{"id":5,"date":"2017-06-30T18:26:59","date_gmt":"2017-06-30T18:26:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/atd-monroecc-physics\/chapter\/chapter-1\/"},"modified":"2019-01-22T17:24:29","modified_gmt":"2019-01-22T17:24:29","slug":"chapter-1","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-monroecc-physics\/chapter\/chapter-1\/","title":{"raw":"Kinematics","rendered":"Kinematics"},"content":{"raw":"

Kinematics<\/h1>\r\n

Concepts and Principles<\/h2>\r\nKinematics is the formal language physicists use to describe motion. The need for a formal language is evidenced by a simple experiment: drop an object from about shoulder height and ask two people to independently describe the motion of the object. Chances are that the descriptions will not be in perfect agreement, even though both observers described the same motion. Obviously, a more formal way of describing motion is necessary to eliminate this type of descriptive ambiguity. Kinematics is the formal method of describing motion.\r\n\r\nThree parameters are carefully defined and used by physicists to describe motion. Specifying these three parameters at all times forms a complete description of the motion of an object.\r\n

Position<\/h3>\r\nThe position of an object is its location relative to a well-defined coordinate system at a particular instant of time. Without a specified coordinate system, position is a meaningless concept. A coordinate system is comprised of a zero<\/em>, a specified positive direction<\/em>, and a scale<\/em>.\r\n\r\nFor example, in the hypothetical experiment in which the object was dropped from shoulder height, a coordinate system could have been defined in which the zero position was at ground level, the positive direction was up, and the scale used was meters. Using this coordinate system, the position of the object could have been specified at any particular instant of time. Of course, choosing the zero at the location at which the object was dropped, the positive direction as down, and the scale in feet is also perfectly acceptable. It doesn\u2019t matter what you choose as a coordinate system, only that you explicitly choose one. Depending on the coordinate system chosen, the position of an object can be positive, negative, or zero.\r\n\r\nWe will use the symbol r to designate position, and measure it in meters (m).\r\n

Velocity<\/h3>\r\nAlthough the word velocity is often used loosely in everyday conversation, its meaning in physics is specific and well-defined. To physicists, the velocity is the rate at which the position is changing. The velocity can be specified at any particular instant of time.\r\n\r\nFor example, if the position is changing quickly the velocity is large and if the position is not changing at all the velocity is zero. A mathematical way to represent this definition is\r\n\r\n\"\"\r\n\r\nThus, velocity is measured in meters per second (m\/s).\r\n\r\nActually, this is the definition of the average<\/em> velocity of the object over the time interval Dt, but as the time interval becomes smaller and smaller the value of this expression becomes closer and closer to the actual rate at which the position is changing at one particular instant of time.\r\n\r\nSince the final position of the object (rfinal) may be either positive, negative, or zero, and either larger, smaller, or the same as the initial position (rinitial), the velocity may be positive, negative, or zero. The sign of the velocity depends on the coordinate system chosen to define the position. A positive velocity simply means that the object is moving in the positive direction, as defined by the coordinate system, while a negative velocity means the object is traveling in the other direction.\r\n\r\n \r\n

Acceleration<\/h3>\r\nAgain, although the word acceleration is often used loosely in everyday conversation, its meaning in physics is specific and well-defined. To physicists, the acceleration is the rate at which the velocity is changing. Again, the acceleration can be specified at any particular instant of time.\r\n\r\nFor example, if the velocity is changing quickly the acceleration is large in magnitude, and if the velocity is not changing the acceleration is zero. If an object has non-zero acceleration, it does not<\/em> mean that the object is speeding up. It simply means that the velocity is changing. Moreover, even if an object has a positive<\/em> acceleration, it does not mean that the object is speeding up! A positive acceleration means that the change in the velocity points in the positive direction. (I can almost guarantee you will experience confusion about this. Take some time to think about the preceding statement right now.)\r\n\r\nA mathematical way to represent acceleration is\r\n\r\n\"\"\r\n\r\nThus, acceleration is measured in meters per second per second (m\/s2).\r\n\r\nAgain, this is actually the average acceleration of the object over the time interval Dt, but as the time interval becomes smaller and smaller, the value of this expression becomes closer and closer to the actual rate at which the velocity is changing at one particular instant of time.\r\n\r\nSince vfinal may be either positive, negative, or zero, and either larger, smaller, or the same as vinitial, the acceleration may be positive, negative, or zero. The algebraic sign of the acceleration depends on the coordinate system chosen to define the position. A negative acceleration means that the change in the velocity points in the negative direction. For example, the velocity could be in the positive direction and the object slowing down or<\/em> the velocity could be in the negative direction and the object speeding up. Both of these scenarios would result in a negative acceleration. Conversely, a positive acceleration means that the change in the velocity points in the positive direction.\r\n\r\nKinematics is the correct use of the parameters position, velocity, and acceleration to describe motion. Learning to use these three terms correctly can be made much easier by learning a few tricks of the trade. These tricks, or analysis tools, are detailed in the following section.\r\n

Analysis Tools<\/h2>\r\n

Drawing Motion Diagrams<\/h3>\r\nThe words used by physicists to describe the motion of objects are defined above. However learning to use these terms correctly is more difficult than simply memorizing definitions. An extremely useful tool for bridging the gap between a normal, conversational description of a situation and a physicists\u2019 description is the motion diagram. A motion diagram is the first step in translating a verbal description of a phenomenon into a physicists\u2019 description.\r\n\r\nStart with the following verbal description of a physical situation:\r\n
\r\n

The driver of an automobile traveling at 15 m\/s, noticing a red-light 30 m ahead, applies the brakes of her car until she stops just short of the intersection.<\/p>\r\n\r\n<\/div>\r\n

Determining the position from a motion diagram<\/h4>\r\nA motion diagram can be thought of as a multiple-exposure photograph of the physical situation, with the image of the object displayed at equal time intervals. For example, a multiple-exposure photograph of the situation described above would look something like this:\r\n\r\n\"\"\r\n\r\nNote that the images of the automobile are getting closer together near the end of its motion because the car is traveling a smaller distance between the equally-timed exposures.\r\n\r\nIn general, in drawing motion diagrams it is better to represent the object as simply a dot, unless the actual shape of the object conveys some interesting information. Thus, a better motion diagram would be:\r\n\r\n\"\"\r\n\r\nSince the purpose of the motion diagram is to help us describe the car\u2019s motion, a coordinate system is necessary. Remember, to define a coordinate system you must choose a zero, define a positive direction, and select a scale. We will always use meters as our position scale in this course, so you must only select a zero and a positive direction. Remember, there is no correct answer. Any coordinate system is as correct as any other.\r\n\r\nThe choice below indicates that the initial position of the car is the origin, and positions to the right of that are positive. (In the text I\u2019ll always use a little diamond to indicate the zero with an arrow pointing in the positive direction.)\r\n\r\n\"\"\r\n\r\nWe can now describe the position of the car. The car starts at position zero and then has positive, increasing positions throughout the remainder of its motion.\r\n

Determining the velocity from a motion diagram<\/h4>\r\nSince velocity is the change in position of the car during a corresponding time interval, and we are free to select the time interval as the time interval between exposures on our multiple-exposure photograph, the velocity is simply the change in the position of the car \"between dots.\" Thus, the arrows (vectors) on the motion diagram below represent the velocity of the car.\r\n\r\n\"\"\r\n\r\nWe can now describe the velocity of the car. Since the velocity vectors always point in the positive direction, the velocity is always positive. The car starts with a large, positive velocity which gradually declines until the velocity of the car is zero at the end of its motion.\r\n

Determining the acceleration from a motion diagram<\/h4>\r\nSince acceleration is the change in velocity of the car during a corresponding time interval, and we are free to select the time interval as the time interval between exposures on our multiple-exposure photograph, we can determine the acceleration by comparing two successive velocities. The change in these velocity vectors will represent the acceleration.\r\n\r\nTo determine the acceleration,\r\n