Homework 1- Introduction to Motion

Transcription

1 Name: Homework 1- Section: In your first Minilab, you will be introduced to a device called a motion detector. Our detectors use sound waves to measure how far away an object is at any given point in time. Through a special computer program, the detector creates a plot of the locations the object has been, and thus maps the object s motion. In order to help you identify and analyze the graphs you will encounter in the lab, this homework assignment will explain the graphs you will see and contains exercises that will teach you the skills you will need in your first lab. This assignment is also designed to help you get used to the conventions that we use in the lab, as well as the general layout of your lab workbook. As will be expected through this course, please be sure to finish this assignment before coming to class so you can be prepared for the labs you will participate in. This being said, welcome to Physics 2015! Graph Type 1: Position-Time Graphs The simplest graph you will see is the position-time graph. In this graph, time is located on the horizontal axis, and your position (or the distance between the object and the detector) is located on the vertical axis. In this course, position is denoted by the letter x. From this type of graph, we can determine: 1) where an object has been, 2) how it changes position, and 3) how quickly or slowly it changes position. Consider the following example below: In the graph on the left, we see a line sloping down x and to the right. We describe this object s motion by saying that it starts far away from the detector (since it is far from x = 0 m, which is the location of the detector), and that it moves closer as time progresses. Since the line has is straight, we can say the object is moving with constant velocity. In this case, that velocity would be negative (since it moves towards the detector). These are conventions that you will keep in the lab itself, so please be sure to remember that, in the case of velocity, negative means towards the detector! We can also use this type of graph to relate the motion of two different objects together. In the graph on the right, we have two objects (object A and object B) that are moving with respect to the detector. We see that the line associated with B is steeper than that of A, so we can say that B is moving faster than A. We also see that A and B are both moving away from the detector, so it makes most sense to say that the object that is ahead is the one that starts farthest away from the detector. Thus, we say that A starts ahead of B since it is higher on the x-axis at t=0. B A With this information, you are now ready to start analyzing position-time graphs. On the following page, you will be asked to identify some of these types of graphs, as well as sketch a few for yourself. Feel free to refer back to this first if you have any questions. Now, please solve Problems 1 through 3 on the following page. University of Utah Department of Physics & Astronomy 1

2 P.1 [.33 pts. per part] Below are 4 different graphs from a motion detector. Describe how you would move in order to generate the following graphs. Note that two of the graphs have 2 different types of motion, so use slower and faster when talking about their speeds. c) a) b) c) P.2 [.25 pts. per part] The following is a distance-time graph of objects A and B created by a motion detector. Please answer the following questions about their motion (Example 2). B A a) Which object is moving faster, A or B? b) Which object has a negative velocity? c) What does the intersection mean? d) Describe the relative motion of the two objects. c) d) P.3 [.5 points per part] Please sketch the following situations on the graphs below. a) The object starts a few meters from the origin and moves with constant velocity towards the origin for 5 seconds, then stands still for 5 seconds. (Note: Each division on the time axis corresponds to 2.5 seconds). b) The object moves away from the origin while continuously speeding up (be careful! This is NOT a situation with constant velocity. You may need to go online for additional help with this problem.) Plot for a) Plot for b) University of Utah Department of Physics & Astronomy 2

3 Graph Type 2: Velocity-Time Graphs The second, and more complicated, type of graph is the velocity-time graph. These graphs tell us how fast the object moves relative to the motion detector. Depending on what you want to examine, such graphs can be more useful for analysis than the position-time graph even though the position-time graph indirectly also contains information about the velocity of the object. When interpreting velocity-time graphs, you must be careful not to confuse them with positiontime graphs. Before we continue, there are a few definitions that must be understood. In your Physics 2010 class, you learn about velocity, which is a vector quantity (something that possesses a length and a direction) that relates how quickly an object moves per a unit of time (usually seconds). In addition, you learn about acceleration, which is also a vector quantity and relates how a velocity changes over a given period of time (or, in plain English, how an object is speeding up or slowing down ). These concepts, at this point, may seem very cryptic and vague. Velocity-time graphs help to cement and clarify these essential concepts. To figure out how position, velocity, and acceleration are related, we must introduce two mathematical ideas. The first of these is called slope, which should be familiar to you from your Algebra classes. In the example on the right, we see a line starting at the origin and ending at a final v velocity of 5 m/sec in 4 seconds. We can calculate the slope using: 5 0 / 1.2 / 4. 0 Note the units we get back are the same units as those for acceleration. Thus, the reason the slope is important on a velocity-time graph is because acceleration is the slope of a velocity-time graph. The second mathematical idea we will use is area under the curve. Although this is a calculus concept, our approach will be purely geometric since the curves we will analyze will be straight lines. If we look at the graph above, we see that our line forms a triangle with a base of 4 (made from the x-axis) and a height of 5 (made from the y-axis). Thus, by using the area formula of a triangle, we see the area under the curve is: sec Note that our units end up as meters, which is a unit of distance. This, in fact, is exactly the total distance travelled during the time period! Thus, area under the curve is important because position is the area under the curve of a velocity-time graph. Before you start then next set of problems, we need to explain one more idea. If the line ends up below the time axis, you need to use negative values (for example, -2 m/sec). Physically, know that velocities above the time axis are moving AWAY from the detector, whereas those below are moving TOWARD the detector. By the way, how would you create the graph above in an experiment? Since you will be asked to do this in the lab, I will explain this for the graph above. You can see that, initially, the speed of the object is 0 m/sec, and, over time, it University of Utah Department of Physics & Astronomy 3

4 increases to 5 m/sec. To physically create this, you would need to start an object at rest and slowly increase speed in the positive direction until it travels at 5 m/sec. Since the velocities are all positive, the object would be moving away from the detector. Thus, to create the graph above, you would start the object at rest and move it faster and faster away from the detector. With this in mind, please solve Problems 4 through 6. Important Points about Velocity-Time Graphs: Velocity-time graphs are NOT the same as position-time graphs! They tell us how fast the object moves, NOT its exact location! An object s acceleration is calculated by finding the slope of a velocity-time graph. An object s position is calculated by finding the area under the curve of a velocity-time graph. Our convention is that positive velocities move AWAY from the detector, and negative velocities move TOWARDS the detector. P.4 [.5 pts. per part] Please describe the type of motion you would need to make in order to create the following velocity-time graphs (again, don t confuse these with position-time graphs!) P.5 [.75 pts. per part] The velocity-time graph of an object is shown below. Calculate the total distance travelled by the object. Show what equations you used to get your answer. (Hint: Remember that distance can be found by calculating area under the curve!) a) 3.0 a) b) b) University of Utah Department of Physics & Astronomy 4

5 P.6 [.75 pts. per part] Sketch velocity-time graphs for the given situations on the graphs provided below. Please reference the examples earlier for help if you get stuck. a) The object moves toward the detector at a constant velocity for half the time of its travel, then stands still for the rest of the time period. b) The object moves away from the detector for half the time, then instantly moves back toward the detector at the same speed as before for the duration of the time period. Relating Position-Time Graphs and Velocity-Time Graphs Now that we understand both position-time and velocity-time graphs, it is time to relate the two together! In order to do this, you will likely have to create both graphs. We recommend you start with the position-time graph, since this is probably the most familiar to you. It is also easiest to create a velocity-time graph from a position-time graph if you know this fact: the slope of a position-time graph is the velocity! This should make sense, especially since this is similar to the relationship between acceleration and the slope of a velocity-time graph. To solidify these concepts, note the following example. Given the following description of an object s motion, you are asked to draw both position- and velocity-time graphs: An object starts at the detector and travels 3 meters from the detector in 2 seconds at a constant velocity. For the next 2 seconds, the object stays at rest. So, where do we start to create these graphs? If we start by drawing the position-time graph, we note the first section of motion (travelling 3 m in 2 sec) relates to a straight line with endpoints at (0,0) and (2,3). Then, for the second section, we know an object at rest on a position-time graph is a horizontal line (the distance from the detector doesn t change). Thus, the graph we draw is the same as is shown on the left. Great! Now, we need to figure out how to create a velocity-time graph. Remembering the hint that velocity is the slope of the distance-time graph, we can find the slope for the two different sections. We can see by looking at the position-time graph that the second section has no slope (it is a flat line), so its slope and, hence, its velocity is equal to zero. This makes sense because the object is standing still during this period. For the first section, we find the slope by using the same formula we used for acceleration: 1.5 /. University of Utah Department of Physics & Astronomy 5

6 Notice that the units we get back are m/sec, which are the correct units for velocity. We can then draw the velocity-time graph, which we see is drawn on the right. Two quick points needs to be made here on notation. Note that we used dotted lines to distinguish a zero velocity/position on the horizontal axis. This is not necessary for you to do in your graphs, but be clear enough in your drawings so your grader knows you are not leaving the axis blank. Also, note that we included vertical lines to indicate where velocities suddenly change. This is called instantaneous change because it happens immediately, or at a single instant. Although this does not happen in real life, it is used in this Physics course because the time it takes for many objects to change is considered negligible or occurring in too short of a time period to worry about calculating. With this introduction, it is now your turn to practice! Note that the two problems at the end are worth more points than those earlier in this homework, which indicates these concepts are the most important for you to know for your lab sections. This rule of thumb will be useful for you to know for the rest of the homework and the labs you will do in this course. With this being said, please answer Problems 7 and 8 below. P.7 [1 pt.] Below, you are given a position-time graph. Fill in the corresponding velocity-time graph with the information you gather from the position-time graph. Note time is in seconds, velocity in meters/second, and distance in meters. P.8 [.25 per part per graph 2pts total] Please draw the distance-time graph and velocity time graph for an object that starts at the origin and travels in the following manner (with correct signs): a) moves away from the origin at a constant velocity of 1 m/sec for the first 5 seconds of travel, b) stands still for the next 5 seconds, c) moves 3 more meters away from the origin in the next 5 seconds at constant velocity, and d) travels back to its starting position at constant velocity in 5 seconds University of Utah Department of Physics & Astronomy 6