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#1 Tutorials One-Dimensional Kinematics with Constant Acceleration Learning Goal: To understand the meaning of the variables that appear in the equations for onedimensional kinematics with constant acceleration. Motion with a constant, nonzero acceleration is not uncommon in the world around us. Falling (or thrown) objects and cars starting and stopping approximate this type of motion. It is also the type of motion most frequently involved in introductory kinematics problems. The kinematic equations for such motion can be written as , , where the symbols are defined as follows: is the position of the particle; is the initial position of the particle; is the velocity of the particle; is the initial velocity of the particle; is the acceleration of the particle. In anwering the following questions, assume that the acceleration is constant and nonzero: . Part A The quantity represented by is a function of time (i.e., is not constant). Register to View Answerfalse One-Dimensional Kinematics with Constant Acceleration Part A Correct Part B The quantity represented by is a function of time (i.e., is not constant). ANSWER: Correct Recall that represents an initial value, not a variable. It refers to the position of an object at some initial moment. Part C The quantity represented by is a function of time (i.e., is not constant). Register to View Answerfalse Correct Part D The quantity represented by is a function of time (i.e., is not constant). ANSWER: One-Dimensional Kinematics with Constant Acceleration Part C Correct The velocity always varies with time when the linear acceleration is nonzero. Part E Which of the given equations is not an explicit function of and is therefore useful when you don't know or don't need the time? ANSWER: Correct Part F A particle moves with constant acceleration . The expression velocity at what instant in time? ANSWER: represents the particle's One-Dimensional Kinematics with Constant Acceleration Part E Correct More generally, the equations of motion can be written as and . Here is the time that has elapsed since the beginning of the particle's motion, that is, is the time at which we start measuring the particle's motion. The , where is the current time and terms and are, respectively, the position and velocity at . As you can now see, the equations , which is a convenient choice if given at the beginning of this problem correspond to the case there is only one particle of interest. To illustrate the use of these more general equations, consider the motion of two particles, A and B. The position of particle A depends on time as . That is, particle A starts moving at time with velocity , from . At time , particle B has twice the . acceleration, half the velocity, and the same position that particle A had at time Part G What is the equation describing the position of particle B? Hint G.1 How to approach the problem One-Dimensional Kinematics with Constant Acceleration Part G Hint not displayed ANSWER: Correct Part H At what time does the velocity of particle B equal that of particle A? Hint H.1 Velocity of particle A Hint not displayed Hint H.2 Velocity of particle B Hint not displayed ANSWER: One-Dimensional Kinematics with Constant Acceleration Part G The two particles never have the same Correct 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. What x vs. t Graphs Can Tell You Part A What is the total distance traveled by the particle? Hint A.1 Total distance Hint not displayed Hint A.2 How to read an x vs. t graph Hint not displayed Express your answer in meters. ANSWER: = 30 Correct Part B What is the average velocity of the particle over the time interval ? Hint B.1 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 Correct 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 of the particle at ? What x vs. t Graphs Can Tell You Part C Hint C.1 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 Correct 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 the velocity and the acceleration, even though they are really the components the of 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 What x vs. t Graphs Can Tell You Part D ANSWER: Graph A Graph B Graph C Graph D Correct 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 What x vs. t Graphs Can Tell You Part D shaded region under the curve? Hint E.1 How to approach the problem Hint not displayed Express your answer in meters. ANSWER: = 30 Correct 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 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 What Velocity vs. Time Graphs Can Tell You 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 Correct ? Part B What is the total distance traveled by the particle? Hint B.1 What Velocity vs. Time Graphs Can Tell You Part A How to approach the problem Hint not displayed Hint B.2 Find the distance traveled in the first 20.0 seconds Hint not displayed Hint B.3 Find the distance traveled in the second 20.0 seconds Hint not displayed Hint B.4 Find the distance traveled in the last 10.0 seconds Hint not displayed Express your answer in meters. ANSWER: = 75 Correct Part C What is the average acceleration Hint C.1 Definition and graphical interpretation of average acceleration Hint not displayed Hint C.2 Slope of a line Hint not displayed Express your answer in meters per second per second. ANSWER: = 0.075 Correct of the particle over the first 20.0 seconds? 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 Velocity vs. Time Graphs Can Tell You Part C What is the instantaneous acceleration of the particle at ? Hint D.1 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 Correct 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 What Velocity vs. Time Graphs Can Tell You 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 Hint E.2 Find the acceleration in the first 20 Hint not displayed Hint E.3 Find the acceleration in the second 20 Hint not displayed Hint E.4 Find the acceleration in the last 10 Hint not displayed What Velocity vs. Time Graphs Can Tell You Part E ANSWER: Graph A Graph B Graph C Graph D Correct 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. A Man Running to Catch a Bus A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration . Use a coordinate system with at the door of the stopped bus. A Man Running to Catch a Bus Part A What is , the position of the man as a function of time? Hint A.1 Which equation should you use for the man's speed? Hint not displayed Answer symbolically in terms of the variables , , and . ANSWER: = Correct Part B What is , the position of the bus as a function of time? Hint B.1 Which equation should you use for the bus's acceleration? Hint not displayed Answer symbolically in terms of and . ANSWER: = Correct Part C What condition is necessary for the man to catch the bus? Assume he catches it at time Hint C.1 How to approach this problem Hint not displayed . A Man Running to Catch a Bus Part C ANSWER: Correct Part D Part E Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ? Hint E.1 What is the general quadratic equation? Hint not displayed ANSWER: No; there is no chance he is going to get aboard. Yes; he will get a second chance Correct ... View Full Document

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