MP2 - MasteringPhysics: Assignment Print View

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Unformatted text preview: MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... [ Print View ] PH1110A2008 Assignment 2 Due at 11:59pm on Wednesday, September 3, 2008 View Grading Details Kinematic Vocabulary One of the difficulties in studying mechanics is that many common words are used with highly specific technical meanings, among them velocity, acceleratio n, position, speed, and displacement. The series of questions in this problem is designed to get you to try to think of these quantities like a physicist. Answer the questions in this problem using words from the following list: A. B. C. D. E. F. G. H. I. J. K. position direction displacement coordinates velocity acceleration distance magnitude vector scalar components Part A Velocity differs from speed in that velocity indicates a particle's __________ of motion. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: B Part B Unlike speed, velocity is a __________ quantity. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: I Part C A vector has, by definition, both __________ and direction. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: H Part D Once you have selected a coordinate system, you can express a two-dimensional vector using a pair of quantities known collectively as __________. Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: K Part E Speed differs from velocity in the same way that __________ differs from displacement. Hint E.1 Definition of displacement Hint not displayed 1 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... Enter the letter from the list given in the problem introduction that best completes the sentence. ANSWER: G Part F Consider a physical situation in which a particle moves from point A to point B. This process is described from two coordinate systems that are identical except that they have different origins. The __________ of the particle at point A differ(s) as expressed in one coordinate system compared to the other, but the __________ from A to B is/are the same as expressed in both coordinate systems. Type the letters from the list given in the problem introduction that best complete the sentence. Separate the letters with commas. There is more than one correct answer, but you should only enter one pair of comma-separated letters. For example, if the words "vector" and "scalar" fit best in the blanks, enter I,J. ANSWER: A,G The coordinates of a point will depend on the coordinate system that is chosen, but there are several other quantities that are independent of the choice of origin for a coordinate system: in particular, distance, displacement, direction, and velocity. In working physics problems, unless you are interested in the position of an object or event relative to a specific origin, you can usually choose the coordinate system origin to be wherever is most convenient or intuitive. Note that the vector indicating a displacement from A to B is usually represented as . Part G Identify the following physical quantities as scalars or vectors. ANSWER: View What x vs. t Graphs Can Tell You To describe the motion of a particle along a straight line, it is often convenient to draw a graph representing the position of the particle at different times. This type of graph is usually referred to as an x vs. t graph. To draw such a graph, choose an axis system in which time is plotted on the horizontal axis and position on the vertical axis. Then, indicate the values of at various times . Mathematically, this corresponds to plotting the variable as a function of . An example of a graph of position as a function of time for a particle traveling along a straight line is shown below. Note that an x vs. t graph like this does not represent the path of the particle in space. Now let's study the graph shown in the figure in more detail. Refer to this graph to answer Parts A, B, and C. 2 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... Part A What is the total distance Hint A.1 traveled by the particle? Total distance traveled by the particle is given by the difference between the initial position at and the position at . In The total distance symbols, . Hint A.2 at How to read an x vs. t graph is plotted on the horizontal axis and position on the vertical axis. For example, in the plot shown in the figure, . Remember that in an x vs. t graph, time Express your answer in meters. ANSWER: = 30 Part B What is the average velocity Hint B.1 of the particle over the time interval ? Definition and graphical interpretation of average velocity Hint not displayed Hint B.2 Slope of a line Hint not displayed Express your answer in meters per second. ANSWER: = 0.600 The average velocity of a particle between two positions is equal to the slope of the line connecting the two corresponding points in an x vs. t graph. Part C What is the instantaneous velocity Hint C.1 of the particle at ? Graphical interpretation of instantaneous velocity The velocity of a particle at any given instant of time or at any point in its path is called instantaneous velocity. In an x vs. t graph of the particle's motion, you can determine the instantaneous velocity of the particle at any point in the curve. The instantaneous velocity at any point is equal to the slope of the line tangent to the curve at that point. Express your answer in meters per second. ANSWER: = 0.600 The instantaneous velocity of a particle at any point on its x vs. t graph is the slope of the line tangent to the curve at that point. Since in the case at hand the curve is a straight line, the tangent line is the curve itself. Physically, this means that the instantaneous velocity of the particle is constant over the entire time interval of motion. This is true for any motion where distance increases linearly with time. Another common graphical representation of motion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of time. In this graph, time is plotted on the horizontal axis and velocity on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line motion, however, these vectors have only one nonzero component in the direction of motion. Thus, in this problem, we will call and the acceleration, even though they are really the components of the velocity and acceleration vectors in the direction of motion. Part D Which of the graphs shown is the correct v vs. t plot for the motion described in the previous parts? Hint D.1 How to approach the problem Hint not displayed the velocity 3 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... ANSWER: Graph A Graph B Graph C Graph D Whenever a particle moves with constant nonzero velocity, its x vs. t graph is a straight line with a nonzero slope, and its v vs. t curve is a horizontal line. Part E Shown in the figure is the v vs. t curve selected in the previous part. What is the area of the shaded region under the curve? Hint E.1 How to approach the problem Hint not displayed Express your answer in meters. ANSWER: = 30 Compare this result with what you found in Part A. As you can see, the area of the region under the v vs. t curve equals the total distance traveled by the particle. This is true for any velocity curve and any time interval: The area of the region that extends over a time interval under the v vs. t curve is always equal to the distance traveled in . What Velocity vs. Time Graphs Can Tell You 4 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... A common graphical representation of motion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of time. In this graph, time is plotted on the horizontal axis and velocity on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line motion, however, these vectors have only a single nonzero component in the direction of motion. Thus, in this problem, we will call the velocity and the acceleration, even though they are really the components of the velocity and acceleration vectors in the direction of motion, respectively. Here is a plot of velocity versus time for a particle that travels along a straight line with a varying velocity. Refer to this plot to answer the following questions. Part A What is the initial velocity of the particle, Hint A.1 Initial velocity Hint not displayed Hint A.2 How to read a v vs. t graph Hint not displayed Express your answer in meters per second. ANSWER: = 0.5 ? Part B What is the total distance Hint B.1 traveled by the particle? How to approach the problem Hint not displayed Part B.2 Find the distance traveled in the first 20.0 seconds Part not displayed Part B.3 Find the distance traveled in the second 20.0 seconds Part not displayed Part B.4 Find the distance traveled in the last 10.0 seconds Part not displayed Express your answer in meters. ANSWER: = 75 Part C What is the average acceleration Hint C.1 of the particle over the first 20.0 seconds? Definition and graphical interpretation of average acceleration of a particle that travels along a straight line in a time interval , or is the ratio of the change in velocity experienced by the The average acceleration particle to the time interval 5 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... . In a v vs. t graph, then, the average acceleration equals the slope of the line connecting the two points representing the initial and final velocities. Hint C.2 Slope of a line Hint not displayed Express your answer in meters per second per second. ANSWER: = 0.075 The average acceleration of a particle between two instants of time is the slope of the line connecting the two corresponding points in a v vs. t graph. Part D What is the instantaneous acceleration Hint D.1 of the particle at ? Graphical interpretation of instantaneous acceleration Hint not displayed Hint D.2 Slope of a line Hint not displayed ANSWER: 1 0.20 = -0.20 0.022 -0.022 The instantaneous acceleration of a particle at any point on a v vs. t graph is the slope of the line tangent to the curve at that point. Since in the last 10 seconds of motion, between and , the curve is a straight line, the tangent line is the curve itself. Physically, this means that the instantaneous acceleration of the particle is constant over that time interval. This is true for any motion where velocity increases linearly with time. In the case at hand, can you think of another time interval in which the acceleration of the particle is constant? Now that you have reviewed how to plot variables as a function of time, you can use the same technique and draw an acceleration vs. time graph, that is, the graph of (instantaneous) acceleration as a function of time. As usual in these types of graphs, time is plotted on the horizontal axis, while the vertical axis is used to indicate acceleration . Part E Which of the graphs shown below is the correct acceleration vs. time plot for the motion described in the previous parts? Hint E.1 How to approach the problem Hint not displayed Part E.2 Find the acceleration in the first 20 Part not displayed Part E.3 Find the acceleration in the second 20 Part not displayed Part E.4 Find the acceleration in the last 10 Part not displayed 6 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... ANSWER: Graph A Graph B Graph C Graph D In conclusion, graphs of velocity as a function of time are a useful representation of straight-line motion. If read correctly, they can provide you with all the information you need to study the motion. Given Positions, Find Velocity and Acceleration Learning Goal: To understand how to graph position, velocity, and acceleration of an object starting with a table of positions vs. time. The table shows the x coordinate of a moving object. The position is tabulated at 1-s intervals. The x coordinate is indicated below each time. You should make the simplification that the acceleration of the object is bounded and contains no spikes. time (s) x (m) 0 0 1 1 2 4 3 9 4 16 5 24 6 32 7 40 8 46 9 48 Part A Which graph best represents the function , describing the object's position vs. time? Hint A.1 Meaning of a bounded and nonspiky acceleration vs. . A bounded and nonspiky acceleration results in a smooth graph of 7 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... ANSWER: 1 2 3 4 Part B Which of the following graphs best represents the function , describing the object's velocity as a function of time? Part B.1 Find the velocity toward the end of the motion Part not displayed Part B.2 What are the implications of zero velocity? Part not displayed Part B.3 Specify the characteristics of the velocity function Part not displayed ANSWER: 1 2 3 4 In principle, you could also just compute and plot the average velocity. The expression for the average velocity is . The notation emphasizes that this is not an instantaneous velocity, but rather an average over an interval. After you compute this, you must put a single point on the graph of velocity vs. time. The most accurate place to plot the average velocity is at the middle of the time interval over which the average was computed. Also, you could work back and find the position from the velocity graph. The position of an object is the integral of its velocity. That is, the area under the graph of velocity vs. time from up to time must equal the position of the object at time . Check that the correct velocity vs. time graph gives you the correct position according to this method. Part C Which of the following graphs best represents the function , describing the acceleration of this object? 8 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... Part C.1 Find the acceleration toward the end of the motion Part not displayed Part C.2 Calculate the acceleration in the region of constant velocity Part not displayed Part C.3 Find the initial acceleration Part not displayed ANSWER: 1 2 3 4 In one dimension, a linear increase or decrease in the velocity of an object over a given time interval implies constant acceleration over that particular time interval. You can find the magnitude of the acceleration using the formula for average acceleration over a time interval: . When the acceleration is constant over an extended interval, you can choose any value of and within the interval to compute the average. Introduction to Projectile Motion Learning Goal: To understand the basic concepts of projectile motion. Projectile motion may seem rather complex at first. However, by breaking it down into components, you will find that it is really no different than the one-dimensional motions that you have already studied. One of the most often used techniques in physics is to divide two- and three-dimensional quantities into components. For instance, in projectile motion, a particle has some initial velocity . In general, this velocity can point in any direction on the xy plane and can have any magnitude. To make a problem more managable, it is common to break up such a quantity into its x component Consider a particle with initial velocity Part A What is the x component of ? and its y component . above the negative x axis. that has magnitude 12.0 and is directed 60.0 Express your answer in meters per second. ANSWER: = -6.00 Part B What is the y component of ? Express your answer in meters per second. ANSWER: = 10.4 Breaking up the velocities into components is particularly useful when the components do not affect each other. Eventually, you will learn about situations in which the components of velocity do affect one another, but for now you will only be looking at problems where they do not. So, if there is acceleration in the x direction but not in the y direction, then the x component of the velocity will change, but the y component of the velocity will not. Part C 9 of 10 10/13/2008 7:57 PM MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assig... Look at this applet. The motion diagram for a projectile is displayed, as are the motion diagrams for each component. The x-component motion diagram is what you would get if you shined a spotlight down on the particle as it moved and recorded the motion of its shadow. Similarly, if you shined a spotlight to the left and recorded the particle's shadow, you would get the motion diagram for its y component. How would you describe the two motion diagrams for the components? ANSWER: Both the vertical and horizontal components exhibit motion with constant nonzero acceleration. The vertical component exhibits motion with constant nonzero acceleration, whereas the horizontal component exhibits constantvelocity motion. The vertical component exhibits constant-velocity motion, whereas the horizontal component exhibits motion with constant nonzero acceleration. Both the vertical and horizontal components exhibit motion with constant velocity. As you can see, the two components of the motion obey their own independent kinematic laws. For the vertical component, there is an acceleration downward with magnitude . Thus, you can calculate the vertical position of the particle at any time using the standard kinematic equation . Similarly, there is no acceleration in the horizontal direction, so the horizontal position of the particle is given by the standard kinematic equation . . Now, consider this applet. Two balls are simultaneously dropped from a height of 5.0 Part D How long Hint D.1 does it take for the balls to reach the ground? Use 10 How to approach the problem at time for the magnitude of the acceleration due to gravity. The balls are released from rest at a height of 5.0 for the balls to reach the ground. . Using these numbers and basic kinematics, you can determine the amount of time it takes Express your answer in seconds to two significant figures. ANSWER: = 1.0 This situation, which you have dealt with before (motion under the constant acceleration of gravity), is actually a special case of projectile motion. Think of this as projectile motion where the horizontal component of the initial velocity is zero. Part E Imagine the ball on the left is given a nonzero initial speed in the horizontal direction, while the ball on the right continues to fall with zero initial velocity. What horizontal speed must the ball on the left start with so that it hits the ground at the same position as the ball on the right? Hint E.1 How to approach the problem Recall from Part B that the horizontal component of velocity does not change during projectile motion. Therefore, you need to find the horizontal component of velocity such that, in a time , the ball will move horizontally 3.0 . You can assume that its initial x coordinate is . Express your answer in meters per second to two significant figures. ANSWER: = 3.0 You can adjust the horizontal speeds in this applet. Notice that regardless of what horizontal speeds you give to the balls, they continue to move vertically in the same way (i.e., they are at the same y coordinate at the same time). Summary 5 of 5 items complete (99.88% avg. score) 49.94 of 50 points 10 of 10 10/13/2008 7:57 PM ...
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This note was uploaded on 10/13/2008 for the course PH 1110 taught by Professor Kiel during the Spring '08 term at WPI.

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