Ch2_GrunfeldM

=**Chapter 2** =

toc

**Constant Speed**


**Lesson 1: Describing Motion with Words**

 * **What (specifically) did you read that you understand well? Describe at least 2 items fully.** In my reading, I was able to make connections to concepts we learned in class. I really understand the concept of displacement and distance, as well as vectors and scalars. By their definitions, vectors distinguish themselves from scalars because they need a direction involved. Before even reading about distance and displacement, I immediately thought of these concepts from class and realized that they can be directly paralleled to the concept of vectors and scalars. Displacement is a vector quantity because it involves direction, while distance is scalar because it does not. [[image:Screen_shot_2011-09-19_at_10.50.07_AM.png width="114" height="138" align="right"]]
 * **What (specifically) did you read that made you feel little confused/unclear/shaky, but further reading helped to clarify? Describe the misconception(s) you were having as well as your new understanding.** At first, I was a little confused about the difference between speed and velocity, but then reading further helped clarify my misunderstanding. Now I realize that scalar and vector quantities come into play again, and I realize the difference in formula for solving for both speed and velocity.
 * **What (specifically) did you read that you don’t understand? Please word these in the form of questions.** Because there is an instantaneous speed that the average speed is based off, is there an instantaneous velocity?
 * **What (specifically) did you read that you thought was pretty interesting, that you didn't know before, or can easily apply to your every day life?** I found it to be interesting that I was able to make up tricks in my mind to help me better remember some concepts. Specifically, I am able to remember that speed is a scalar quantity because they both start with S, and velocity is vector because they both start with V. This connection helps me and shows that if I continue to think and make connections like these, I can apply physics to everyday life.

**Lesson 2: Describing Motion with Diagrams**

 * **What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.** In our class discussion, we talked about kinematics, which describes the motion of objects. This concept provides a visual understanding for people to better absorb the subject matter. The introduction of Lesson 2 stressed the importance of using diagrams for visual aid, which I already knew from today in class. I also knew about the vector diagrams, which use arrows to imply the direction, magnitude, and velocity/acceleration of an object.
 * **What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.** Today in class, I was confused as to why a large distance between dots meant the object was moving fast, rather than the other way around. The reading helped to clarify my confusion and it makes sense now, because the object's position was changing more rapidly, which is what a ticker tape diagram represents.


 * **What (specifically) did you read that you still don’t understand? Please word these in the form of a question.** After reading through Lesson 2, I understand everything and do not need further clarification.
 * **What (specifically) did you read that was not gone over during class today?** In class, we only briefly touched upon the ticker tape diagrams, so the details were not gone over during class. I did not know that the distance between the dots on a ticker tape diagram represents the object's position change during that time interval, I only thought about speed.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 130%;">**Lab: The Speed of a Constant Motion Vehicle (CMV)**
<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Maxx Grunfeld and Tim Hwang** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**9/9/11**

<span style="font-family: 'Comic Sans MS',cursive;"> **Purpose:** The purpose of this lab is to figure out how to calculate the distance traveled by an object, as well as the displacement, the speed, and the velocity of an object at constant speed. We will also generate and interpret the equation of the lines from the experimental data.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**<range type="comment" id="485775">Objectives:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%; line-height: 23px;">**1) How precisely can you measure distances with a meterstick?** Using a meterstick, I can measure the distance as small as to the nearest millimeter, and as large as the standard meter. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%; line-height: 23px;"> **2) How fast does your CMV move?** Our CMV will move about 2 seconds per foot (30.48 cm) because when we turned it on, it seemed about 2 seconds for it to move past the length of my binder, which appears to be about 1 foot. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%; line-height: 23px;"> **3) What information can you get from a position-time graph?** You can get the position of the CMV at each time, which may be relative to the average speed.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%; line-height: 27px;">**Materials:** spark timer + spark tape, meterstick, masking tape, CMV

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%; line-height: 27px;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: large; line-height: 27px;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Discussion Questions:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**1. Why is the slope of the position-time graph equivalent to average velocity?** In the position-time graph, the y-axis represents distance and the x-axis represents time. The equation to find the slope of a line is the change in y over the change in x. The equation to find the average velocity of an object is the change in distance over the change in time. Therefore, because the y-axis of this graph represents distance and the x-axis represents time, the equations of slope and average velocity are equivalent. <span style="font-family: 'Comic Sans MS',cursive;">** 2. Why is it average velocity and not instantaneous velocity? ** It is average velocity and not instantaneous velocity because we observed how fast the car was moving in a period of time, instead of looking at how fast the car was moving at a certain instant. <span style="font-family: 'Comic Sans MS',cursive;">** 3. Why was it okay to set the y-intercept equal to zero? ** It was okay to set the y-intercept equal to zero because in a graph, the y-intercept is the starting point. We started our CMV at zero seconds, which caused it to start at a position of 0 cm as well. <span style="font-family: 'Comic Sans MS',cursive;">** 4. What is the meaning of the R2 value? ** The meaning of the R2 value of the graph depicts how far off we were from getting our line exactly accurate to constant speed. Because we had an R2 value of 0.99477, that is the percentage of how close our line was to being constant. <span style="font-family: 'Comic Sans MS',cursive;">** 5. 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? ** It would have a smaller slope and therefore the line would be flatter. My current graph would be steeper because for each tenth of a second, our car traveled a greater distance.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">**Conclusion:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Our results showed that our car traveled at an average velocity of 58.021 cm/s. My hypothesis was wrong because our car traveled much faster than I expected. I predicted that it would travel about 2 seconds per foot (30.48 cm), but our car traveled about a foot in 0.5 seconds. Some sources of error may have contributed to inaccuracies of this experiment. If the surface wasn't perfectly level, it would have caused the car to either slow down or speed up at certain times. The meterstick could have shifted as well, and our estimations could have been off. If we had to redo this lab, I would minimize several of these issues by using measuring tape instead of a meterstick because it is flat and easier to get the exact line, compared to a meterstick which is raised and off the ground, and therefore harder to tell. I would also use the last ten dots from the ticker tape, instead of having to guess when the speed was finally constant.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Graph Shapes (At rest and Constant speed)
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<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">**Lab Representing Constant Motion**
<span style="font-family: 'Comic Sans MS',cursive;">**Constant Fast** <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;">**Constant Slow** <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;">At Rest <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;">** Discussion Questions **
 * 1) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**How can you tell that there is no motion on a…**
 * 2) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**position vs. time graph –** the line will be completely straight
 * 3) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**velocity vs. time graph** – the line will be completely straight
 * 4) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**acceleration vs. time graph** – the line will be completely straight


 * 1) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**How can you tell that your motion is steady on a…**
 * 2) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**position vs. time graph** – the line will be proportional and going up/down at a constant speed
 * 3) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**velocity vs. time graph –** dots from both lines are generally around 0 because it is moving at a constant speed
 * 4) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**acceleration vs. time graph** – dots from both lines are generally around 0 because it is moving at a constant speed


 * 1) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**How can you tell that your motion is fast vs. slow on a…**
 * 2) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**position vs. time graph** – the line will have a steeper slope with faster motion
 * 3) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**velocity vs. time graph** – the dots of the line of a slower motion will be higher than the dots of the line of a faster motion
 * 4) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**acceleration vs. time graph** – there will be more erraticism or inconsistency at the beginning of the line of a faster motion


 * 1) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**How can you tell that you changed direction on a…**
 * 2) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**position vs. time graph** – the line will go in an opposite way
 * 3) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**velocity vs. time graph –** the line will be either positive or negative (positive = away, negative = towards)
 * 4) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**acceleration vs. time graph** – the erraticism will also start on opposite sides


 * 1) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**What are the advantages of representing motion using a…**
 * 2) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**position vs. time graph** – easy to see the slope/shape of the line
 * 3) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**velocity vs. time graph** – you can see a positive or negative velocity
 * 4) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**acceleration vs. time graph** – you can also see an increase or decrease in velocity


 * 1) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**What are the disadvantages of representing motion using a…**
 * 2) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**position vs. time graph** – it is not easy to determine the speed of the object, only its relative position
 * 3) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**velocity vs. time graph** – you do not see the slope of a line
 * 4) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**acceleration vs. time graph** – you do not see the slope of a line


 * 1) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**Define the following:**
 * 2) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**No motion** – the object is at rest and therefore not moving
 * 3) <span style="color: black; font-family: 'Comic Sans MS',cursive;">**Constant speed** – the object is traveling at the same speed the entire time

<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">**Lesson 2: Describing Motion with Words (1e)**

 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.** From our class discussion, I already knew the formula for calculating acceleration, and I understood well how to apply it to certain problems, as done with the homework. Also, I was familiar with the fact that acceleration is a vector quantity and it must need direction indicated with it. I was aware that the indication of direction, same with as velocity, is shown with a + or -, representing whether it is traveling towards or away from you. Also, when an object is speeding up, the +/- sign will be the same as its velocity, but if it is slowing down it will be the opposite.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.** At first, I was a little confused about how an object traveling at constant speed could still be accelerating. The reading helped clarify and it makes sense to me now. I realized that basically all objects traveling at constant speed are accelerating, unless it is at rest, no matter if the speed is increasing or decreasing. However, not all accelerations must be at constant speed. Objects traveling at constant speed (excluding objects at rest) are changing velocity, and therefore accelerating.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you still don’t understand? Please word these in the form of a question.** Everything is clear and I understand the material.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that was not gone over during class today?** In class today, we did not discuss free-falling objects. Therefore, we did not discuss the square relationship between the distance traveled and the total time traveled when an object starts at rest and moves with constant acceleration.

<span style="font-family: 'Comic Sans MS',cursive;">The Big 5
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<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">**Acceleration Graphs**
<span style="font-family: 'Comic Sans MS',cursive;">**Increasing Towards and Away** <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;">**Decreasing Towards and Away** <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;">Increasing and Decreasing Speed Graphs
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<span style="font-family: 'Comic Sans MS',cursive; font-size: 16px;">**Lesson 3: Describing Motion with Position v. Time Graphs**
<span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;">I realized that any object that has a changing velocity, instead of being constant, means that the object is accelerating.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.** I have already known from our class discussion, as well as math courses in the past, a lot about slope. I am very familiar with the equation and solving for the slope of two points, which was a significant part of lesson 3c. Additionally, from our class discussion, I learned how to describe position, velocity, and acceleration-time graphs by looking at the slope or curve, and I was able to accurately describe the examples given in the reading.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.** I was aware of the definitions about both velocity and acceleration, and I knew that they had to do with one another, but before the reading I was a little bit confused about their relationship. This graph shows the difference between an object moving at constant motion and an object that is accelerating:
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you still don’t understand? Please word these in the form of a question.** Everything is clear and I understand the material.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that was not gone over during class today?** In class, we did not discuss determining the slope of a line. However, I have already been familiar with this for a while, and the reading was merely a review.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 16px;">**Lesson 4: Describing Motion with Velocity v. Time Graphs**

 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.** From our class discussion, I really understood the significance of slope on a v-t graph. I could also relate the shape of the graph to the motion of the object, as we did in class.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.** At first, I was a little confused about determining whether the line on a v-t graph was positive or negative, and increasing or decreasing acceleration, but the reading helped clarify. When the line is above 0, it has a positive velocity. It could still be decreasing acceleration when it is above 0, but that is determined by what direction the line is moving.
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that you still don’t understand? Please word these in the form of a question.** Is the + or - in acceleration based on whether the object is moving towards or away from you?
 * <span style="font-family: 'Comic Sans MS',cursive;">**What (specifically) did you read that was not gone over during class today?** We did not learn how to determine the area of a v-t graph. However, it is simple because all you have to do is figure out what shape your graph makes (rectangle, triangle, trapezoid, etc.) and solve using the area formula for that particular shape.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**9/15/11** <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 17px;">**Lab: Acceleration Graphs of Cart on Incline**
<span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**Maxx Grunfeld and Tim Hwang** <span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**9/15/11**

<span style="font-family: 'Comic Sans MS',cursive;">**Purpose:** The purpose of this lab is to investigate and make a claim about the straight-line motion of an object in different situations, including increasing and decreasing horizontal motion. We will calculate the constant velocity, average velocity, or constant acceleration of the object. We will also interpret the meaning of the slope of the graph, and generate and interpret the equations of the lines from the experimental data.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**<range type="comment" id="485775">Objectives:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**1) What does a position-time graph for increasing speed look like?** The graph will have a line going from left to right, rising, with a positive and steep slope. <span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**2) What information can be found from the graph?** From the graph, you can find the position of the cart at each time, as well as the velocity.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**Materials:** spark tape, spark timer, track, dynamics cart, ruler/meter/measuring tape

<span style="font-family: 'Comic Sans MS',cursive;">** Procedure :** <span style="font-family: 'Comic Sans MS',cursive;">1) Set the track on a textbook to give an incline. Place the spark timer at the top of the incline and tape the spark tape from the timer onto the cart. <span style="font-family: 'Comic Sans MS',cursive;">2) Release the cart to test the acceleration. <span style="font-family: 'Comic Sans MS',cursive;">3) Measure the distance between 10 dots on the spark tape and record in Excel. <span style="font-family: 'Comic Sans MS',cursive;">4) Move the timer to the bottom of the incline and push the cart upwards, attached to the ticker tape. <span style="font-family: 'Comic Sans MS',cursive;">5) Measure the distance between 10 dots on the spark tape and record in Excel.

<span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: large;">

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<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Analysis:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**a) Interpret the equation of the line (slope, y-intercept) and the R2 value.** The equation of our line was y = 13.114x2 + 24.279x. 24.279 was our initial velocity, and 13.114 x 2 is our acceleration because alone, it is equal to 1/2 acceleration. When our line was a linear trend, the R2 value was around 96%, but when we changed it to a polynomial trend line, it became more accurate and increased to around 99%. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**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.)** Halfway point: 45 cm/s. Endpoint: 50 cm/s. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**c) Find the average speed for the entire trip.** The average speed is 37.5 cm/s.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**Discussion Questions:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">**1. What would your graph look like if the incline had been steeper?** The curve would have been steeper as well because it would most likely accelerate faster. <span style="font-family: 'Comic Sans MS',cursive;">** 2. What would your graph look like if the cart had been decreasing up the incline? ** The curve goes in the opposite direction and is flatter than the other curve. <span style="font-family: 'Comic Sans MS',cursive;">** 3. Compare the instantaneous speed at the halfway point with the average speed of the entire trip. ** The instantaneous speed at the halfway point is 36 cm/s, while the average speed of the entire trip is 37.5. If the speed at the halfway point is so close to the average speed, and the endpoint had a much higher speed at about 100 cm/s, the cart must have accelerating increasingly faster at a very high rate. <span style="font-family: 'Comic Sans MS',cursive;">** 4. Explain why the instantaneous speed is the slope of the tangent line. In other words, why does this make sense? ** <span style="font-family: 'Comic Sans MS',cursive;">This makes sense because the slope always represents the speed at that instant, as does the instantaneous speed. The curve of the graph prevents it from having a slope, but if you draw a tangent line, you can use that slope to figure out the instantaneous speed as well. <span style="font-family: 'Comic Sans MS',cursive;">** 5. Draw a v-t graph of the motion of the cart. Be as quantitative as possible. **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: 16px;">**Conclusion:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 14px;">Our results showed that our cart traveled at an average speed of 37.5 cm/s. It had an initial velocity of 24.279, and an acceleration of 13.114. My hypothesis was right to some degree, that the curve would have a positive and steep slope; however, I didn't know how to find the slope at the time so it was more of a guess. I also didn't state in my hypothesis that we would be able to find the acceleration just by graphing, without much more calculations. The estimations of values may have accounted for some sort of error. Also, we may have started at a point at the ticker tape where the acceleration was still erratic, so I would have taken the 10 middle dots if I redid this trial.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 130%;">Lab: A Crash Course in Velocity
<span style="font-family: 'Comic Sans MS',cursive;">**Maxx Grunfeld, Tim Hwang, Ryan Luo, Max Llewellyn** <span style="font-family: 'Comic Sans MS',cursive;">**9/25/11**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Purpose:** The purpose of this lab is to determine the position where 2 CMV's will be at the same time when moving towards each other and also when moving in the same direction. We will experimentally verify mathematical predictions.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Objectives:** <span style="font-family: 'Comic Sans MS',cursive;">

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<span style="font-family: 'Comic Sans MS',cursive;">**Hypothesis:** <span style="font-family: 'Comic Sans MS',cursive;">- The CMVs will meet at 211.01 cm in 6.67 seconds (6.67,211.01) if they start 600 cm apart and move towards each other, starting simultaneously. <span style="font-family: 'Comic Sans MS',cursive;">- The faster CMV will catch up with the slower CMV at 221.64 cm in 3.82 seconds (3.82,221.64) if they start 100 m apart and move in the same direction, starting simultaneously

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Materials:** Constant Motion Vehicle, tape measure and/or metersticks, masking tape (about 30 cm/group), stopwatch, spark timer and spark tape

<span style="font-family: 'Comic Sans MS',cursive;">** Procedure: **

<span style="font-family: 'Comic Sans MS',cursive;">media type="file" key="600cm Crash Course.mov" width="300" height="300"

<span style="font-family: 'Comic Sans MS',cursive;">media type="file" key="1m Overtake Video.mov" width="300" height="300"


 * <span style="font-family: 'Comic Sans MS',cursive;">Analysis: **

<span style="font-family: 'Comic Sans MS',cursive;">Percent Error

<span style="font-family: 'Comic Sans MS',cursive;">Percent Difference

<span style="font-family: 'Comic Sans MS',cursive;">**Data:**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Discussion Questions:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**1) Where would the cars meet if their speeds were exactly equal?** If the cars' speeds were exactly equal, they would meet at exactly 300 meters because it is halfway in between the entire 600. Additionally, in the overtake, the cars would not crash at any point because they were going at the same speed. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**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: 'Comic Sans MS',cursive; font-size: 110%;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**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?** No, you cannot find the points when the cars are at the same place at the same time from a velocity-time graph because a velocity-time graph doesn't show the position of an object, only its velocity. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">

<span style="font-family: 'Comic Sans MS',cursive;">In our hypothesis, we predicted that our faster blue car, traveling at 58.021 cm/s, would crash into our slower yellow car, traveling at 31.875 m/s, at 211.01 cm (6.67,211.01) when traveling in opposite directions going towards each other. The experimental results we obtained for the crashing part of this lab were generally higher than the our theoretical value of 211.01 cm. The average was 236.09, and we can tell that there must have been a significant amount of human error involved because the percent error of our average experimental value was 11.89%. We also predicted that our faster blue car would overtake or catch up to our slower yellow car at 221.64 (3.82,221.64) when traveling in the same direction. These results seemed to be a little more accurate. We calculated an average of 230.60, which is not so far off compared to our predicted value of 221.64. Our average percent error is less as well, at only 4.04%. Nevertheless, error is always involved in every experiment and can be reduced in several ways. <span style="font-family: 'Comic Sans MS',cursive;">One source of error may include that the two people who turned the cars on did so either before or after the other person. Also, the cars we experimented with did not travel in a completely straight line which may have caused inaccuracy within our results. If I were to redo this, I would make sure I was on a completely flat surface or even put the cars on a track if necessary to get the exact distances. I also would have horizontal indicators of the distances, as well as the measuring tape that we laid vertically, to try to get a better estimation of the distances.
 * <span style="font-family: 'Comic Sans MS',cursive;">Conclusion: **

<span style="font-family: 'Comic Sans MS',cursive;">Egg Drop Analysis
<span style="font-family: 'Comic Sans MS',cursive;">Partner: Gabby Leibowitz <span style="font-family: 'Comic Sans MS',cursive;">9/30/11

<span style="font-family: 'Comic Sans MS',cursive;">The egg was placed in a cradle of straws suspended with rubber bands wrapped around the straws, attached by sewing strings to a parachute of aluminum foil.

<span style="font-family: 'Comic Sans MS',cursive;">After dropping our egg in its contraption from the window 8.5 m above the ground, our egg remained intact. When we calculated the acceleration, it came out to be 9.33 m/s2:

<span style="font-family: 'Comic Sans MS',cursive;">d = 8.5 m <span style="font-family: 'Comic Sans MS',cursive;">v0 = 0 <span style="font-family: 'Comic Sans MS',cursive;">t = 1.35 s <span style="font-family: 'Comic Sans MS',cursive;">a = ?

<span style="font-family: 'Comic Sans MS',cursive;">d = v0t + 1/2at2 <span style="font-family: 'Comic Sans MS',cursive;">8.5 = 1/2a(1.35)2 <span style="font-family: 'Comic Sans MS',cursive;">8.5 = .91125a <span style="font-family: 'Comic Sans MS',cursive;">a = 9.33 m/s2

<span style="font-family: 'Comic Sans MS',cursive;">Compared to the normal force of gravity, the acceleration we calculated is less than 9.8 m/s2 but relatively close. This means that when we dropped our egg, it came out to be close to but with slightly less of an acceleration than when an object falls only by the force of gravity. If I could redo this activit y<span style="font-family: 'Comic Sans MS',cursive;">, I would try to make it lighter by possibly reducing the amount of straws or incorporating a cone shape to our final project.

<span style="font-family: 'Comic Sans MS',cursive;">Quantitative Graph Interpretation
<span style="font-family: 'Comic Sans MS',cursive;">9/19/11

<span style="font-family: 'Comic Sans MS',cursive;">
 * <span style="font-family: 'Comic Sans MS',cursive;">D. **

<span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;">**E.** <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;">

<span style="font-family: 'Comic Sans MS',cursive;">Free Fall = any object is only acted on by the force of gravity <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;"> <span style="font-family: 'Comic Sans MS',cursive;">
 * <span style="font-family: 'Comic Sans MS',cursive;">Cannot have an initial or final velocity of zero
 * <span style="font-family: 'Comic Sans MS',cursive;">Acceleration = (-)9.8 m/s2
 * <span style="font-family: 'Comic Sans MS',cursive;">v = 0

<span style="font-family: 'Comic Sans MS',cursive;">Lesson 5: Free Fall and the Acceleration of Gravity

 * <span style="color: #009acd; font-family: 'Comic Sans MS',cursive; font-size: 14pt;">Introduction to Free Fall **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">A free falling object is an object that is falling under the sole influence of gravity. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">There are two important motion characteristics that are true of free-falling objects:

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Free-falling objects do not encounter air resistance.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Because free-falling objects are accelerating downwards at a rate of 9.8 m/s/s, a [|ticker tape trace] or dot diagram of its motion would depict an acceleration. The dot diagram at the right depicts the acceleration of a free-falling object. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Topic sentence:** Free falling objects have only the force of gravity acting upon them, which is calculated to always be (-)9.8 m/s2, due to the lack of human or any other force.


 * <span style="color: #009acd; font-family: 'Comic Sans MS',cursive; font-size: 130%;">The Acceleration of Gravity **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">It was learned in the [|previous part of this lesson] that a free-falling object is an object that is falling under the sole influence of gravity. A free-falling object has an acceleration of 9.8 m/s/s, downward (on Earth). <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">This numerical value for the acceleration of a free-falling object is such an important value that it is given a special name. It is known as the ** acceleration of gravity. ** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">There are slight variations in this numerical value (to the second decimal place) that are dependent primarily upon on altitude.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Recall from an [|earlier lesson] that acceleration is the rate at which an object changes its velocity. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">To accelerate at 9.8 m/s/s means to change the velocity by 9.8 m/s each second.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">If the velocity and time for a free-falling object being dropped from a position of rest were tabulated, then one would note the following pattern.


 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">Time (s) || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">Velocity (m/s)  ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">0 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">0  ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">1 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">- 9.8  ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">2 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">- 19.6  ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">3 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">- 29.4  ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">4 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">- 39.2  ||
 * <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">5 || <span style="display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">- 49.0  ||

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Observe that the velocity-time data above reveal that the object's velocity is changing by 9.8 m/s each consecutive second.


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Topic Sentence: **<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Because free falling objects travel at a set acceleration of (-)9.8 m/s2, it means that the object is traveling at or changing by (-)9.8 m/s2 each second; however, the acceleration of gravity can slightly alter in different places of the Earth.

<span style="color: #009acd; font-family: 'Comic Sans MS',cursive; font-size: 130%;">**Representing Free Fall by Graphs**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">A position versus time graph for a free-falling object is shown below. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Since a free-falling object is undergoing an acceleration (g = 9.8 m/s/s), it would be expected that its position-time graph would be curved. The negative slope of the line indicates a negative (i.e., downward) velocity.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">A velocity versus time graph for a free-falling object is shown below. <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%; line-height: 0px; overflow-x: hidden; overflow-y: hidden;"> <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Since a free-falling object is undergoing an acceleration (g = 9,8 m/s/s, downward), it would be expected that its velocity-time graph would be diagonal.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Since the slope of any velocity versus time graph is the acceleration of the object, the constant, negative slope indicates a constant, negative acceleration. This analysis of the slope on the graph is consistent with the motion of a free-falling object - an object moving with a constant acceleration of 9.8 m/s/s in the downward direction.

<span style="font-family: 'Comic Sans MS',cursive;">** Topic Sentence: ** Free falling objects can be represented by the use of graphs; showing its position, it will be curved with a negative slope, starting out slower and increasing speed, and showing its velocity, it will be a straight line starting at rest with a negative slope as well.

<span style="color: #009acd; font-family: 'Comic Sans MS',cursive; font-size: 140%;">**How Fast? and How Far?**

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Free-falling objects are in a state of [|acceleration]. The velocity of a free-falling object is changing by 9.8 m/s every second.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">The formula for determining the velocity of a falling object after a time of t seconds is <span style="color: red; display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">vf = g * t

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">The distance that a free-falling object has fallen from a position of rest is also dependent upon the time of fall. <span style="color: red; display: block; font-family: 'Comic Sans MS',cursive; font-size: 110%; text-align: center;">d = 0.5 * g * t2 <span style="font-family: 'Comic Sans MS',cursive;">**Topic Sentence:** In free fall, the velocity and distance of an object that is dropped from a position at rest are both dependent on the time that it falls.
 * <span style="color: #009acd; font-family: 'Comic Sans MS',cursive; font-size: 14pt;">The Big Misconception **

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Free-fall is the motion of objects that move under the sole influence of gravity; free-falling objects do not encounter air resistance. More massive objects will only fall faster if there is an appreciable amount of air resistance present.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">The acceleration of an object is directly proportional to force and inversely proportional to mass. Increasing force tends to increase acceleration while increasing mass tends to decrease acceleration. <span style="font-family: 'Comic Sans MS',cursive;">Thus, the greater force on more massive objects is offset by the inverse influence of greater mass. Subsequently, all objects free fall at the same rate of acceleration, regardless of their mass.

<span style="font-family: 'Comic Sans MS',cursive;">**Topic Sentence:** The mass of a free falling object is irrelevant to its acceleration; only surface area and possibly air resistance needed to be taken into account.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 130%;">Lab: What is the acceleration of a falling body?
<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Maxx Grunfeld and Tim Hwang <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">10/6/11

<span style="font-family: 'Comic Sans MS',cursive;">** Objective: ** What is the acceleration of a falling body?

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Materials**: Ticker Tape Timer, Timer tape, Masking tape, Mass, clamp, meterstick.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Hypothesis:** The acceleration of a falling body should be (-)9.8 m/s2, which is the acceleration of objects that are acted upon by only the force of gravity. The x-t graph should be a curve going upward with a positive slope, and the v-t graph should be a line going upward with a positive slope as well.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Data: **

<span style="font-family: 'Comic Sans MS',cursive;">**Class Data:** <span style="font-family: 'Comic Sans MS',cursive;">Period 4


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">Graphs: **



<span style="font-family: 'Comic Sans MS',cursive;">**Analysis:** <span style="font-family: 'Comic Sans MS',cursive;">Our v-t graph has an R2 value of 0.99909, which is very close to 1 and therefore is almost perfect in accordance with the data. However, we reached such a high R2 value because we deleted one of our data points, since it seemed to be an extreme outlier and threw the data and the line off completely. This may have occurred due to us possibly have skipping a dot, although we were careful, or the ticker tape may have gotten caught or tangled while we dropped the falling body from the railing. Nevertheless, after deleting that last point our data became significantly more accurate. Our hypothesis was also right because the graph was a linear with a positive slope. The slope of our line was 805, which is equal to the experimental acceleration. It is not too far off from the theoretical acceleration, which equals 981 cm/s2. The b value, or y-intercept in the x-t graph is -26.011. The y-intercept for the v-t graph is 60.036. We did not set the y-intercept for this experiment equal to zero because when we dropped the weight we might not have dropped it exactly at the moment when the spark timer made its first dot. Our x-t graph was also relatively accurate with an R2 value of 0.98386. Our hypothesis was right in that the graph was a curve with a positive slope.







<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Discussion Questions:** <span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">**1) Does the shape of your v-t graph agree with the expected graph? Why or why not?** The shape of our v-t graph agrees with the graph we predicted because it is linear, going upward, with a positive slope. We did not use negative numbers in our data, although the falling body was dropped downward, which is why the line did not have a negative slope. <span style="font-family: 'Comic Sans MS',cursive;">** 2) Does the shape of your x-t graph agree with the expected graph? Why or why not?** The shape of our x-t graph also agrees with the expected graph because it came out to be a curve, going upward, with a positive slope as well. We knew that the object would gain speed as it fell, which is shown with the curve of our graph. <span style="font-family: 'Comic Sans MS',cursive;">** 3) How do your results compare to that of the class? (Use Percent difference to discuss quantitatively.)** The experimental acceleration we obtained from our results came out to be 805 cm/s2. The average class acceleration was 834.03 cm/s2, so our results were a little lower. We calculated a percent difference of only 3.48%, so our results were relatively close to the class average. <span style="font-family: 'Comic Sans MS',cursive;">**4) Did the object accelerate uniformly? How do you know?** Yes, the object accelerated uniformly because our v-t graph shows a constant acceleration. It did not accelerate at exactly 981 cm/s2, which is the acceleration of a typical free falling body, but it still accelerated at a constant rate which was close to that number. <span style="font-family: 'Comic Sans MS',cursive;">** 5) What factor(s) would cause acceleration due to gravity to be higher than it should be? Lower than it should be?** If the person dropping the falling body were to accidentally use force, the acceleration would be higher than it should be. Friction of the ticker tape going through the spark timer could have caused the acceleration to be lower than it should be.

<span style="font-family: 'Comic Sans MS',cursive;">**Conclusion:** <span style="font-family: 'Comic Sans MS',cursive;">My hypotheses for both graphs were accurate because they both went upward with positive slopes, although the x-t graph was a curve shape and the v-t was linear. Our final results show that the weight fell at an acceleration of 805 cm/s2, which is not too far off from the theoretical value of 981 cm/s2. Our hypothesis was also somewhat correct that the acceleration of the falling body should have been 981 cm/s2, but due to possible sources of error, the results we obtained were off a bit. Our percent error was 17.9%, which is not an unreasonable value but could have definitely been more accurate. Our percent difference was much lower at only 3.48%, meaning it was very close to the class average and therefore reassuring that our results were pretty accurate. Some possible sources of error can be accidental human force when dropping the weight, or possibly friction from the ticker tape going through the spark timer. There may also have been some inaccuracies in measuring the distances between dots on the ticker tape, because the meterstick or measuring tape may have shifted in the process. Measuring tape is definitely a better method of measuring for this particular experiment, due to the flatness of the tape, and taping it down may fix and prevent some of the errors and alterations that could have taken place.