Chapter 2 Physics Homework

9/2/2019 CH 02 HW CH 02 HW Due: 3:00pm on Monday, September 9, 2019 Position versus Time Graphs Conceptual Question The motions described in each of the questions take place at an intersection on a two-lane road with a stop sign in each direction. For each motion, select the correct position versus time graph. For all of the motions, the stop sign is at the position , and east is the positive x direction. Part A A driver ignores the stop sign and continues driving east at constant speed. Hint 1. Determining velocity from a position versus time graph The slope on a position versus time graph is the rise (change in position) over the run (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed. Hint 2. Driving east Since east is defined as the positive x direction, a car traveling east must have a positive velocity. A positive velocity is represented as a positive slope on a position versus time graph. Hint 3. Constant speed Since velocity is represented by the slope on a position versus time graph, a car moving at constant speed must be represented by a line of constant slope. ANSWER: A B C D E F 1/50 9/2/2019 CH 02 HW Correct Part B A driver ignores the stop sign and continues driving west at constant speed. Hint 1. Determining velocity from a position versus time graph The slope on a position versus time graph is the rise (change in position) over the run (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed. Hint 2. Driving west Since east is defined as the positive x direction, a car traveling west must have a negative velocity. A negative velocity is represented as a negative slope on a position versus time graph. Hint 3. Constant speed Since velocity is represented by the slope on a position versus time graph, a car moving at constant speed must be represented by a line of constant slope. ANSWER: A B C D E F Correct Part C A driver, traveling west, slows and stops at the stop sign. Hint 1. Determining velocity from a position versus time graph The slope on a position versus time graph is the rise (change in position) over the run (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed. Hint 2. Driving west Since east is defined as the positive x direction, a car traveling west must have a negative velocity. A negative velocity is represented as a negative slope on a position versus time graph. Hint 3. Acceleration Since velocity is represented by the slope on a position versus time graph, a car that accelerates must be represented as a curve with changing slope. If a car slows, then the slope of the graph must approach zero. If a car's speed increases, the slope must become more positive or more negative (depending upon which direction it is moving). 2/50 9/2/2019 CH 02 HW ANSWER: A B C D E F Correct Part D A driver, after stopping at the stop sign, travels east with a positive acceleration. Hint 1. Determining velocity from a position versus time graph The slope on a position versus time graph is the rise (change in position) over the run (change in time). In physics, the ratio of change in position over change in time is defined as the velocity. Thus, the slope on a position versus time graph is the velocity of the object being graphed. Hint 2. Driving east Since east is defined as the positive x direction, a car traveling east must have a positive velocity. A positive velocity is represented as a positive slope on a position versus time graph. Hint 3. Acceleration Since velocity is represented by the slope on a position versus time graph, a car that accelerates must be represented as a curve with changing slope. If a car slows, then the slope of the graph must approach zero. If a car's speed increases, the slope must become more positive or more negative (depending upon which direction it is moving). ANSWER: A B C D E F Correct ± Average Velocity from a Position vs. Time Graph Learning Goal: 3/50 9/2/2019 CH 02 HW To learn to read a graph of position versus time and to calculate average velocity. In this problem you will determine the average velocity of a moving object from the graph of its position as a function of time . A traveling object might move at different speeds and in different directions during an interval of time, but if we ask at what constant velocity the object would have to travel to achieve the same displacement over the given time interval, that is what we call the object's average velocity. We will use the notation to indicate average velocity over the time interval from to . For instance, is the average velocity over the time interval from to . Part A Consulting the graph shown in the figure, find the object's average velocity over the time interval from 0 to 1 second. Answer to the nearest integer. Hint 1. Definition of average velocity Average velocity is defined as the constant velocity at which an object would have to travel to achieve a given displacement (difference between final and initial positions, which can be negative) over a given time interval, from the initial time to the final time . The average velocity is therefore equal to the displacement divided by the given time interval. In symbolic form, average velocity is given by . ANSWER: = 0 Correct Part B Find the average velocity over the time interval from 1 to 3 seconds. Express your answer in meters per second to the nearest integer. Hint 1. Find the change in position The final and initial positions can be read off the y axis of the graph. What is the displacement during the time interval from 1 to 3 seconds? Express your answer numerically, in meters ANSWER: = 40 Hint 2. Definition of average velocity 4/50 9/2/2019 CH 02 HW Average velocity is defined as the constant velocity at which an object would have to travel to achieve a given displacement (difference between final and initial positions, which can be negative) over a given time interval, from the initial time to the final time . The average velocity is therefore equal to the displacement divided by the given time interval. In symbolic form, average velocity is given by . ANSWER: = 20 Correct A note about instantaneous velocity. The instantaneous velocity at a certain moment in time is represented by the slope of the graph at that moment. For straight-line graphs, the (instantaneous) velocity remains constant over the interval, so the instantaneous velocity at any time during an interval is the same as the average velocity over that interval. For instance, in this case, the instantaneous velocity at any time from 1 to 3 seconds is the same as the average velocity of . Part C Now find . Give your answer to three significant figures. Hint 1. A note on the displacement Since the object's position remains constant from time 0 to time 1, the object's displacement from 0 to 3 is the same as in Part B. However, the time interval has changed. ANSWER: = 13.3 Correct Note that is not equal to the simple arithmetic average of and , i.e., , because they are averages for time intervals of different lengths. Part D Find the average velocity over the time interval from 3 to 6 seconds. Express your answer to three significant figures. Hint 1. Determine the displacement What is the displacement? Answer to the nearest integer. ANSWER: = -40 Hint 2. Determine the time interval 5/50 9/2/2019 CH 02 HW What is the time interval? Answer to two significant figures. ANSWER: = 3.0 ANSWER: = -13.3 Correct Part E Finally, find the average velocity over the whole time interval shown in the graph. Express your answer to three significant figures. Hint 1. Determine the displacement What is the displacement? Answer to the nearest integer. ANSWER: = 0 Correct ANSWER: = 0 Correct Note that though the average velocity is zero for this time interval, the instantaneous velocity (i.e., the slope of the graph) has several different values (positive, negative, zero) during this time interval. Note as well that since average velocity over a time interval is defined as the change in position (displacement) in the given interval divided by the time, the object can travel a great distance (here 80 meters) and still have zero average velocity, since it ended up exactly where it started. Therefore, zero average velocity does not necessarily mean that the object was standing still the entire time! 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 vs. 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 vs. 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. 6/50 9/2/2019 CH 02 HW Part A What is the overall displacement of the particle? Express your answer in meters. Hint 1. Definition of displacement The displacement of the particle is given by the difference between the initial position . In symbols, at and the position at . Hint 2. How to read an x vs. t graph Remember that in an vs. plot shown in the figure, graph, time at is plotted on the horizontal axis and position . on the vertical axis. For example, in the ANSWER: = 30 Correct In this example, the magnitude of the displacement is also equal to the total distance traveled by the particle (30 ). Part B What is the average velocity of the particle over the time interval ? Express your answer in meters per second. Hint 1. Definition and graphical interpretation of average velocity The average velocity of a particle that undergoes a displacement along a straight line in a time interval is defined as . In an vs. graph, then, the average velocity equals the slope of the line connecting the initial and final positions. Hint 2. Slope of a line The slope "run," or of a line from point A, with coordinates , to point B, with coordinates , is equal to the "rise" over the 7/50 9/2/2019 CH 02 HW . 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 vs. graph. Part C What is the instantaneous velocity of the particle at ? Express your answer in meters per second. Hint 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 vs. 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. ANSWER: = 0.600 Correct The instantaneous velocity of a particle at any point on its vs. 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 vs. 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 of the velocity and acceleration vectors in the direction of motion. Part D Which of the graphs shown is the correct vs. plot for the motion described in the previous parts? 8/50 9/2/2019 CH 02 HW Hint 1. How to approach the problem Recall your results found in the previous parts, namely the fact that the instantaneous velocity of the particle is constant. Which graph represents a variable that always has the same constant value at any time? ANSWER: Graph A Graph B Graph C Graph D Correct Whenever a particle moves with constant nonzero velocity, its curve is a horizontal line. vs. graph is a straight line with a nonzero slope, and its vs. Part E Shown in the figure is the vs. curve selected in the previous part. What is the area of the shaded region under the curve? Express your answer in meters. 9/50 9/2/2019 CH 02 HW Hint 1. How to approach the problem The shaded region under the vs. curve is a rectangle whose horizontal and vertical sides lie on the axis and the axis, respectively. Since the area of a rectangle is the product of its sides, in this case the area of the shaded region is the product of a certain quantity expressed in seconds and another quantity expressed in meters per second. The area itself, then, will be 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 vs. curve equals the overall displacement of 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 vs. curve is always equal to the displacement over . Analyzing Position versus Time Graphs: Conceptual Question Two cars travel on the parallel lanes of a two-lane road. The cars’ motions are represented by the position versus time graph shown in the figure. Answer the questions using the times from the graph indicated by letters. Part A At which of the times do the two cars pass each other? Hint 1. Two cars passing 10/50