Ch2_ChiavelliJ

= = = Chapter 2 Wiki Page = toc

Class Notes: Constant Speed/Motion/Ticker Tape Diagrams
Constant speed means the speed is always the same value Average speed is distance/time Instantaneous speed is the speed at a specific point in time (ex: what the speedometer reads)

**Types of Motion** - at rest - constant speed- not changing - increasing speed} - decreasing speed} acceleration (changing speed) = = **//Motion Diagrams//** v=0 a=0 ->v->v->v a=0 ->v-->v---> +a-> --->v-->v->v -a <- same applies vertically and in opposite directions* = = Ticker Tape Diagrams | . . . . . . . .| |... . . . . . . . .|

Offer precise measurements and clear display of motion, but no source of direction Signs are arbitrary (x-axis/y-axis)

= Lesson 1 (a,b,c,d, //and// e) Notes = a. ** Mechanics ** - the study of the motion of objects

** Kinematics ** - the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations – to represent real-world applications

b. 1. As learned in class, the use of words (distance, displacement, up, down, left, right) to communicate motion is integral in physics, and that each word has a specific definition.

** Scalars ** - quantities that are fully described by a magnitude (or numerical value) alone

** Vectors ** - quantities that are fully described by both a magnitude and direction

c. ** Distance ** - is a //scalar// quantity that refers to “how much ground an object has covered” throughout motion

** Displacement ** - is a //vector// quantity that refers to “how far out of place an object is”, or the object’s overall change in position

2. In class, I was not yet confident concerning the variations between distance and displacement and how to determine them. The concept became clearer as I sifted through the examples on the webpage and was able to better grasp the material. I had difficulty differentiating between both terms and understanding the distinct qualities and key points that define them. The reading clarified my issue and enabled me to better comprehend the differences and structure of the terms.

Displacement (vector qt.) is **direction aware**, while distance (scalar qt.) is unaffected by direction of motion

Displacement takes changes in direction into account Heading in the opposite direction will begin to cancel any displacement already in effect

d. ** Speed ** - a scalar quantity, the rate at which an object covers distance, or “how fast an object is moving”

** Velocity ** - a vector quantity defined as “the rate at which an object changes position”, is **direction aware**, and requires proper direction information

Ex: A person takes 5 steps forward, then 5 steps back. V= 0 ft/s

** Average Speed= Distance Traveled/Time of Travel **

** Average Velocity= Displacement (Δ distance)/time elapsed **

3. What is the importance of average velocity in general, and in terms of direction? What information can be learned from average velocity?

** Instantaneous Speed ** - the speed at any given instant in time

** Average Speed ** - the average of all instantaneous speeds (speedometer example)

4. I read about instantaneous velocity, which was not covered in class, but I had developed a brief background while completing the laboratory.

** Acceleration **

A vector quantity that is described as the rate at which an object changes its velocity, so an object is accelerating if its velocity is changing

** Constant Acceleration ** - when an object changes its velocity by the same amount each second --- an object with a **//constant//** velocity is not accelerating

As an object falls, it will typically accelerate Free-falling object: accelerates at constant rate, covering differing distances each second

*Notice Pattern: square relationship- the total distance covered after 2s is 4 times the total distance covered after 1s, etc….

** Avg. Acceleration ** = Δvelocity/time OR = //vf –vi///time

Common Acceleration Values: m/s/s, mi/h/s, km/h/s, m/s­2

Direction of Acceleration Vector Depends On: 1. Is object speeding up/slowing down?

2. Is object moving in (+)/(-) direction?

*If an object is slowing down, its acceleration is the opposite of its motion.

*If an object is speeding up, the acceleration is the same direction as the velocity

Positive v. Negative Acceleration

1. I had previously understood average acceleration and constant acceleration, and that fact that constant velocity is not accelerating.

2. The reading further clarified average acceleration, and the direction of acceleration in a vector diagram. The reading aided in simplifying the concept and making it easier to understand.

3. I am still not completely confident with free-falling acceleration and the square relationship in the table above discussed in the reading. How does the square relationship affect acceleration in the table above?

4. Most of the information had been previously, but briefly discussed in class, except for free-falling acceleration and the square relationship present in some acceleration concepts.

= Lab: A Crash Course in Velocity Part (1) =
 * Objective: ** What is the speed of a Constant Motion Vehicle (CMV)?

Hypothesis: The CMV is moving at 17cm/s.
 * How fast does a CMV move? **

Hypothesis: Distances can be measured to a tenth of the unit being used.
 * How precisely can distances be measured? **

Hypothesis: A position-time graph portrays the distance and time of a desired constant motion experiment, ultimately showing average velocity.
 * What does a position-time graph tell you? **

__ Data Table of Time and Position of CMV __






 * Discussion Questions **

Slope represents the change in y over the change in x, which can represent the change in position over the change in time. The slope of the trend line is an average of distance and time, and therefore, the slope of the linear function that the position-time graph portrays is equal to the average velocity.
 * Why is the slope of the position-time graph equivalent to average velocity? **

Instantaneous velocity describes the measured motion of an object from one point to another, in a given period of time. However, average velocity is defined as displacement over the change in time over a general period of time. The constant motion of the CMV enables us to calculate a trend line, and ultimately the average velocity of the vehicle. We assume that the CMV is consistently in constant motion at a controlled velocity, and not slowing down or speeding up irrelevantly.
 * Why is it average velocity and not instantaneous velocity? What assumptions are we making? **

It is appropriate to the set the y-intercept to zero because the CMV was at first motionless at a specific position, where the measurements are started from. Since no distance has been changed or time elapsed, it is correct to assume the y-intercept is zero.
 * Why was it okay to set the y-intercept equal to zero? **

The R2 value can be defined as the percentage of points described by the trend line, and a testament to how precise the line of best fit is as compared to 100% accuracy.
 * What is the meaning of the R2 value? **

The graph of the slower CMV would have a smaller and more gradual slope, which would lie beneath the slope of the faster CMV.
 * If you were to add the graph of another CMV that moved more slowly on the same axes as your current graph, how would you expect it to lie relative to yours? **

**Conclusion** The results of the laboratory concluded that the CMV was moving at 29.06cm/s. My hypothesis of 17.00cm/s was not accurate, as the CMV was about 12.06cm/s faster than I expected it to be. During the laboratory, the estimation of hundredths when measuring between spark dots on the tape could have been a primary source of error. The bulkiness of the measuring device could have also been a factor, with a tendency to slip or shift on the spark tape, therefore tainting calculations. Also, if the spark tape poorly captures points of constant speed, it would diminish the accuracy of the results. Lastly, if the batteries of the CMV were faltering, it would have consequently created errors in the measurements of the lab. The use of fresh batteries could avert a chance of power reduction, as well as the implementation of flat measuring tape or a thin ruler to better ensure precise measurements. To establish consistency in terms of results, it would also be wise to run the procedure multiple times as a preventative measure against erroneous results.

Class Notes: Graph Shapes (At Rest/Constant Speed)


Velocity vs. time graphs convey the most information: speed, acceleration=slope, and displacement=area

= Lesson 2 (a,b,c) = ** Ticker Tape- ** a span of dots that represent an object’s motion, the distance between dots is reflective of the object’s position change during that time gap

- Wide distance=fast movement during that interval - Short distance=slow movement during that interval Portrays constant velocity or acceleration (as per dot spacing)

** Vector Diagrams ** - to show the direction and relative magnitude of a vector quantity using an arrow (size matters)- velocity vector

*** “When an object is slowing down, the direction of the acceleration is in the opposite direction of the object’s motion”


 * 1) After completing the CMV lab, involving a ticker tape analysis and further class discussion, I was confident in my understanding of its ability as a tool and how to make conclusions by scanning it. In class, I grasped the concept certain aspects of vector diagrams such as the size of the arrow and its importance, as well as direction.
 * 2) In class, I was not fully confident concerning the relationship between velocity and acceleration as it concerns to vector diagrams. However, learning from an animation and reading the “rule of thumb” was able to lead me on the right path to comprehension.
 * 3) What is the best way to draw and interpret vector diagrams? Can I perform some more practice examples?
 * 4) Everything in the reading had already been discussed in class today.

**Class Notes: The Big 5 of Kinematics**
Acceleration (a) m/s 2

V=Δd/Δt *only for constant speed averages

The rate at which velocity is changing

V=//vi// +//vf// /2 *only for average speed

a=//vf// -//vi// /Δt …also written as…

// vf // = //vi +// at Δd= ½(//vi + vf)//t Δd= //vi//t + ½ at 2 // vf 2 = vi 2 // + 2aΔd =Class Activity: Graphing Acceleration=

Graphs: At Rest: Constant Speed Fast: Constant Speed Slow:

Changing Directions:


 * Objectives:**
 * What is the difference between static and dynamic equilibrium?
 * How is “at rest” represented on a position vs. time graph? On a velocity vs. time graph? On an acceleration vs. time graph?
 * How is constant speed represented on a position vs. time graph? On a velocity vs. time graph? On an acceleration vs. time graph?
 * How are changes in direction represented on a position vs. time graph? On a velocity vs. time graph? On an acceleration vs. time graph?
 * Discussion Questions**
 * Discussion Questions**

How can you tell that there is no motion on a… On all three graphs, the line is a positively sloped linear horizontal line. However, for a velocity and acceleration vs. time graph, the line lies on the x-axis.
 * 1) position vs. time graph
 * 2) velocity vs. time graph
 * 3) acceleration vs. time graph

How can you tell that your motion is steady on a… On a position vs. time graph, steady motion is portrayed as a positively sloped linear line with points evenly spaced throughout. Steady motion on a velocity vs. time graph is displayed as a linear horizontal line close to the x-axis. On acceleration vs. time graphs, steady motion is depicted as a linear horizontal line on or about the x-axis.
 * 1) position vs. time graph
 * 2) velocity vs. time graph
 * 3) acceleration vs. time graph

How can you tell that your motion is fast vs. slow on a… On a position vs. time graph, the change in slope, as well as the spacing between coordinates depicts whether or not motion is fast or slow. The farther the distance between points, the slower the motion, while the closer the distance, the faster the motion. Aside from a slight difference in the distance between points, both velocity and acceleration vs. time graphs do little to communicate slow or fast motion.
 * 1) position vs. time graph
 * 2) velocity vs. time graph
 * 3) acceleration vs. time graph

How can you tell that you changed direction on a… On a position vs. time graph, there is a linear positively sloped line (walking away), shifting point of ups and downs, then a linear negatively sloped line (walking towards). On a velocity and acceleration time graph, the linear horizontal lines stay close to the x-axis, except for fluctuations in the center, and then a horizontal line representing the change in direction.
 * 1) position vs. time graph
 * 2) velocity vs. time graph
 * 3) acceleration vs. time graph

What are the advantages of representing motion using a… Using a position vs. time graph enables you to represent steady, fast, slow, no, or changes in motion, and well as speed. Velocity vs. time graphs can communicate a change in direction through its fluctuations, speed, displacement, and acceleration through its slope. The only advantage for using acceleration vs. time graphs, is to show a change in the direction of motion.
 * 1) position vs. time graph
 * 2) velocity vs. time graph
 * 3) acceleration vs. time graph

What are the disadvantages of representing motion using a… There are really no disadvantages to using a velocity vs. time graph, and the same basically applies to a position vs. time graph for representing constant motion. Acceleration vs. time graphs are not beneficial when representing steady, fast, slow, or no motion.
 * 1) position vs. time graph
 * 2) velocity vs. time graph
 * 3) acceleration vs. time graph


 * 1) Define the following:
 * 2) No motion- an object is at rest
 * 3) Constant speed- an object maintains a constant pace while in motion

=Class Notes: Increasing/Decreasing Speed Graphs = =Lesson 3 Notes=
 * a. **

Position v. Time Graphs

The slope of a line is reflective of velocity (+,-, curved, straight) “As the slope goes, so goes the velocity”

Curved lines= changing slope, straight lines= constant slope

The slope of a line of a position time graph is equal to the velocity of the object
 * b. **

Is used to interpret graphs and determine the value of the velocity
 * c. **

1. I had already understood the way constant and changing velocity looked on a position vs. time graph. In addition, the use of slope of a position vs. time graph line was something learned and applied in the classroom. 2. The misconception I was having concerned the visualization of distance and time on the graph, ultimately portraying the value of velocity through slope. By analyzing the first two graphs again and discussing it briefly in class, the concept became less obscure and more easily understandable. 3. What is the proper way to analyze a position vs. time graph like the examples above? What information do you need to interpret and are there any other important ideas related to reading these graphs? 4. The majority of the information was not further dissected in class today.

= Lesson 4 Notes = Velocity v. Time Graphs- another way to describe motion
 * a. **



*The slope of the line on a velocity v. time graph reveals important information about the acceleration of the object.

Whether or not velocity is positive or negative can be determined by whether the line lies in a positive or negative region of the graph

Whether or not an object is speeding up/slowing down can be determined by the numerical change in velocity. (further from x-axis= speeding up, closer=slowing down) *The slope of the line on a velocity v. time graph is the same as the acceleration of the object
 * b. **

MANY EXAMPLE DIAGRAMS OF MOTION RELATIONSHIPS
 * c. **

On a velocity vs. time graph, find the slope by picking two points, and use the slope formula, ultimately finding the acceleration
 * d. **

In a velocity v. time graph, the area between the line and axis represents the displacement of the object during a certain time period
 * e. **

Area Formulas: Rectangle- bxh Triangle- ½ bxh Trapezoid- ½ b x (h1+h2)

You can also break a trapezoid into a rectangle and triangle to make solving simpler if need be

EXAMPLE PHOTOS:

 1. This reading had not yet been covered in class, but I understood the basic fundamentals of the velocity vs. time graph, as I had performed tasks involving it on the laboratory at home. Specifically, graphs with objects moving with a positive velocity. 2-3. The majority of this reading had not yet been discussed, besides from basic graphs. When reading, the concept of acceleration being equal to the slope of a velocity vs. time graph line was not fully comprehended, but the class discussion clarified it. However, the velocity vs. time graphs concerning negative velocity and acceleration are still unclear and need further explaining. How should I analyze a velocity vs. time graph? Are there any specific steps that need to be taken? 4. This reading was not fully covered in class yet, especially the variety of graphs attributed to velocity vs. time and how to further interpret them.

= Lab: Acceleration on an Incline = 9/14/11 Lab Partner: Sammy Caspert


 * Objectives and Hypostheses: **


 * ** What does a position-time graph for increasing speeds look like? **

I predict that a position-time graph for increasing speeds will look like a curved line. The line will curve upwards and away from the origin.


 * ** What information can be found from the graph? **

The average speed, instantaneous speed, velocity, and acceleration can be determined from the graph.


 * Procedure: **


 * 1) Align track and physics textbook to form incline, which forms the necessary ramp
 * 2) Pull spark tape through spark timer on top of incline and attach with a piece of tape to the back of the dynamics cart at top of track
 * 3) Hold ticker tape and “feed” to machine for best results
 * 4) Turn on spark timer and select 10 Hz
 * 5) Hold the spark tape so that it will easily move through the machine as the dynamics cart is in motion
 * 6) Measure the distance in cm using a ruler or tape measure from each point on the spark tape from “zero” according to the start of the ticker tape
 * 7) Using an Excel spreadsheet, input the time in seconds as the x-value and distance in centimeters as the y-value


 * Increasing Down Incline Decreasing Up Incline **


 * Data: **

a) ** Interpret the equation of the line (slope, y-intercept) and the R2 value. ** The equation of the line y=Ax2+Bx can be interpreted as Δd= ½at 2 +//vi//t. The slope of the line is representative of the speed of the dynamics cart. The y-intercept is zero, which means the velocity starts at zero. The R2 value is defined as the percentage of points described by the line. It represents how close to 100% the equation of the trend line best fits the points. For both graphs of increasing and decreasing speed had R2 values of about 0.999, which is indicative of a good fit.
 * Analysis: **

b) ** Find the instantaneous speed at halfway point and at the end. (You may find this easier to do on a printed copy of the graph. Just remember to take a snapshot of it and upload to wiki when you are done.) ** Increasing Down Incline: Instantaneous Speed Halfway Point: 25.0 cm/s  Instantaneous Speed End: 33.0 cm/s

Decreasing Up Incline: Instantaneous Speed Halfway Point: 58.3 cm/s Instantaneous Speed End: 33.3 cm/s

c) ** Find the average speed for the entire trip. ** The average speed for the increasing down incline was 20.3 cm/s. The average speed for the decreasing up incline was 59.5 cm/s

The polynomial function of a line would have a steeper curve upwards away from the origin and x-axis, as it ascended down the incline. We had done a second procedure to determine the graph of the cart decreasing up the incline. It has a line that curves downwards and towards the origin.
 * Discussion Questions: **
 * What would your graph look like if the incline had been steeper? **
 * What would your graph look like if the cart had been decreasing up the incline? **

For the increasing down incline dynamics cart, the halfway point instantaneous speed is 25.0 cm/s, while the average speed is 20.3 cm/s. They are relatively close in proximity, but the instantaneous speed of the halfway point is greater. However, this makes sense as the average speed takes in a greater number of points into account, as it is the total change in distance over the total change in time. For the decreasing up incline, the halfway point instantaneous speed is 58.3 cm/s, while the average speed was 59.5 cm/s. These figures are very similar, which is indicative of the fact that the instantaneous speed of the center of the function is relatively equivalent to the average speed.
 * Compare the instantaneous speed at the halfway point with the average speed of the entire trip. **

Instantaneous speed is described as the speed of an object at a specific moment in time. The slope of the tangent line is the instantaneous speed, and can be measured from any point on the line. It isolates a particular moment in time, therefore conveying the instantaneous speed.
 * Explain why the instantaneous speed is the slope of the tangent line. In other words, why does this make sense? **


 * Draw a v-t graph of the motion of the cart. Be as quantitative as possible. **



**Conclusion:** With respect to my hypotheses concerning the shape of a position vs. time graph for increasing speeds, I was correct in describing a curved upwards line moving away from the origin. The shape of a position vs. time graph for decreasing speeds was also determined during the laboratory. In terms of the fit of our trend lines, they were very precise. The results for instantaneous speed for both increasing and decreasing go as follows: 25.0 cm/s (halfway increasing down incline), 33.0 cm/s (end increasing down incline), 58.3 cm/s (halfway decreasing up incline), and 33.3 cm/s (end decreasing up incline). The average speed for the increasing down incline was 20.3 cm/s and the average speed for the decreasing up incline was 59.5 cm/s. Throughout the laboratory, we used the graph to find average speed, instantaneous speed, and velocity, which meant my hypothesis was generally accurate. A first source of error could be that the wheels of the cart were faulty and would then lead to erroneous results, in which case new and sound wheels should be installed for better results. Secondly, the spark timer is not always exact when determining the start of motion of an object. The cluster of dots in the beginning of the tickercould be misleading in general and when measuring. We used both a ruler and measuring tape to ensure the best possible measurements of our ability. However, a finer measuring tool would be more efficient in making more precise measurements. Another option is to record the measurements a number of time and take the average of the measurements to avert any outliars.

=Lab: A Crash Course in Velocity (Part II)= 9/21/11 Lab Partners: Sammy Caspert, Andrew Chung, and Amanda Fava

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The purpose of the laboratory is to determine at what point two CMV's will crash/meet when spaced 600cm apart, as well as what point they will cross when spaced 100cm apart, but going in the same direction.



<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Procedure: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">1. Use algebraic problem solving to determine the timing and distance of both scenarios <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">2. Solve graphically to fulfill graphing requirements and compare results to algebraic method <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">3. Using a tape measure, measure the 600cm distance and place the CMV's at their respective locations <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">4. Film the crash scenario at least once <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">5. Collect data of where the CMV's collide and repeat 5 times to ensure best results <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">6. Using a tape measure, measure 100cm distance between the CMV's and place them in their respective locations according to speed <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">7. Film the catch-up scenario at least once <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">8. Collect data of where the faster CMV catches up to the slower CMV and repeat 5 times to ensure best results <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">9. Graph the necessary velocity vs. time graph <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">10. Calculate percent error and difference <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">11. Write an informative conclusion



<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Percent Error/Difference Calculations

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**Discussion Questions** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">If the speeds were exactly equal and coming from opposite ends (one at 0cm and the other at 600cm), the cars would meet at 300cm. If the cars were traveling the same speed and starting at different locations (one at 0cm and the other at 100cm), they would never meet. **<span style="font-family: 'Times New Roman',Times,serif;">2. Sketch position-time graphs to represent the catching up and crashing situations. Show the point where they are at the same place at the same time. ** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">See graphs pictured above **<span style="font-family: 'Times New Roman',Times,serif;">3. Sketch velocity-time graphs to represent the catching up situation. Is there any way to find the points when they are at the same place at the same time? ** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">It is not possible to interpret the location (points) where the CMV's are in the same place at the same time from analyzing this velocity vs. time graph. This graph does not portray position well in relativity especially to the fact that the Blue CMV is starting at 0, while the Yellow CMV is really beginning 100cm away from 0 in the positive direction.
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">1. Where would the cars meet if their speeds were exactly equal? **

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Videos:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Collision <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">media type="file" key="good vid crash.mov" width="300" height="300"

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Catch-Up <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">media type="file" key="good catch up.mov" width="300" height="300"

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**Conclusion** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The results we got were solved for algebraically and graphically, with regard to when the Blue and Yellow CMV would collide, as well as when the faster Blue CMV would "pass" the slower Yellow CMV. The Blue CMV had a velocity of 64.62 cm/s and the Yellow CMV had a velocity of 29.06 cm/s. In 6.4s (t), the two cars collided at 186.1cm (d) from 0cm. This was supported by our algebra, and experimental data reflected in the position vs. time graph. When positioned 100cm apart, the faster Blue CMV passed the slower Yellow CMV after 2.81s (t), and at 181.72cm (d) from 0cm. These results are also validated by the position vs. time graph and original algebra. Analyzing the substance of our results as a whole, the percent error for the collision experiment was 2.26%, while for the catch-up was .01%. Both figures represent accurate results. As far as percent difference goes, the values pictured above show an overall portrayal of very good precision. A major source of error was the fact that some of the CMV's would turn as they continued to speed along. This would throw off the logistics of the experiment, requiring a decent estimation, and ultimately degrading results. To prevent this, we lined the measuring tape along the wall as to prevent such swerving from occurring. To improve the lab, we could use a long track equipped with the proper measuring device. This would avert any such swerving. Also, the status of the CMV battery from the first experiment could be different, and therefore could affect the velocity of the CMV. To address this source of error, using a fresh battery each time would avert differences in power, in order to ensure a consistent velocity. Another source of error could be the accuracy and precision of the results of the counterpart CMV lab. When measuring distance in that lab, the measuring tape could naturally shift when meticulously measuring the distance of the spark tape dots. It would be beneficial to re-check and confirm the results from that lab as to not also taint the results of this lab. If I were to redo the lab, I would take these issues into consideration as to aim for better and more sound results.

=<span style="font-family: 'Times New Roman',Times,serif;">Egg Drop Project Results =



<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The results of the egg drop test were successful, as my design kept the egg unharmed from the impact of an 8.5m fall. As expected, the parachute created air resistance, ultimately slowing down the free-fall of the egg and the contraption. The device landed on its cushioned underbelly to protect the bottom of the egg, and predictably jerked to one of its sides. The protective wings on the sides of the contraption served as a buffer to absorb the impact of stopping. Rubber bands successfully kept the spiked cover in place, which also secured the parachute in place.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**Calculations for Acceleration**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The device successfully brought the egg to safety from the fall and impact for three key reasons. First, the parachute used created air resistance, which slowed down the contraption. The schematics of the parachute also prevented the device from flipping over in midair or becoming tangled, when dropped properly. Secondly, the design of the device was based around the concept of securing the egg in a way that would not be affected by the fall or impact. Lastly, the buffers and cushions strategically attached to the main component were employed with the purpose of protecting the egg against any location of impact. The top of the device had spiked buffers, while the sides and bottom were lined with thick protective wads of straws.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">If I were to complete the project again, I might explore a design that would be lighter in mass. I would start by cutting down the unnecessary use of straws, and then proceed to use tape, instead of hot glue to hold the device together. Also, I would make the parachute as big as possible within the confines of the limitations and construct it with better precision to maximize performance. Lastly, I would attempt to implement different techniques of cushioning that would weigh less.

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Quantitative Graph Interpretation =

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=Lesson 5 Notes= Summary Method 1

Free Fall

Gravity is the only force that acts upon a free-falling object, air resistance is irrelevant, and acceleration of 9.8m/s2 downwards dictates the basic specifications for free fall.

An object in free fall is only being acted upon by the force of gravity

- Not encountered by air resistance - Accelerate downwards at a rate of 9.8m/s2

A ticker tape diagram would convey acceleration (the free-falling object is accelerating downwards)

The acceleration of gravity is a critical figure also represented by g, but for the purpose of conceptual analysis can be approximated.

** Acceleration of Gravity ** = 9.8m/s2 = g

Commonly approximated to: 10m/s2

** g = 9.8 m/s/s, downward ( ~ 10 m/s/s, downward) **

Interpreting position and velocity v. time graphs for free fall, is similar to the analysis previously learned and applies the same concepts.

Position v. Time Graph- Free-Falling Object

Velocity v. Time Graph- Free-Falling Object

The formula provided below can be used to find the velocity of a falling object, after a specified amount of time. v//f// = g * t

Formula used to determine the velocity of a falling object, after a certain amount of time

A more massive object does not accelerate faster than a less massive object.

Although air resistance can have an effect

= Falling Object Lab = 10/4/11 Partner: Sammy Caspert

The purpose of this laboratory is to determine the acceleration of gravity.

**Hypothesis:**I predict that the acceleration of gravity will be <span style="font-family: 'Times New Roman',Times,serif; font-size: 15px;"> 980cm/s^2 (9.8m/s^2 ). This is the known and accepted value, and therefore should reflect our results. I believe the velocity vs. time graph will depict a negatively-sloped linear line accelerating/ speeding up away from the origin if velocity is considered negative. If velocity is considered positive, then it will be a positively-sloped linear line accelerating/speeding up away from the origin. I predict "g" will be equal to the value of the slope of the line on the v-t graph.

**Data:**


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Sample Calculations: **


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Graphs: **



<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Velocity vs. Time Graph: The shape of the v-t graph is a positively sloped linear line representing speeding up away from the origin. The equation of the trend line is y=891.38x-26.952 (y=mx+b). The slope or //m// of the equation represents the acceleration of gravity on a free-falling object. Ideal results would yield a slope of 980.00 cm/s^2 (9.8m/s^2), but our results were not exact. In reality, our percent error was 9.08%. The y-intercept or //b// of the trend line is -26.952. The reason the y-intercept is not set to zero is because our initial velocity is not guaranteed to be zero. This is due to the fact that the "dropping phase" of the laboratory is not controlled enough to interpret the y-intercept as zero. The R^2 value is defined as the percentage of points described by the line. It represents how close to 100% the equation of the trend line best fits the points. The R^2 value of the graph is 0.997, which represents a good fit.
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Analysis: **

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Position vs. Time Graph: The shape of the x-t graph is positively curved line representing acceleration or speeding up away from the origin. The equation of the trend line is y=(445.18x^2)-27.085 (y=Ax+Bx). The A of the equation is indicative of //half// the acceleration of the object in free fall. This value is half of the slope of the v-t graph. This equation is reflective of the kinematics equation: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">change in distance= (1/2at^2)+(initial velocity)t. The B of the equation represents initial velocity and is also comparable to the kinematics equation previously stated. As stated above, the R^2 value of the trend line is defined as the percentage of points which represents how close to 100% the equation of the trend line best fits the points.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**Discussion Questions:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**//1. Does the shape of your v-t graph agree with the expected graph? Why or why not?//** <span style="font-family: 'Times New Roman',Times,serif;">The shape of the v-t graph agrees with my expected graph. My prediction and the actual graph description both are a linear lines representing speeding up/acceleration in the positive region of the graph. I took the signs of velocity into account in my hypothesis, but for the sake of the laboratory it was considered positive.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**//2. Does the shape of your x-t graph agree with the expected graph? Why or why not?//** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Yes, my hypothesis agrees with the shape of the x-t graph. My prediction and the actual graph are both positively curved lines representing acceleration or speeding up away from the origin.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**//3. How do your results compare to that of the class? (using % difference to discuss quantitatively)//** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The percent difference between the class average experimental value and our individual experimental value is 6.20%. So, our value for acceleration of gravity was 6.20% more than the class average.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**//4. Did the object accelerate uniformly?//** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Yes, because starting from rest, the object speeds up in equal intervals, from equal intervals of time. To put this assertion in perspective, the percent error for our v-t trend line was 9.08%. So, the object did in fact accelerate very uniformly in terms of experimental data and taking into account the possible sources of error.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**//5. What factor(s) would cause acceleration due to gravity to be higher than it should be? Lower than it should be?//** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">The acceleration due to gravity could be higher than it should be if the spark timer made a mistake or was fed kinked spark tape, therefore throwing off the spark dot measurements and the acceleration of gravity result. The acceleration due to gravity could be lower than it should be if the spark tape experience increased friction when being pulled through the spark timer. This "dragging" would slow down the acceleration of the object in free fall.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**Conclusion** <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Our first hypothesis for the acceleration of gravity was somewhat correct. The value produced was 891.38cm/s^2 or 8.91m/s^2. When compared to the known and predicted value of 980cm/s^2 or 9.80m/s^2 the results are good. The percent error of our experimental results was 9.08% in relation to the accepted value of the acceleration of gravity (g). The second component of our hypothesis was also correct. The actual graph displayed a linear line accelerating or speeding up away from the origin and we predicted the same if the velocity was considered positive. We were also successful in determining "g" from the slope of the v-t graph. This is evident from out acceleration of gravity result of 891.38cm/s^2 or 8.91m/s^2. One possible source of error could have been if the spark tape experienced friction when passing through the spark timer. This would slow down the acceleration of the object in free fall. To address this source of error, there should be no kinks or dents, but a spark tape that is as smooth as possible. As far as reducing the friction, there is no concrete way of doing so, besides little things like written above, as well as "feeding the tape" and not letting it slide through and uncurl itself. Another source of error could be the shifting of the measuring tape when measuring the dots of the spark tape. In order to prevent this occurrence, it pays to tape the spark tape and measuring tape along side one another with precision and possibly even use a ruler to meticulously glide across points and measure. Lastly, the initial velocity of the object in free fall is not guaranteed to be zero. However, the class does not have an apparatus that would make it possible to do so. So, the timing of the spark timer and human reflex must be relied on for sound results.

Class Notes: Free Fall
Example Free Fall Problems:

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