Buggy Gone Wild
A Study of the Effect of Time on the Position of a Buggy
Researchers: Emmy Xu, Grace Chung, Jongwoo Park
Lab Conducted on: September 16, 2021
Research Question
Variables
Controls
This experiment aimed to deduce how the amount of time a buggy traveled for would affect its position.
Independent variable: time elapsed (seconds)
Dependent variable: position of the buggy (cm)
For the purpose of this experiment, time was defined in seconds, and "position" was defined as the displacement of the buggy in centimeters from its initial position.
Controls: the buggy itself, the surface on which the buggy traveled
For this experiment, there were different buggies available that traveled at varying speeds. Using buggies with varying speeds would result in inconsistent data, thus it was imperative that the same buggy be used for all trials of the experiment. The surface that the buggy ran on could also potentially lead to inconsistent data given that different surfaces provide differing levels of friction and resistance. Thus, all trials were conducted on a smooth, hard surface.
Method -- Procedure
METHOD
The experiment was conducted in two parts: first with the buggy traveling towards the North, then with the buggy traveling towards the South. In the first part, data was collected at six points between the range of 2 and 15 seconds: 2, 5, 7, 10, 12, 15. Two trials were conducted for each time point, and the distance traveled by the buggy for each of these three trials was recorded using a meter stick then averaged to deduce the typical distance traveled by the buggy for each time point. In the second part, data was collected at five points: 1, 2, 3, 4, 5 (seconds), and each point only consisted of one trial. Data points from the first part were then recorded into the Logger Pro software and plotted on a graph, separately from data collected from the second part. Researchers then determined a line of best fit for both graphs. From there, the slope, y-intercept, and equation of the line for each graph were deduced and used to analyze the effect of time elapsed on the position of the buggy.
PROCEDURE
1. Start the buggy above the ground with its back wheels hovering above the start line in the selected direction, and set it down when the stopwatch starts timing.
2. Let the buggy run until the stopwatch indicates that the set time has elapsed.
3. Stop the buggy in its place and turn it off immediately.
4. Use the meter stick to measure the distance between the start line and the front wheels of the buggy.
5. Repeat steps 1-4 two times for each time point. In total, there should be 12 data points collected, two for each time point for the first part.
6. Repeat steps 1-4, this time going in the opposite direction, conducting one trial for each time point. There should be a total of five data points collected for the second part.
*ITEMS NOT DRAWN TO SCALE
PART 1: Graph Analysis
The graph formed by the data from part 1 is best described as a linear progression with the equation:
x = 41t + 25.29
x - position (cm) t - time (seconds)
The slope of the line of best fit is 41 cm/second. This means that the buggy was traveling, on average, at a constant velocity of 41 centimeters per second North.
The y-intercept is 25.29 cm. This should indicate the starting position of the buggy, also known as the origin.
Likely Error: The true value of the origin should be 0 cm; however, given that the buggy's back wheels were lined up with the starting line instead of the front wheels, the length of the buggy likely accounts for the increased value of the y-intercept. This error does not affect the velocity data collected for the experiment.
PART 2: Graph Analysis
The graph formed by the data from part 2 is best described as a linear progression with the equation:
x = -37.95t - 20.25
x - position (cm) t - time (seconds)
The slope of the line of best fit is -37.95 cm/second. This means that the buggy was traveling, on average, at a constant velocity of 37.95 centimeters per second South.
The y-intercept is -20.25 cm. This should indicate the starting position of the buggy, also known as the origin.
Likely Error: The true value of the origin should be 0 cm; however, given that the buggy's back wheels were lined up with the starting line instead of the front wheels, the length of the buggy likely accounts for the decreased value of the y-intercept. This error does not affect the velocity data collected for the experiment.
Graphing Analysis
Based on the two graphs developed and the equations of their best fit lines, a general equation that could be used to describe the relationship between velocity, time, and position would be:
x = vt + i
x - position
v - velocity
t - time
i - initial position
In both part 1 and part 2 of the experiment, the slope of the best fit line was indicative of the constant velocity of the buggy, and the y-intercept was indicative of the initial position of the buggy. Using this general equation, the position of an object moving at a constant speed (a linear progression) can be determined, given that variables v, t, and i are known.
Conclusion
The purpose of this experiment was to investigate how the amount of time a buggy traveled for would affect its position. Through analysis of the data collected in both Part 1 and Part 2, it can be concluded that the relationship between time and the buggy’s position is linear, meaning that the buggy travels at a constant velocity. As time progresses, the buggy’s distance from its initial position increases at a constant rate.
A piece of evidence that further supports the conclusion of a linear relationship is the correlation value determined by Logger Pro. Correlation quantifies the strength of the linear relationship between a pair of variables; in this case, those variables would be time elapsed(t) and position of buggy (x). The closer the points lie to a straight line, the stronger the linear relationship between two variables. The correlation value for the Part 1 graph of Time vs. Position is .9978. This value is very close to 1 and indicates a strong positive linear relationship. This would suggest that for every set increase in the x value, the y value increases by a set value as well.
From this experiment, conclusions can be drawn about the relationship between time and position as well as time and velocity.
First, time and position will be considered in regard to speed and direction.
Direction in a position vs. time graph can be deduced by whether the slope is positive or negative. In this experiment, a “positive” direction was assigned to the North and a “negative” direction was assigned to the South. In the Part 1 Graph, where the buggy was traveling north, the value of the slope was positive, at 41cm/s. Comparatively, in the Part 2 Graph, where the buggy was traveling south, the value of the slope was negative, at -37.95cm/s. Direction is essential to calculating position. Distance, defined as change in location regardless of direction, measures only the total units traveled. Distance is not effective in determining position, because depending on which direction the buggy traveled in, and whether or not the buggy changed direction at all, the position will be altered. Displacement, defined as the overall change in position with respect to direction, is a much better measurement of position. In this experiment, if direction had not been taken into consideration, the data would’ve reflected that the buggy’s end position in Part 2 was between the start line and the end position of Part 1. This is evidently not true, as during Part 2, the buggy ended at -206cm, which is 832cm away from the ending position in Part 1.
In a position vs. time graph, which is depicted by the Part 1 Graph, speed can be deduced by the slope; specifically, the steepness or the shallowness of the slope. If a slope is steeper, the speed is faster, and if a slope is shallower, the speed is slower. In a position time graph with a steep slope, the y value is increasing by a greater amount for each increase in x value compared to a graph with a shallower slope. In the Part 1 Graph, the slope is 41cm/s; thus, for every 1 unit of increase in the x value, the y value increases by 41. If the buggy were traveling faster, say, at 100 cm/s, the slope would be much steeper as for every 1 unit of increase in the x value, the y value would increase by 100. If the buggy were traveling slower, say, at 20cm/s, the opposite would be true. A shallower slope would be depicted as the y value would only increase by 20 for every increase of 1 in the x value.
To go further in depth regarding slope, an analysis of the relationship between time and velocity can be conducted.
Direction in a velocity vs. time graph can be deduced by whether the y-value, representing velocity, is positive or negative. In this experiment, slope in the position-time graph is a representation of the velocity of the buggy. Because both position-time graphs for this experiment demonstrate a linear relationship, this means that the slope, or the velocity, will always be constant. Thus, as mentioned previously, a negative slope is indicative of a “negative” direction, which in this case, means the buggy was traveling south. Looking at the graphical representation provided below, the velocity of the buggy in both Part 1 and Part 2 are depicted on the velocity-time graph. The graph for the buggy in Part 1 is a flat line resting at 41cm above the x-axis. It is a positive value because in Part 1 the buggy was traveling North. The graph for the buggy in Part 2 is a flat line resting at -37.95cm. It sits below the x-axis because its negative value indicates that it was traveling south.
Speed in a velocity vs. time graph is represented by the absolute value of the y-value. Referring to the graphical representation below, the buggy in Part 1 traveled at a constant speed of 41cm/s, thus it is represented by a flat line that sits at “41” on the y axis. In Part 2, the buggy traveled at a constant speed of 37.95cm/s, thus it is represented by a flat line that sits at “-37.95” on the y axis. It is important to note that in both cases the velocity is represented by a flat line. This is indicative of the constant speed of the buggy, which supports the conclusion that there is a linear relationship between time and position in this experiment.
Evaluating Procedures
Improving the Investigation
The first greatest source of uncertainty in this lab is the human error involved with stopwatch timing. A big component of this has to do with reaction time. In this experiment, one researcher directed "GO" as another researcher pressed start on the stopwatch. The disparity that is created by the reaction time and the inconsistency from trial to trial likely prevented the data from having a perfect linear correlation.
The second greatest source of uncertainty in this lab was the starting and the stopping of the buggy. The buggy travels at quite a fast pace. Additionally, the switch is underneath the buggy and the device is not easily deterred. Thus, there were times when the buggy was somewhat difficult to stop. It is likely that there were times when the buggy's position was shifted in the process, leading to inaccuracies in the data that would have prevented a perfect linear correlation value of 1.
A third source of uncertainty in this experiment was the measurement process. Because the researchers did not use a tape measure, the alternative was measuring a meter stick length, then marking the end point and shifting the meter stick to meet the end point, repeating this process until the meter sitck reached the buggy. This measurement method holds room for error in that the meter stick was slightly tilted at times, resulting in inaccuracies. Additionally, the end point was often marked by a researcher's finger, reducing the precision of the measurement.
A fourth uncertainty in this experiment was the buggy's course. Although great efforts were made to set the buggy on a straight course forward, it often traveled somewhat diagonally. This shift in direction would have resulted in a slightly differing position measurement compared to a truly straight course. Additionally, the degree of shift differed from trial to trial, resulting in inconsistency within the experiment.
--------------------------------------
Given the uncertainties listed above, the most accessible improvement would likely be the measurement. With a tape measurer, the error from marking end points and tilted meter sticks could be reduced significantly. It would also provide more consistency within the data, as the errors involved with marking end points and tilted meter sticks differed from trial to trial.
Cart Gone Wild: Journey Down the Ramp
Researchers: Emmy Xu, Elle Grillo, Ibrahim Kahn
Lab conducted on: September 23, 2021
Research Question
Variables
Controls
This experiment aimed to deduce how the amount of time a cart traveled for would affect its position.
Independent variable: time elapsed (seconds)
Dependent variable: position of the cart (cm)
For the purpose of this experiment, time was defined in seconds, and "position" was defined as the displacement of the cart in centimeters from its initial position.
Controls: the cart itself, the height of the ramp, the length of the ramp, the stability of the camera (not handheld)
For this experiment, the position of the cart was tracked through the movie software on Logger Pro. Thus, it was imperative that the camera used to film remain stable. The camera used was propped onto a computer (instead of being held by hand) to maintain stability. Additionally, the same cart and same ramp (in terms of length and elevation level) were used throughout the experiment.
Method - Procedure
METHOD
The challenge with this experiment was that it would have been difficult to stop the cart (and subsequently take measurements) as it rolled down the ramp down to the precise second due to the momentum of the cart. Thus, a different method was developed. A camera was placed at a direct side view of the cart (to prevent perspective distortion), and a clip was filmed of the cart rolling down the ramp from beginning to end. A meter stick was placed parallel to the side of the ramp to offer a metric of measurement. This video was then plugged into Logger Pro's "movie" software. The meter stick in the video was set to 100cm, and the origin was set to the location of the front tip of the cart before it started moving. As the video played frame by frame, points were plotted based on the position of the front tip of the car as it traveled down the ramp.
PROCEDURE
1. Set up the camera in a stable position with a direct side view of the ramp.
2. Set up a meter stick parallel to the side of the ramp
3. The researcher holding the cart in position at the top of the ramp releases the cart as soon as the researcher behind the camera starts recording.
4. The researcher positioned at the bottom of the ramp stops the cart as it reaches the bottom.
5. The video is plugged into the Logger Pro software to be used for data collection.
*items not drawn to scale
Data Table
The data table to the left shows two sets of data: the position values recorded, in centimeters, and the velocity values, in centimeters per second, relative to time.
Because there was only one trial, there was no processing of data. The raw data represents the data used for analysis.
**Calculation error in graphic above
The acceleration should be approximately 2 * (a-value) not .5 * (a-value).
Acceleration = 2 * 22.29 = 44.58 cm/s/s
Conclusion
The purpose of this experiment was to investigate how the amount of time a cart traveled for would affect its position. Through analysis of the data collected using Logger Pro’s movie software, it can be concluded that the relationship between time and the cart’s position is quadratic with a positive “b” value, meaning the cart in this experiment travels with continually increasing velocity. This is supported by the velocity data collected using Logger Pro, which shows that the relationship between time and the cart’s velocity is linear, meaning the cart is accelerating at a constant rate.
From this experiment, conclusions can be drawn about the relationship between position, velocity, and acceleration. One of the most effective ways of drawing such conclusions is through analysis of the position - time, velocity - time, and acceleration - time graphs.
From a position vs. time graph, the velocity can be deduced by the slope of the graph. The most basic principle: if the slope of the position - time graph is constant, the velocity is constant. This was demonstrated in the previous buggy lab, in which the slope of the position - time graph was 41 cm/s, indicating that the buggy lab traveled at a constant rate of 41 cm/s North throughout the duration of its run. In this lab, the slope of the cart’s position - time graph displays a curvature, indicating that the cart’s velocity is not constant, but rather, changing. In other words, the cart is accelerating. The equation for the position time graph, -22.9(t^2) + 33.13t - 44.58, represents the initial velocity of the cart with its b-value “33.13”, meaning the cart started traveling at a rate of 33.13 cm/second. This means that initially, for every second that passed, the cart traveled 33.13 centimeters. The c-value (divided by 2) “-44.58” represents the acceleration of the cart, *44.58 cm/second/second. This means that for every second that passed, the velocity of the car increased by 44.58 seconds. Thus, at t=2, the velocity of the car should’ve increased: 33.13 + 44.58 = 77.71 cm/second. When the position of an object travels at a changing velocity, its position - time graph is best modeled by a quadratic equation, where the slope of the graph displays a curvature.
*flipped video problem -- acceleration appears negative, even though it was positive
In a velocity vs. time graph, the displacement (and therefore position) can be deduced by calculating the area under the graph, and the acceleration can be deduced by calculating the slope of the graph. The position concept is applicable because by multiplying the velocity by the amount of time an object spent traveling at that velocity, the displacement can be determined. If an object travels at a constant velocity, the displacement would simply be calculated by the product of the velocity and the total time traveled. If an object’s velocity is changing, the velocity graph can be divided into geometric shapes, and the areas of those individual pieces can be calculated and then added together. In this experiment, the velocity graph forms a triangular shape. Thus, the displacement is calculated by using the equation .5 x base x height. It is important to note that the area above the x-axis is positive, and the area below is negative. Thus, in calculating distance, the total area above and below should be added together. However, if calculating displacement, the area below must be subtracted from the area above.
The velocity vs. time graph also indicates the acceleration of the cart. Similar to the relationship between the slope of the position - time graph and velocity, the slope of a velocity - time graph represents the acceleration of the object. In this experiment, the velocity - time graph is best modeled by a linear regression model, meaning that the slope is constant, and therefore, the acceleration is constant as well. In the analysis of the velocity - time graph, it is indicated that the slope was -43.12cm/s/s. This means that for every second that passes, the velocity increases by *43.12 cm/s.
*flipped video problem -- acceleration appears negative, even though it was positive
Evaluating Procedures
Improving the Investigation
The first greatest source of uncertainty in this experiment was the lack of repeated trials. One of the most important factors to consider in determining confidence in results is the number of trials conducted. Ideally, there would be many repeated trials that all yielded the same results. Having many repeated trials with data that is in consensus would strengthen the conclusion reached and the confidence in the conclusion. In this experiment, only one trial was conducted. Thus, since there were no repeated trials, there was only one set of data used for graphing, analysis and determining results. This aspect of the experimental set up weakens the conclusion reached.
The second greatest source of uncertainty in this experiment was the camera angle at which the set up was filmed from. Camera angle can have a significant influence on the measurements taken by Logger Pro. This is also a disadvantage of collecting data on a digital software rather than in real time. A camera placed at an upward angle would distort the proportions of the objects being measured in the video, as would a camera placed at a lower angle. However, it is difficult to achieve a position that views the set up from a truly parallel side angle. Thus, the inevitable distortion would likely lead to inaccuracies in the data collection step of the experiment, as it can distort the true position of the cart.
---------------------------
One way in which the experiment could be improved is by running more trials and determining whether the data collected from those trials is in consensus with the data from the first trial, and therefore, supports the conclusion reached. If the results of the repeated trials differed from that of the first, further experiments could be conducted to investigate the disparity and reach a new conclusion. If the results support the conclusion reached, it would add confidence to the experiment.