Ch2_malleys

=Constant Speed =

toc Speed of a CMV Lab
**Objective:** What is the speed of a Constant Motion Vehicle (CMV)? **Hypothesis:** The CMV will move about 43 cm/s, based on how fast we saw it move before. Distances can be measured fairly precisely, though it’s doubtful that they will always be 100% accurate. A position-time graph can tell you the velocity of an object. Speed of a CMV  

**Discussion questions**
 * 1) **Why is the slope of the position-time graph equivalent to average velocity?**
 * 2) The slope of a line is equal to change in y divided by change in x. In this case, that would be change in position divided by change in time. That is the formula for velocity.
 * 3) **Why is it average velocity and not instantaneous velocity? What assumptions are we making?**
 * 4) We are trying to find the speed of the car through time. If we found the instantaneous velocity, we would merely be looking for the velocity at a set time – for example, .5 seconds. The assumption we are making is that the car is moving at a constant speed.
 * 5) **Why was it okay to set the y-intercept equal to zero?**
 * 6) We started our experiment at point zero with zero seconds; hence, our starting point, and therefore the y-axis, was equal to zero.
 * 7) **What is the meaning of the R2 value?**
 * 8) The R2 value shows the ability of the trendline to pass through all the points. By extension, it can show us the accuracy of the trendline.
 * 9) **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?**
 * 10) <span style="font-family: Georgia,serif; font-size: 14px;">I would expect that line to be parallel to ours; it would be moving for the same amount of time, just slower.

<span style="font-family: Georgia,serif; font-size: 14px;">**Conclusion** <span style="font-family: Georgia,serif; font-size: 14px;">We discovered that our CMV moved at 46.703 cm/s. I hypothesized that it would move at 43 cm/s. My answer was off, but only by several centimeters. I was correct when I hypothesized that the position-time graph could show us velocity. There were likely slight inaccuracies in our data; as a human, it is possible to be 100% accurate all of the time. For example, the ruler we were using could have shifted as we were measuring or the dots on the paper could not have been steady at the point that we measured. As well as this, there were more precise tools that we could have used; for example, a tape measure is flat, whereas the rulers we used were about half a centimeter thick. This could have contributed to issues in perspective, effectively skewing our data. We also could have made sure that all of the CMVs had fresh batteries, to improve their consistency.

<span style="font-family: Georgia,serif;">** HW 9/8/11 **

 * 1) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
 * 2) <span style="font-family: Georgia,serif; font-size: 14px;">Before this reading, I had already understood the difference between distance and displacement, as well as the difference between speed and velocity. Distance is a scalar quality, and it refers to the amount of ground covered when an object is moving. Displacement, on the other hand, is a vector quality, and therefore refers to how far the object has moved; simply put, it is the object's total change in position. Speed and velocity, while very similar, are not the dame. Speed is a scalar quality, and therefore does not require a direction. A speed would be, for example, 30 mi/hr. Velocity, however, is a vector quality, and so it does require a direction; for example, 20 mi/hr, west.
 * 3) <span style="font-family: Georgia,serif; font-size: 14px;">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 as your new understanding.
 * 4) <span style="font-family: Georgia,serif; font-size: 14px;">I was not aware of the difference between scalar and vector. While I had heard the words before, I kept swapping their definitions. This article helped me to clarify that information. I now know that scalars can be fully described by magnitude alone, whereas a vector also needs a direction.
 * 5) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
 * 6) <span style="font-family: Georgia,serif; font-size: 14px;">What situations would require average speed, and which would require instantaneous?
 * 7) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that was not gone over during class today?
 * 8) <span style="font-family: Georgia,serif; font-size: 14px;">I don't believe we went over kinematics (the word itself). It was one that I had not heard before; however, the text cleared it up quite nicely.

<span style="font-family: Georgia,serif; font-size: 120%;">**Constant Speed Notes** <span style="font-family: Georgia,serif; font-size: 110%;">**Types of Motion** <span style="font-family: Georgia,serif; font-size: 110%;">- At rest – no movement <span style="font-family: Georgia,serif; font-size: 110%;">- Constant speed – speed is not changing <span style="font-family: Georgia,serif; font-size: 110%;">- Increasing speed <span style="font-family: Georgia,serif; font-size: 110%;">- Decreasing speed

<span style="font-family: Georgia,serif;">** Motion Diagrams ** <span style="font-family: Georgia,serif; font-size: 110%;">The following image is an example of a motion diagram, which can be helpful in determining velocity and acceleration. <span style="font-family: Georgia,serif; font-size: 14px;"> <span style="font-family: Georgia,serif; font-size: 14px;">- Signs are arbitrary; however, we usually use Decartes's version of things

<span style="font-family: Georgia,serif;">** Ticker Tape Diagrams ** <span style="font-family: Georgia,serif; font-size: 14px;">Ticker tape diagrams, too, can be useful in determining speed, velocity, and acceleration. Ex: <span style="font-family: Georgia,serif;">

<span style="font-family: Georgia,serif;">** HW 9/9/11 ** <span style="font-family: Georgia,serif;">Five Kinematics Equations <span style="font-family: Georgia,serif;">a= Vf- Vi/ Δt <span style="font-family: Georgia,serif;">Vf= Vi +aΔt <span style="font-family: Georgia,serif;">Δd= 1/2( Vi + Vf)t <span style="font-family: Georgia,serif;">Δd= Vit + 1/2at 2 <span style="font-family: Georgia,serif;">Vf 2 =Vi 2 +2aΔd
 * 1) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
 * 2) <span style="font-family: Georgia,serif; font-size: 14px;">Before this reading, I was already clear on ticker tape diagrams and vector diagrams. I know that vector diagrams can depict both the direction and the velocity and acceleration of an object in motion, and that ticker tape diagrams can only detect its velocity and acceleration - not its direction.
 * 3) <span style="font-family: Georgia,serif; font-size: 14px;">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 as your new understanding.
 * 4) <span style="font-family: Georgia,serif; font-size: 14px;">Visually, I wasn't too clear on what a vector diagram would look like with arrows pointing up or down, or in what situation it would be useful. However, this image: [[image:U1L2c3.gif]]made it much more clear.
 * 5) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
 * 6) <span style="font-family: Georgia,serif; font-size: 14px;">Everything that I read was clear.
 * 7) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that was not gone over during class today?
 * 8) <span style="font-family: Georgia,serif; font-size: 14px;">I don't believe there was any new material!

<span style="font-family: Georgia,serif;">** Graphical Representation of Equilibrium Lab ** <span style="font-family: Georgia,serif;">**Objectives:**
 * <span style="font-family: Georgia,serif;">What is the difference between static and dynamic equilibrium?
 * <span style="font-family: Georgia,serif;">How is “at rest” represented on a position vs. time graph? On a velocity vs. time graph? On an acceleration vs. time graph?
 * <span style="font-family: Georgia,serif;">How is constant speed represented on a position vs. time graph? On a velocity vs. time graph? On an acceleration vs. time graph?
 * <span style="font-family: Georgia,serif;">How are changes in direction represented on a position vs. time graph? On a velocity vs. time graph? On an acceleration vs. time graph?

<span style="font-family: Georgia,serif;">** Graphs: ** <span style="font-family: Georgia,serif;">

<span style="font-family: Georgia,serif;">** Discussion Questions ** <span style="color: #000000; font-family: Georgia,serif;">On all of the above, there will simply be a straight line.
 * 1) <span style="color: #000000; font-family: Georgia,serif;">How can you tell that there is no motion on a…
 * 2) position vs. time graph
 * 3) velocity vs. time graph
 * 4) acceleration vs. time graph

<span style="color: #000000; font-family: Georgia,serif;">2. How can you tell that your motion is steady on a... <span style="color: #000000; font-family: Georgia,serif;">a. position vs. time graph: There is a linear graph demonstrating that as the time changed, so did the position. It was a relatively straight line. <span style="color: #000000; font-family: Georgia,serif;">b. velocity vs. time graph: There is a relatively straight line that is directly above the x-axis. <span style="color: #000000; font-family: Georgia,serif;">c. acceleration vs. time graph: There is a relatively straight line that is on the x-axis.

<span style="color: #000000; font-family: Georgia,serif;">3. How can you tell your motion is fast vs. slow on a... <span style="color: #000000; font-family: Georgia,serif;">a. position vs. time graph: On a graph that shows a fast motion, the position will move much more quickly over a certain amount of time than a graph showing a slow motion. In other words, a graph that shows a faster motion will have a line that is more vertical than one that doesn’t. <span style="color: #000000; font-family: Georgia,serif;">b. velocity vs. time graph: There is no way to tell - the velocity should remain constant throughout. <span style="color: #000000; font-family: Georgia,serif;">c. acceleration vs. time graph: Again, there is no way to tell - there shouldn't be any acceleration.

<span style="color: #000000; font-family: Georgia,serif;">4. How can you tell that you changed direction on a... <span style="color: #000000; font-family: Georgia,serif;">a. position vs. time graph: The position goes up and then goes back down; in this case, it went down to zero, indicating no net change in position. <span style="color: #000000; font-family: Georgia,serif;">b. velocity vs. time graph: <span style="color: #000000; font-family: Georgia,serif;">c. acceleration vs. time graph: <span style="color: #000000; font-family: Georgia,serif;">There isn't really any way to tell on either of the above graphs; the velocity and acceleration remained constant, as was only to be expected.

<span style="color: #000000; font-family: Georgia,serif;">5. What are the advantages of representing motion using a... <span style="color: #000000; font-family: Georgia,serif;">a. position vs. time graph: There are a lot of advantages - it can tell you when you've sped up or slowed down, when you've changed direction, and the approximate speed. <span style="color: #000000; font-family: Georgia,serif;">b. velocity vs. time graph: Using this graph can tell you the approximate velocity of an object, obviously; in this case, however, it was only able to tell us that the velocity remained constant in every scenario. <span style="color: #000000; font-family: Georgia,serif;">c. acceleration vs. time graph: This graph is the most helpful in determining when an object has changed speeds.

<span style="color: #000000; font-family: Georgia,serif;">6. What are the disadvantages of representing motion using a... <span style="color: #000000; font-family: Georgia,serif;">a. position vs. time graph: This kind of graph is not as able as the acceleration graph to depict changes in speed. <span style="color: #000000; font-family: Georgia,serif;">b. velocity vs. time graph: It has only a limited application; it can only tell you when an object's velocity changes. Other than that, it just shows a straight line, representing that the velocity remained constant. <span style="color: #000000; font-family: Georgia,serif;">c. acceleration vs. time graph: This graph is only useful when the object's speed is changing; other than that, it should just show a straight line.

<span style="color: #000000; font-family: Georgia,serif;">7. Define the following: <span style="color: #000000; font-family: Georgia,serif;">a. no motion - at rest; no movement <span style="color: #000000; font-family: Georgia,serif;">b. constant speed - speed is not changing

<span style="font-family: Georgia,serif;">Acceleration <span style="color: #000000; font-family: Georgia,serif;">** HW 9/13/11 **


 * 1) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
 * 2) <span style="font-family: Georgia,serif;">I previously understood that acceleration was not what occurred when an object was speeding up, but is really when an object changes speed in general. I was also aware of the definition of constant acceleration; it is when an object gains or loses speed at the same rate throughout a certain interval of time.
 * 3) <span style="font-family: Georgia,serif; font-size: 14px;">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 as your new understanding.
 * 4) <span style="font-family: Georgia,serif;">I was unclear on why we were using the units that we were using, but the reading cleared that up for me. Because acceleration is a change in velocity over time, it only makes sense that the velocity units are divided by a unit of time to get units for acceleration; for example, m/s/s.
 * 5) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
 * 6) <span style="font-family: Georgia,serif;">Everything was clear.
 * 7) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that was not gone over during class today?
 * 8) <span style="font-family: Georgia,serif;">This reading went over the way to determine the vector quantity of acceleration, which I do not believe was something that we covered today in class. I learned that the general rule of thumb is that if an object is slowing down, then its acceleration is in the opposite direction of its motion, and vice versa.

<span style="font-family: Georgia,serif;">** Lab: Acceleration Graphs **
<span style="font-family: Georgia,serif; font-size: 120%;">Lab Partner: Ryan Hall <span style="font-family: Georgia,serif;">**Objectives:**
 * <span style="font-family: Georgia,serif;">What does a position-time graph for increasing speeds look like?
 * <span style="font-family: Georgia,serif;">What information can be found from the graph?

<span style="font-family: Georgia,serif;">**Available Materials:** Spark tape, spark timer, track, dynamics cart, ruler/meterstick/measuring tape

<span style="font-family: Georgia,serif;">**Hypothesis**: Instead of a straight line, a position-time graph showing increasing speed will show a line that is curved. From that graph, we can determine average speed, acceleration (whether it was gaining speed or losing speed). By using the change in speed over the change in distance, we can find the acceleration.

<span style="font-family: Georgia,serif;">**Procedure:** <span style="font-family: Georgia,serif;">1. Using an educated guess, write a hypothesis answering the questions in the objective. <span style="font-family: Georgia,serif;">2. Place the ramp on top of the textbook, measure a piece of ticker tape, and attach that to the car. <span style="font-family: Georgia,serif;">3. Start the ticker timer, let the car go down the ramp, and stop it when it gets to the end. <span style="font-family: Georgia,serif;">4. Measure the distance between each point on the ticker tape, and record those on an Excel spreadsheet.

<span style="font-family: Georgia,serif;">**Observations/Data:** <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;">**Analysis:** <span style="font-family: Georgia,serif;">**1. Interpret the equation of the line (slope, y-intercept) and the R2 value.** <span style="font-family: Georgia,serif;">a). Increasing Down Incline: The equation of this line is y = 12.3x 2 + 1.88x. In this case, the A value would equal half of the acceleration (making the actual acceleration 24.6cm/s/s). B represents the initial velocity; in this case, 1.88 cm/s. The R 2 value represents the accuracy of the trendline in decimal form. Here, it was .999, making it very accurate. <span style="font-family: Georgia,serif;">b). Decreasing Up Incline: The equation of this line is y = -26.7x 2 + 113.7. The A value represents half the acceleration (making the real acceleration -53.4 cm/s/s), and the B value represents the initial velocity; in this situation, it was 113.7 cm/s. The R 2 value represents the accuracy of the trendline in decimal form. Here, it was .999, making it very accurate. <span style="font-family: Georgia,serif;">**2. Find the instantaneous speed at halfway point and at the end.** <span style="font-family: Georgia,serif;">The instantaneous speed at the halfway point is approximately 16.1 cm/s. The instantaneous speed at the end is about 36.3 cm/s. <span style="font-family: Georgia,serif;">**3. Find the average speed for the entire trip.** <span style="font-family: Georgia,serif;">a). Increasing Down Incline: The average speed was 15.4 cm/s. <span style="font-family: Georgia,serif;">b). Decreasing Up Incline: The average speed was 74.3 cm/s.

<span style="font-family: Georgia,serif;">**Discussion Questions:** <span style="font-family: Georgia,serif;">1. What would your graph look like if the incline had been steeper? <span style="font-family: Georgia,serif;">On both graphs, the line would have curved more sharply. <span style="font-family: Georgia,serif;">2. What would your graph look like if the cart had been decreasing up the incline? <span style="font-family: Georgia,serif;">Exactly like the blue line in this graph: <span style="font-family: Georgia,serif;">

<span style="font-family: Georgia,serif;">3. Compare the instantaneous speed at the halfway point with the average speed of the entire trip. <span style="font-family: Georgia,serif;">The instantaneous speed of the car gaining speed at its halfway point was approximately 16.1 cm/s. Its average speed of the trip was about 15.4 cm/s. Though these numbers are close, they are not the same. This is only to be expected - the latter is the average of all the instantaneous speeds.\ <span style="font-family: Georgia,serif;">**Decreasing Up Incline** <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;">4. Explain why the instantaneous speed is the slope of the tangent line, why does this make sense? <span style="font-family: Georgia,serif;">The tangent line intersects the line at one point only. Therefore, the slope of the tangent line is the slope of only one point on the other line. Instantaneous speed is the speed of an object at a certain interval in time. By using only one point on the original line, we are able to determine this. <span style="font-family: Georgia,serif;">5. Draw a velocity-time graph of the motion of the cart. Be as quantitative as possible. <span style="font-family: Georgia,serif;">

<span style="font-family: Georgia,serif;">**Conclusion:** For the most part, our hypothesis was correct; the resulting graph was curved, not straight, as predicted, and did indeed provide all of the information that we originally believed it would. We determined the average speeds of both graphs (14.4 cm/s and 74.3 cm/s) and acceleration of both (24.6 cm/s/s and -54.3 cm/s/s). From this experiment, we learned that when an object is gaining speed, the resulting graph will be one of a curve pointing upwards, and vice versa. Human error is always a contributor to sources of error in any experiment; this one was no exception. For example, when releasing the car down the ramp, we may have used some extra force, therefore increasing its speed and, by extension, its velocity. As well as this, the rulers we used could have led to inaccurate measurements of the distances between ticker tape dots. In order to minimize these issues, we could have made sure not to use so much force - or any force at all, really - when releasing the car down the track. We could also use more efficient measuring devices.

<span style="font-family: Georgia,serif;">** HW 9/15/11 **
<span style="font-family: Georgia,serif;">**Lesson 3:**


 * 1) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
 * 2) <span style="font-family: Georgia,serif;">Although we spent virtually no time going over it in class, I was already aware of how to find the slope of a line (rise/run). I was also aware of the fact that the slope of a P-T graph was equal to the velocity.
 * 3) <span style="font-family: Georgia,serif; font-size: 14px;">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 as your new understanding.
 * 4) <span style="font-family: Georgia,serif;">I am not a very visual person, and as such, despite the myriad times we've gone over them in class, was having trouble interpreting the P-T graphs. However, being able to see and read about the graphs made them much more clear in my mind.
 * 5) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
 * 6) <span style="font-family: Georgia,serif;">Everything was clear.
 * 7) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that was not gone over during class today?
 * 8) <span style="font-family: Georgia,serif;">I believe we've gone over everything in class - there was no new material.

<span style="font-family: Georgia,serif;">**Lesson 4:**
 * 1) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
 * 2) <span style="font-family: Georgia,serif;">I knew before reading that a graph showing a constant, positive velocity would result in a line with a slope of zero (a horizontal line). I also knew that a car with a positive, changing velocity would result in a sloped line. Finally, I was also aware that when the velocity is positive, only positive points are plotted, and vice versa.
 * 3) <span style="font-family: Georgia,serif; font-size: 14px;">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 as your new understanding.
 * 4) <span style="font-family: Georgia,serif;">I was still unclear on the idea that the slope of a V-T graph was equal to the acceleration. Seeing it written down helped to clarify it for me.
 * 5) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
 * 6) <span style="font-family: Georgia,serif;">Everything was clear.
 * 7) <span style="font-family: Georgia,serif; font-size: 14px;">What (specifically) did you read that was not gone over during class today?
 * 8) <span style="font-family: Georgia,serif;">Though we may have touched upon it briefly in class, I was not really aware how to use a V-T graph to determine displacement. Now, I can find the displacement using three different shapes: a rectangle, a triangle, and a trapezoid.

<span style="font-family: Georgia,serif;">Lab: A Crash Course in Velocity (Pt. 2)
<span style="font-family: Georgia,serif;">**Lab Partners:** Ryan Hall, Jenna Malley, Nicole Kloorfain <span style="font-family: Georgia,serif; font-size: 90%;">**Objectives**: Both algebraically and graphically, solve the following 2 problems. Then set up each situation and run trials to confirm your calculations. <span style="font-family: Georgia,serif; font-size: 90%;">**Calculations:** <span style="font-family: Georgia,serif; font-size: 90%;"> <span style="font-family: Georgia,serif; font-size: 90%;">**Available Materials**: Constant Motion Vehicle, Tape measure and/or metersticks, Masking tape (about 30 cm/group), Stop watch, spark timer and spark tape <span style="font-family: Georgia,serif;">**Procedure:** <span style="font-family: Georgia,serif;">**Part a:** <span style="font-family: Georgia,serif;">media type="file" key="Movie on 2011-09-21 at 08.57.mov" width="300" height="300" <span style="font-family: Georgia,serif;">**Part b:** <span style="font-family: Georgia,serif;">media type="file" key="Movie on 2011-09-21 at 09.06.mov" width="300" height="300"

<span style="font-family: Georgia,serif; font-size: 90%;">**Data Collected:** <span style="font-family: Georgia,serif; font-size: 90%;"> <span style="font-family: Georgia,serif; font-size: 90%;">**Discussion questions:**
 * 1) <span style="font-family: Georgia,serif; font-size: 90%;">Where would the cars meet if their speeds were exactly equal?
 * 2) <span style="font-family: Georgia,serif; font-size: 90%;">For the first problem, they would meet in the exact center. In the second problem, they would never meet.
 * 3) <span style="font-family: Georgia,serif; font-size: 90%;">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.
 * 4) <span style="font-family: Georgia,serif; font-size: 90%;">Crashing:
 * 5) <span style="font-family: Georgia,serif; font-size: 90%;">[[image:Screen_shot_2011-09-21_at_10.23.49_AM.png caption="Crashing Data Table"]]
 * 6) [[image:Screen_shot_2011-09-21_at_11.59.32_AM.png]]
 * 7) <span style="font-family: Georgia,serif;">The point where they meet is where the lines cross, which is at approximately 9.33 seconds.


 * 1) <span style="font-family: Georgia,serif;">Catching up:
 * 2) [[image:Screen_shot_2011-09-21_at_12.59.33_PM.png caption="Catching Up CMV Data Table"]]
 * 3) [[image:Screen_shot_2011-09-21_at_12.59.16_PM.png]]
 * 4) <span style="font-family: Georgia,serif;">The point where the blue car caught up to the yellow car is where the lines crossed, which is at approximate 3.44 seconds.
 * 5) <span style="font-family: Georgia,serif; font-size: 90%;">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?
 * 6) [[image:Screen_shot_2011-09-21_at_2.12.45_PM.png width="611" height="407"]]
 * 7) <span style="font-family: Georgia,serif; font-size: 90%;">Using only the graph, no, I do not believe that it is possible.

<span style="font-family: Georgia,serif;">**Calculating Percent Difference and Percent Error** <span style="font-family: Georgia,serif;">The formula for percent error is (|theoretical-experimental|/theoretical) * 100 <span style="font-family: Georgia,serif;">So, in the case of the crash course, that would have been: <span style="font-family: Georgia,serif;">164.5-160.7/164.5 *100 <span style="font-family: Georgia,serif;">2.5% <span style="font-family: Georgia,serif;">The formula for percent difference is (average experimental-individual experimental)/average experimental * 100 Calculations:

<span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;">**Conclusion:** The calculations we started out with before starting the experiment turned out to be relatively correct. For the crash course problem, we calculated that the cars would meet at about 164.5 cm; in actuality, they met at 160.7, giving us only a 2.5% margin of error. The largest percent difference we had between each of the trials was 1.26%. For the catching up problem, we estimated that the blue car would catch up with the yellow car at 160.6 cm, though they really did meet at 162.9 cm. This is only a 1.43% margin of error. The largest percent difference we had between each of the trials was 5.44%. There were, as always, several sources of human error that could have contributed to the percent errors and percent differences. For example, the blue car traveled more diagonally than straight, which could have caused its speed to be slightly off. As well as this, we may have thought the cars met a few tenths of a centimeter before they actually did. The measuring tape we used could have shifted during the experiment, despite the fact that we taped it to the floor. In order to make the results more accurate, it would definitely help to have a car that moved straight, and perhaps there should have been two of us marking off the point where the car met, as opposed to only one. We also could have checked to make sure the ruler was straight after every trial that we did.

<span style="font-family: Georgia,serif;">Egg Drop Project
<span style="font-family: Georgia,serif;">Picture <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;">Discussion

<span style="font-family: Georgia,serif;">Our egg drop project worked perfectly. I believe this was because of the several layers of straws on the bottom, protecting it from the full force of the impact. If I had to do anything differently, I would probably have made the base a little bit smaller - it did not need to be so big in order to work properly, and could have taken the weight down a few grams as well.

<span style="font-family: Georgia,serif;">Calculations for a <span style="font-family: Georgia,serif;">d = 8.5 m <span style="font-family: Georgia,serif;">v1 = 0 m/s <span style="font-family: Georgia,serif;">t = 1.61 s <span style="font-family: Georgia,serif;">a = ? <span style="font-family: Georgia,serif;">d = v1*t+1/2at^2 <span style="font-family: Georgia,serif;">8.5 = 1/2a*1.61^2 <span style="font-family: Georgia,serif;">8.5 = 1.3a <span style="font-family: Georgia,serif;">a = 6.56 m/s/s

=Free Fall=

<span style="font-family: Georgia,serif;">**Lesson Five: Free Fall and the Acceleration of Gravity**
<span style="font-family: Georgia,serif;">**Introduction to Free Fall**

<span style="font-family: Georgia,serif;">A free falling object is an object that is falling under the sole influence of gravity. There are two important motion characteristics that are true of free-falling objects:


 * <span style="font-family: Georgia,serif;">Free-falling objects do not encounter air resistance.
 * <span style="font-family: Georgia,serif;">All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s/s (often approximated as 10 m/s/s)

<span style="font-family: Georgia,serif;">A [|ticker tape trace] or dot diagram of an object in free-fall would depict acceleration. Recall that if an object travels downward and speeds up, then its acceleration is downward.

<span style="font-family: Georgia,serif;">**The Acceleration of Gravity**

<span style="font-family: Georgia,serif;">A free-falling object has an acceleration of 9.8 m/s/s, downward (on Earth). It is known as the acceleration of gravity. Physicists have a special symbol to denote it - the symbol g. There are slight variations in this numerical value (to the second decimal place) that are dependent primarily upon on altitude.

<span style="font-family: Georgia,serif;">**Representing Free Fall by Graphs**

<span style="font-family: Georgia,serif;">A position versus time graph for a free-falling object is shown below.



<span style="font-family: Georgia,serif;">A curved line on a position versus time graph signifies an accelerated motion. 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. A further look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object, the small initial slope indicates a small initial velocity and the large final slope indicates a large final velocity. Finally, the negative slope of the line indicates a negative (i.e., downward) velocity.

<span style="font-family: Georgia,serif;">A velocity versus time graph for a free-falling object is shown below.



<span style="font-family: Georgia,serif;">A diagonal line on a velocity versus time graph signifies an accelerated motion. A further look at the velocity-time graph reveals that the object starts with a zero velocity and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object that is moving in the negative direction and speeding up is said to have a negative acceleration.

<span style="font-family: Georgia,serif;">**How Fast? And How Far?**

<span style="font-family: Georgia,serif;">The velocity of a free-falling object that has been dropped from a position of rest is dependent upon the time that it has fallen. The formula for determining the velocity of a falling object after a time of t seconds is

<span style="display: block; font-family: Georgia,serif; text-align: center;">**vf = g * t**

<span style="font-family: Georgia,serif;">The above equation can be used to calculate the velocity of the object after any given amount of time when dropped from rest.

<span style="font-family: Georgia,serif;">**The Big Misconception**

<span style="font-family: Georgia,serif; font-size: 16px;">The answer to the question (doesn't a more massive object accelerate at a greater rate than a less massive object?) is absolutely not!

<span style="font-family: Georgia,serif; font-size: 16px;">The actual explanation of why all objects accelerate at the same rate involves the concepts of force and mass. At that time, you will learn that 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. 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: Georgia,serif; font-size: 16px;">Fr<span style="font-family: Georgia,serif; font-size: 90%;">ee Fall Lab
<span style="display: block; font-family: Georgia,serif; font-size: 14px; text-align: left;">**Objective**: What is the acceleration of a falling body? <span style="font-family: Georgia,serif; font-size: 14px;">**Hypothesis:** The V-T graph will have a constant, negative slope. The slope will be 981 cm/s/s. <span style="font-family: Georgia,serif; font-size: 14px;">**Materials**: Ticker Tape Timer, Timer tape, Masking tape, Mass, clamp, meterstick. **<span style="font-family: Georgia,serif;">Class Data: **
 * || <span style="font-family: Georgia,serif; font-size: 90%;">** Period 2 ** ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">754.43 ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">856.73 ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">851.07 ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">891.38 ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">891.12 ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">798.13 ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">710.65 ||
 * || <span style="color: #000000; display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">755.87 ||
 * || <span style="display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">**// 659.39 //** ||
 * || <span style="display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">**// 1225.4 //** ||
 * <span style="color: #000000; font-family: Georgia,serif; font-size: 90%;">Average || <span style="display: block; font-family: Georgia,serif; font-size: 90%; text-align: right;">** 839.417 ** ||
 * <span style="font-family: Georgia,serif; font-size: 90%;">Our Data: **
 * <span style="font-family: Georgia,serif;">Calculations: **

<span style="font-family: Georgia,serif; font-size: 90%;"> <span style="font-family: Georgia,serif; font-size: 90%;"> <span style="font-family: Georgia,serif;">**Analysis**: The equation of the line in the position-time graph takes the shape of y=ax <span style="font-family: Georgia,serif; vertical-align: super;">2 <span style="font-family: Georgia,serif;">+bx, which is very similar to the formula we use in class, g=1/2at <span style="font-family: Georgia,serif; vertical-align: super;">2 <span style="font-family: Georgia,serif; font-size: 14px;">+V<span style="font-family: Georgia,serif; vertical-align: sub;">i t. This means that the initial velocity is the a value - in this case, 29.75 /s. It also states that the acceleration is 1226 m/s/s. The equation of the velocity-time graph takes the form of y=mx+b. The acceleration is the same as the slope on a VT graph, so it would be 1225.4. Both accelerations that the graphs provide are very similar to each other. **<span style="font-family: Georgia,serif;">Discussion Questions: ** **<span style="font-family: Georgia,serif;">Conclusion: **<span style="font-family: Georgia,serif;">Our results did not quite match up with our hypothesis. We predicted that a velocity-time graph of an object in free-fall would show a negative slope (and therefore a negative acceleration); however, our graph did not show this. I know now that that is simply because we didn't include negative measurements in our Excel spreadsheet. The results we obtained for the acceleration should have been 981 m/s/s or less; the result we obtained is technically impossible. (Our percent difference was 24.9%). However, there are myriad sources of error that could have contributed to this, as there are in every experiment. While measuring a very long piece of paper with a very long ruler, it's easy to obtain imprecise measurements. The ruler could have slid back and forth without our noticing. As well as this, there were some points on the ticker-tape that were a little smudged - it was sometimes hard to tell where, exactly, we should start our measurements. The friction of the tape in the ticker timer could also have contributed to the errors in our measurements and calculations. To fix this, we could have used the motion detector instead of the ticker tape. We could also have done more trials in order to test the experiment's accuracy and consistency.
 * 1) <span style="font-family: Georgia,serif; font-size: 14px;">Does the shape of your v-t graph agree with the expected graph? Why or why not?
 * 2) <span style="font-family: Georgia,serif; font-size: 90%;">No; I expected our graph to show a negative velocity. However, this is only because we did not enter negative numbers into our Excel spreadsheet. Technically, the velocity //is// negative; our graph just does not show this.
 * 3) <span style="font-family: Georgia,serif; font-size: 14px;">Does the shape of your x-t graph agree with the expected graph? Why or why not?
 * 4) <span style="font-family: Georgia,serif; font-size: 90%;">Yes, it does. It shows a relatively straight line of increasing speed.
 * 5) <span style="font-family: Georgia,serif; font-size: 14px;">How do your results compare to that of the class? (Use Percent difference to discuss quantitatively.)
 * 6) <span style="font-family: Georgia,serif; font-size: 90%;">Our results are very different than the rest of the class's - our individual result had a percent difference of 46%. The data we collected is technically impossible. However, there were many reasons that this could have occurred - several sources of error are discussed in the conclusion.
 * 7) <span style="font-family: Georgia,serif; font-size: 14px;">Did the object accelerate uniformly? How do you know?
 * 8) <span style="font-family: Georgia,serif; font-size: 90%;">Yes, it did - the velocity increased at a constant rate; we can tell by the relatively straight line.
 * 9) <span style="font-family: Georgia,serif; font-size: 14px;">What factor(s) would cause acceleration due to gravity to be higher than it should be? Lower than it should be?
 * 10) <span style="font-family: Georgia,serif; font-size: 90%;">The friction of the tape in the ticker timer could have contributed to this issue.