Kinematics Lab #1: Walking the Graphs Results and Discussion By: Teacher: Mr. Chung Due: October 28, 2010 SPH3U1-01 1
Introduction The goal of this lab was to match, as accurately as possible, three position-time graphs and three velocity-time graphs by walking back and forth in front of a Vernier motion detector. Then, the motion detector transmitted the collected data to a laptop where it was plotted in real-time over the original graph that was supposed to be matched. The following two sections describe the results of the experiment and the conclusions that can be drawn from it. You don t need this since you have Graph 1 in the title. Results Graph 01b The first graph was a position-time graph. After walking this graph, the Logger Pro software outputted the following graph. The blue labels were added to facilitate the explanation of specific sections of the graph. Good Job!! Change to Figure 1 will be better. Description is good. Graph 1: The position-time graph obtained from walking graph 01b. In order to correctly walk this graph, one had to start at a distance of 1 m from the motion detector. After the motion detector started recording his movements, the person remained stationary at 1 metre for 1 second, represented by the fact that the line for section 1 has a slope of 0. Then the person started walking away from the motion detector in the positive direction, travelling 1.5 metres in 2 seconds. This brought the person to a position of 2.5 metres away from the motion detector. He remained there for 3 seconds. At the end of the 3 seconds, the person started to walk in the negative direction towards the motion detector, travelling a total of 0.75 metres in 1.4 seconds. This left the person at a position of 1.75 m from the origin, where he remained stationary for the remaining 2.6 s of the graph. Page 2
Graph 01c The next graph that had to be matched was similar to the first one in the sense that it was also a position-time graph: Graph 2: The position-time graph obtained from walking graph 01c. For this graph, the person had to start at 3.0 metres from the motion detector. However, the red line indicates that the person actually started at about 3.2 metres away from the motion detector. Then, immediately after the motion detector started recording, he walked approximately 1.5 metres towards the motion detector in a period of about 3.0 seconds. Now the person was about 1.5 metres away from the motion detector, where he remained at rest for a period of about 1.0 second. After that period of rest, he took 1.0 second to further walk 1.0 metre towards the origin. He remained there for about 2.0 seconds. After two seconds at rest, the person walked 2.5 metres away from the motion detector in a period of about 3.0 seconds. By the time the person finished walking this graph, he ended up at about 3.0 metres from the origin, the same as his starting position. Page 3
Graph 01d Graph 01d was different from the first two graphs because it was a velocity time graph. Instead of the y-axis representing the position, the y-axis now represents the velocity of the person walking the graph. Graph 3: The velocity-time graph obtained from walking graph 01d. In order to walk this graph, the starting position of the walker did not matter because the graph only specified the velocity of the person and not his position. Therefore, an arbitrary position in front of the motion was chosen to be the starting position. From that starting position, the person remained stationary for the first 2 seconds of the graph. This rest period is represented by the fact that, in section 1 of the graph, the line remains constant at a velocity of 0.0 m/s during this time. After this period of rest, the person started to walk away from the motion detector at a relatively constant velocity of 0.5 m/s for about 3 seconds. After the 3 seconds of motion, the person stopped moving for 2 seconds, causing the velocity to return to 0.0 m/s in section 3 of the graph. Then the walker reversed his direction and started walking towards the motion detector in section 4 at a constant velocity of 0.5 m/s for the remaining 3 seconds. Page 4
Graph 01e Graph 4: The velocity-time graph obtained from walking graph 01e. This velocity-time graph was more difficult to match than the previous velocity-time graph due to the constant change in velocity during section 1 of the graph. The person walking the graph started at a velocity of 0.0 m/s. The he started to walk away from the motion detector while increasing his velocity. This acceleration is represented by the fact that the line in section 1 is sloped, denoting a gradual increase in the person s velocity. He continued to accelerate away from the motion detector for 4.0 seconds. By then the person s velocity had reached a velocity 0.5 m/s. The walker continued to move away from the motion detector for 2.0 seconds, but now his velocity was constant at 0.5 m/s as one can tell by the fact that the line in section 2 has a slope of 0. At the 6-second mark, the walker reversed his direction and started walking towards the motion detector at a constant velocity of about -0.4 m/s for 3 seconds. Then he stopped and remained stationary for the remaining 1.0 second. However, because of the experimental error associated with this lab, the experimental result for this graph deviates considerably from the graph that was supposed to be matched. Page 5
Graph 01f Graph 5: The velocity-time graph obtained from walking graph 01f. Walking this graph was similar to walking the first two graphs because it was also a position-time graph. Starting at a position of about 0.9 metres from the motion detector, the walker walked a total of 1.0 metre away from the origin in 3.4 seconds in section 1 of the graph. Then he stopped walking and remained stationary for about 3.3 seconds. The person then continued to walk away from the origin for section 3 of the graph. He walked 1.4 metres away from the origin in 3.3 seconds. Graph 01g Graph 6: The velocity-time graph obtained from walking graph 01g Page 6
This was the last graph that had to be matched. The graph was not matched by walking; it was matched by holding up a binder in front of the motion detector and moving it back and forth according to the velocity. Since this was a velocity-time graph, the binder s initial position was arbitrarily chosen. Section 1 of the graph involved moving the binder away from the motion detector at a velocity of about 0.3 m/s for about 3 seconds. Then, for section 2 of the graph, the binder was moved towards the motion detector at a velocity of about 0.3 m/s for about 4 seconds. The binder was then held stationary for the remaining 3 seconds of the graph. The data collected in the results section will be further discussed in the discussion section of this lab report. Discussion This discussion section will be focused on three main topics: graph translations, experimental errors and real-world applications of the physics concepts that are used in this lab. Graph translations This part of the discussion section will be focused mainly on two of the graphs from the results section: graph 01b, and graph 01e. Graph 1: The position-time graph obtained from walking graph 01b. According to the textbook Physics: concepts and connections, one can calculate the velocity of a particular object by calculating the slope of its position-time graph (1). The generic formula for the slope of a line is: Page 7
Slope = rise / run In the case of a position-time graph, however: Since the slope = velocity, Velocity = Δd / Δt By applying this formula to graph 01b, the velocity of section 1 is 0 m/s because the line in section 1 has a slope of 0 since the line has a change in position but not a change in time. Then, by applying this same formula to section 2 of the graph, one sees that: The velocity of section 2 = rise / run = Δd / Δt = 1.5 m / 2.0 s = 0.75 m/s The velocity of section 2 is also constant since the slopes for every point on the line are equal to each other. And because the velocity of section 2 is positive, it means that the person was moving away from the motion detector during that period of time. Like section 1, section 3 of the graph also has a velocity of 0 m/s since it has a slope of 0. Section 4, on the other hand, has a slope of -0.54. Therefore, the straight line in section 4 of the graph represents a constant velocity of 0.54 m/s, meaning that the person is now moving towards the motion detector at a speed of 0.54 m/s. During section 5, the walker was at rest for the remaining 2.6 seconds of the graph. One can then use this information to plot a velocity-time graph that describes a motion that is equivalent to the motion described in the position-graph (1): 1 0.8 0.6 Velocity (m/s) 0.4 0.2 0-0.2-0.4-0.6-0.8 0 2 4 6 8 10 Time (seconds) Graph 7: The velocity-time graph derived from Graph 1. Page 8
After creating the velocity-time graph, one can then go a step further and use the velocity-time graph to plot an acceleration-time graph. Acceleration is simply the rate at which the velocity of an object changes. According to Physics: concepts and connections, the formula to calculate acceleration is: a = (v2 v1) / (t2 t1) or a = Δv / Δt Another way of calculating the acceleration is by calculating the slope of the line in a velocity-time graph (1): Since acceleration = (v2 v1) / (t2 t1), and slope = (v2 v1) / (t2 t1) Then acceleration = slope. However these equations are not necessary for calculating the acceleration in graph 7 because during the entire 10 seconds of the graph the walker was either at rest or moving at a constant velocity. A sustained constant change in velocity did not occur. Therefore, the acceleration for graph is 0.0 m/s^2: 1 0.75 Acceleration (m/s^2) 0.5 0.25 0-0.25-0.5-0.75-1 0 2 4 6 8 10 Time (seconds) Graph 8: The acceleration-time graph derived from Graph 7. Graph 01e is trickier to translate because it is a velocity-time graph: Page 9
Graph 4: The velocity-time graph obtained from walking graph 01e. In order to translate a velocity-time graph into a position-time graph, one must first find the area under the graph, in other words, the area of the shape formed by each section of the graph and the x-axis (1). One does this by using two formulas: area = length x width for rectangles Or area = (base x height)/2 for triangles These formulas allow for the calculation of the displacement. For example, section 1 of the graph forms a triangular shape with the x-axis. Therefore: area of section 1 = (base x height)/2 = (4.0 s x 0.5 m/s)/2 = 2.0 m / 2 = 1.0 m Notice how the calculation of the area of section 1 results in the s unit being cancelled out, and leaves a figure that represents a distance. Therefore, the displacement of the motion in section 1 of the graph is 1 metre in the positive direction, away from the motion detector. If the formula for rectangles is applied with section 2 of the graph: area of section 2 = l x w = 2.0 s x 0.5 m/s = 1.0 m The displacement for section 2 is 1.0 m away from the motion detector. Using the same formula: Page 10
area of section 3 = 3 s x 0.4 m/s = -1.2 m The displacement for section 3 is 1.2 m, meaning 1.2 metres closer to the origin. The displacement for section 4 is 0.0 m because the velocity is 0.0 m/s. When plotting the position-time graph, it is important to add each successive displacement value to the values that come before it. For example the net displacement of sections 1 and 2 is 2.0 m since 1.0 m + 1.0 m = 2.0 m. The net displacement of sections 1, 2, and 3 is 0.8 m since 1.0 m + 1.0 m + (-1.2 m) = 0.8 m. The new position-time graph translated from Graph 4 will look like this (note, the starting position is assumed to be at 0.0 m): Graph 9: The position-time graph derived from the Graph 4. Notice how the first part of the graph is curved. This is because the velocity during section 1 of the velocity-time graph is not constant. It get progressively greater as time passes. Therefore, the slope of section 1 of the position-time graph must get progressively steeper as time passes. The lines for the rest of the sections are all straight because their velocities are all constant (1). Next, one can plot an acceleration-time graph using the information given in Graph 4. As mentioned earlier, the slope of the line in a velocity-time graph equals the acceleration. Therefore: Acceleration of section 1 = Δv / Δt = (0.5 m/s) / (4.0 s) = 0.13 m/s^2 Page 11
The acceleration for section 1 is constant because the line in section 1 is straight, meaning that the velocity increases by the same amount each second. The acceleration for sections 2, 3, and 4 is 0.0 m/s^2, since the slopes of those sections are all 0.0. The new acceleration-time graph would look something like this: 0.5 Acceleration (m/s^2) 0.25 0-0.25 0 2 4 6 8 10-0.5 Time (Seconds) Graph 10: The acceleration-time graph derived from Graph 4. Experimental Error Experimental error was rife in this lab. In the results section, none of the experimental lines match the theoretical lines in any of the six graphs. Some experimental lines deviate quite considerably from the theoretical lines, especially among the velocity-time graphs. There were many causes of experimental error in this lab. Most of the experimental error was due to the fact that human movements are very imprecise. Our bodies lack the precision needed to move perfectly according to the motions described in the graphs (2). As a result, our bodies cannot change velocity instantly and are unable to maintain a perfectly constant velocity or a perfectly constant acceleration. Also, in addition to our innate muscular imprecision, the malfunctioning of the Vernier motion detector could have also played a role in producing experimental errors. In Graph 2, near the beginning and the end of the graph, small blips are clearly visible within the experimental line. These were most likely the result of the motion detector malfunctioning and producing incorrect measurements. Perhaps the walker inadvertently kept on shifting the binder that he was holding up, causing the ultrasound waves from the motion detector to bounce off the binder at various angles, away from the motion detector. These erratic movements could have caused the blips in Graph 2 (3). Page 12
Real-World Applications The concepts used in this lab have many applications in the real world. For example, evolution has bestowed the gift of echolocation upon bats. Bats use their ability of echolocation to navigate and hunt for prey in pitch darkness. Echolocation is the process that occurs when a bat sends out pulses of ultrasound into its surroundings that then bounce back from obstacles or prey (4). The bat then is able to interpret the ultrasound waves reflected back at it and paint a sonic picture of its surroundings. The basic operating principle of echolocation is not unlike that of the Vernier motion detector. They both bounce ultrasound waves off of an object (3). However, a bat s echolocation is much more advanced than the Vernier motion detector. The motion detector can only measure the position of a single point on an object, whereas a bat can measure the position of many points on many objects in its surroundings, enough to paint a picture of its surroundings. Basically, one can say that the bat is combining many individual positiontime graphs into one grand position-time graph that allows it to paint a real-time picture of its surroundings. Page 13
References: (1) Nowikow, I. & Heimbecker, B. (2001). Physics: concepts and connections. Toronto, Ontario: Irwin Publishing. (2) Duffy, V. D. (Ed.). (2007). Digital human modeling: First International Conference on Digital Human Modeling, ICDHM 2007, held as part of HCI International 2007, Beijing, China, July 22-27, 2007 : proceedings. Berlin: Springer. (3) (n.a.). (February 17, 2010). Motion Detector 2. Retrieved October 28, 2010, from http://www2.vernier.com/booklets/md-btd.pdf (4) (n.a.). (May 01, 1950). Science: Bat Sonar. Time Magazine. Retrieved October 27, 2010, from http://www.time.com/time/magazine/article/0,9171,812332,00.html Page 14