a cart to carry a pumpkin down a ramp for the farthest distance, but with the lowest weight and acceleration.


 * Our Project:**




 * Description:** In order to achieve an ultra-light weight, we wanted to minimize what was needed to hold and secure a pumpkin. Lightweight wheels, a light wood chassis, and a plastic container were all the parts we determined were necessary for carrying a pumpkin.

Acceleration ∆d = ½ (vf+ vo) t 8.8 = ½ (0 + vo) 4.44 17.6 = 4.44vo vo = 3.964 m/s
 * Calculations: **

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Velocity at bottom of ramp <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">vf = vo + at <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">0 = 3.964 + a (4.44) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">-3.964 = 4.44a <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">a = -.893 m/s2


 * Possible changes:** In order to improve our project, we would want to make it travel farther than it did, as well as accelerate slower. In order to achieve this, we would use larger wheels, lubricate the axes to minimize friction, as well as attach the axles to the frame to be 100% sure that they are perfectly even, so that our project travels in a straight line.

=**Lab: Ball in Cup**=

<span style="font-family: Tahoma,Geneva,sans-serif;">At medium range, the initial velocity was 52.51 cm/s. <span style="font-family: Tahoma,Geneva,sans-serif;">Our cup would need to be placed 227.4 cm horizontally away from the launcher in order for the ball to land inside.
 * <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 where to place the cup on the floor so the ball lands inside. **


 * Part 1 (a & b) + Data table w/ calculations**


 * Part 1 calculations continued + Part c**

Our percent error was very low at 0.74%. This indicates that we took very accurate measurements, and calculated all of our values properly. At 0.74%, this indicates that our theoretical value should yield a successful result. However, if it did not, then we would know that the 0.74% was actually significant in obtained a positive result, and we would need to re-measure more accurately in order to achieve an even lower percent error.

media type="file" key="My First Project - Medium.m4v" width="300" height="300"
 * Video**

=Lab: Shoot Your Grade=


 * Partners:** Rachel Knapel and Andrea Aronsky


 * Purpose/Procedure:** We must shoot a ball at 25 degrees on medium power so that it flies through 5 hoops and lands in a cup. The hoops will hang from the ceiling, and we will determine where to place them based on our calculations.


 * Hypothesis:** Our calculations will yield accurate results, and therefore our projectile will pass through 5 hoops and land in the cup.


 * Materials:** Ball, Shooter, Clamp, Measuring tape, Masking tape, String, Paper clamps, Carbon Paper & Plumb bob


 * Method:** By calculating height of the projectile at a given horizontal distance, we could set up our rings by hanging them from the ceiling at a height that corresponds to our values calculated.


 * Initial velocity**


 * Hoop Placement**


 * Cup Placement**

media type="file" key="New Project - Medium.m4v" width="300" height="300"
 * Video:**

Data Table & Percent Error

Our Data Table provides a simple way to see the difference between our theoretical and experimental hoop heights at given horizontal distances. Our ball did not go into the cup, so an experimental height is not provided. The height was calculated by adding the displacement from the initial shot height to the distance the launch point was from the ground. Our percent error mirrors the parabolic trajectory of the ball. Due to vertical acceleration (-9.8 m/s/s), the vertical position of the ball changes more dramatically farther away from the maximum height, and at the maximum height does not change at all. This means that our positions towards opposite end of the trajectory had a greater chance of being inaccurate, in comparison to at the top. As can be seen above, our percent error around the maximum height was at its lowest, and then increases away from that point. In theory, the percent error for the cup, had we gotten it in, should have been the highest.

Our hypothesis was mostly correct. Specifically, we hypothesized that our calculations would allow us to get the ball through 5 hoops and the cup. Our experiment allowed us to pass through 5 hoops, but not the cup. Taking everything into account, our hypothesis was correct. By calculating where the ball would be at different positions (horizontal and vertical displacement), we were able to accurately place the hoops at correct locations for the ball to pass through. For example, we calculated that at a horizontal displacement of 99.6 cm, the ball would be 136.3 cm above the ground. In reality, it was 137.4 cm above the ground. With only a 0.88% error, it is clear that the method of solving for position using physics equations will successfully yield an accurate prediction of where the ball will be.
 * Conclusion:**

Experimental errors resulted from the fact that while on paper our calculations are accurate, real life has more factors and forces that can affect the trajectory of the ball. The first source of error was the 3rd dimension, or the Z-axis. This meant that, while we may have positioned the hoop perfectly along the x and y axises, the z axis could cause our ball to miss. A z-axis value was something that we were unable to calculate for with our given information. The second source of error was the air resistance. A projectile by definition has no other forces acting upon it besides gravity. However, in testing our experiment, it was clear that this wasn't the case. Air resistance, along with a breeze from an air vent caused our projectile to not follow a perfectly predicted course. Finally, the launcher itself was a major source of error. Our calculations were based of an initial velocity that fluctuates. Since a spring heats up, it becomes more flexible and does not fire the projectile as far with continued use. This caused our projectile to, literally, fall short of its intended target.

In order to address the error, we could use a heavier projectile to reduce the effect of air resistance. Also, by using a "cooling system" such as a small fan behind the launcher, we could negate the effect of heat increasing flexibility of the spring and reducing initial velocity. One relevant real-life application of this is in the army. Snipers, for example, need to accurately shoot their bullet to hit the intended target. Factors such as wind, air resistance, gravity, and initial velocity all affect where the bullet will go, and the gunner must take these things into account in a split second. Especially out in environments where the wind speed is constantly fluctuating, this can be a difficult task. They overcome the error with extremely high-powered guns, which shoot the projectile with enough force to reduce the impact of outside factors.

=Activity: Vector Mapping=


 * Graphical & Analytical analysis:**
 * Graphical Results: 3.5 meters @ 197 degrees**


 * Analytical Results: 4.25 meters @ 196.3 degrees**


 * Percent Error calculations:**
 * Shown above are our percent error calculations for the analytical & graphical methods. Clearly, the analytical method yielded a much more accurate answer, in comparison to the graphical method. Our analytical percent error was 2.4%, while the graphical was 15.7%. This is because the graphical method leaves much more room for human error, since you must physically draw the vectors. Those were errors on our part in drawing. The error in the analytical method is only from not having perfectly accurate measurements.**

=Notes 10/14=

=Notes 10/21=

=Notes 10/26=