11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 1/29 HW_Week2 Due: 11:59pm on Friday, September 12, 2014 Speed of a Bullet A bullet is shot through two cardboard disks attached a distance apart to a

11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 1/29 HW_Week2 Due: 11:59pm on Friday, September 12, 2014 Speed of a Bullet A bullet is shot through two cardboard disks attached a distance apart to a shaft turning with a rotational period , as shown. Part A Derive a formula for the bullet speed in terms of , , and a measured angle between the position of the hole in the first disk and that of the hole in the second. If required, use , not its numeric equivalent. Both of the holes lie at the same radial distance from the shaft. measures the angular displacement between the two holes; for instance, means that the holes are in a line and means that when one hole is up, the other is down. Assume that the bullet must travel through the set of disks within a single revolution. The relative position of the holes can be used to find the bullet’s speed. Remember, the shaft will have rotated while the bullet travels between the disks. The disks rotate by 2 in time . How long will it take them to rotate by ? Give your answer in terms of , , and constants such as . If your formula is correct, when you plug 2 in for , your answer will be . ANSWER: 2 J R J J R J J R R J 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 2/29 ANSWER: Correct Exercise 2.14 A race car starts from rest and travels east along a straight and level track. For the first 5.0 of the car’s motion, the eastward component of the car’s velocity is given by . Part A is the acceleration of the car when = 14.8 ? Express your answer with the appropriate units. ANSWER: Correct Motion of Two Rockets Learning Goal: To learn to use images of an object in motion to determine velocity and acceleration. image where a stroboscope has illuminated the rockets at the uniform time intervals indicated. J J T 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 3/29 Part A At what time(s) do the rockets have the same velocity? The diagram shows position, not velocity. You can’t find instantaneous velocity from this diagram, but you can determine the average velocity between two times and : . Note that no position values are given in the diagram; you will need to estimate these based on the distance between successive positions of the rockets. ANSWER: Correct Part B ANSWER: at time only at time only at times and at some instant in time between and at no time shown in the figure 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 4/29 Correct Part C At what time(s) do the two rockets have the same acceleration? The velocity is related to the spacing between images in a stroboscopic diagram. Since acceleration is the rate at which velocity changes, the acceleration is related to the how much this spacing changes from one interval to the next. ANSWER: Correct Part D The motion of the rocket labeled A is an example of motion with uniform (i.e., constant) __________. ANSWER: at time only at time only at times and at some instant in time between and at no time shown in the figure at time only at time only at times and at some instant in time between and at no time shown in the figure and nonzero acceleration velocity displacement time 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 5/29 Correct Part E The motion of the rocket labeled B is an example of motion with uniform (i.e., constant) __________. ANSWER: Correct Part F At what time(s) is rocket A ahead of rocket B? You can answer this question by looking at the diagram and identifying the time(s) when rocket A is to the right of rocket B. ANSWER: Correct Velocity from Graphs of Position versus Time the figure. and nonzero acceleration velocity displacement time before only after only before and after between and at no time(s) shown in the figure 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 6/29 Part A During which trial or trials is the object’s velocity not constant? Check all that apply. slope of the curve at that point. The slope of a line is its rise divided by the run: . ANSWER: Correct The graph of the motion during Trial B has a changing slope and therefore is not constant. The other trials all have graphs with constant slope and thus correspond to motion with constant velocity. Part B During which trial or trials is the magnitude of the average velocity the largest? 0 U Trial A Trial B Trial C Trial D 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 7/29 Check all that apply. Recall that . Then note that the question asks only about the magnitude of the velocity. ANSWER: Correct While Trial B and Trial D do not have the same average velocity, the only difference is the direction! The magnitudes are the same. Neither one is “larger” than the other, and it is only because of how we chose our axes that Trial B has a positive average velocity while Trial D has a negative average velocity. In Trial C the object does not move, so it has an average velocity of zero. During Trial A the object has a positive average velocity but its magnitude is less than that in Trial B and Trial D. ± Average Velocity from a Position vs. Time Graph Learning Goal: 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 achieve the same displacement over the given time interval, 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  UJNF Y U Trial A Trial B Trial C Trial D 0 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 8/29 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. 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: 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. interval from 1 to 3 seconds? Express your answer numerically, in meters ANSWER: 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: 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 9/29 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 straightline 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. 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: 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. is the displacement? Answer to the nearest integer. ANSWER: NT N 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 10/29 is the time interval? Answer to two significant figures. ANSWER: ANSWER: Correct Part E Finally, find the average velocity over the whole time interval shown in the graph. Express your answer to three significant figures. is the displacement? Answer to the nearest integer. ANSWER: ANSWER: Correct 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! Given Positions, Find Velocity and Acceleration NT 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 11/29 Learning Goal: To understand how to graph position, velocity, and acceleration of an object starting with a table of positions vs. time. indicated below each time. You should make the simplification that the acceleration of the object is bounded and contains no spikes. time (s) 0 1 2 3 4 5 6 7 8 9 Part A Which graph best represents the function , describing the object’s position vs. time? A bounded and nonspiky acceleration results in a smooth graph of vs. . ANSWER: Correct Part B Which of the following graphs best represents the function , describing the object’s velocity as a function of 4 0 1 2 3 4 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 12/29 time? Velocity is the time derivative of displacement. Given this, the velocity toward the end of the motion is __________. ANSWER: Two of the possible velocity vs. time graphs indicate zero velocity between and . would the corresponding position vs. time graph look like in this region? ANSWER: The problem states that “the acceleration of the object is bounded and contains no spikes.” This means that the velocity ___________. ANSWER: positive and increasing positive and decreasing negative and increasing negative and decreasing a horizontal line straight but sloping up to the right straight but sloping down to the right curved upward curved downward 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 13/29 ANSWER: Correct 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? has spikes has no discontinuities has no abrupt changes of slope is constant 1 2 3 4 0 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 14/29 Acceleration is the time derivative of velocity. Toward the end of the motion the acceleration is __________. ANSWER: is the acceleration over the interval during which the object travels at constant speed? Answer numerically in meters per second squared. ANSWER: Acceleration is the time derivative of velocity. Initially the acceleration is _________. ANSWER: ANSWER: zero positive negative zero positive negative 1 2 3 4 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 15/29 Correct 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. Velocity and Acceleration of a Power Ball Learning Goal: To understand the distinction between velocity and acceleration with the use of motion diagrams. defined concepts that are not at all synonymous. Distinguishing clearly between them is a prerequisite to understanding is sketched at several equally spaced instants of time, and these sketches (or snapshots) are combined into one single picture. In this problem, we make use of these concepts to study the motion of a power ball. This discussion assumes that we have already agreed on a coordinate system from which to measure the position (also called the position vector) of objects as a function of time. Let and be velocity and acceleration, respectively. Consider the motion of a power ball that is dropped on the floor and bounces back. In the following questions, you will describe its motion at various points in its fall in terms of its velocity and acceleration. Part A You drop a power ball on the floor. The motion diagram of the ball is sketched in the figure . Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing. 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 16/29 By definition, the velocity is the ratio of the distance traveled to the interval of time taken. If you interpret the vector displacement as the distance traveled by the ball, the length of is directly proportional to the length of . Since the length of displacement vectors is increasing, so is the length of velocity vectors. ANSWER: Correct While the ball is in free fall, the magnitude of its velocity is increasing, so the ball is accelerating. Part B Since the length of is directly proportional to the length of , the vector connecting each dot to the next could represent velocity vectors as well as displacement vectors, as shown in the figure here . Indicate whether the velocity and acceleration of the ball are, respectively, positive (upward), negative, or zero. zero, respectively. Separate the letters for velocity and acceleration with a comma. The acceleration is defined as the ratio of the change in velocity to the interval of time, and its direction is given by the quantity , which represents the change in velocity that occurs in the interval of time . ANSWER: Correct increasing decreasing not changing N,N 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 17/29 Part C Now, consider the motion of the power ball once it bounces upward. Its motion diagram is shown in the figure here . Indicate whether the magnitude of the velocity of the ball is increasing, decreasing, or not changing. By definition, the velocity is the ratio of the distance traveled to the interval of time taken. If you interpret the vector displacement as the distance traveled by the ball, the length of is directly proportional to the length of . Since the length of displacement vectors is decreasing, so is the length of velocity vectors. ANSWER: Correct Since the magnitude of the velocity of the ball is decreasing, the ball must be accelerating (specifically, slowing down). Part D The next figure shows the velocity vectors corresponding to the upward motion of the power ball. Indicate whether its velocity and acceleration, respectively, are positive (upward), negative, or zero. acceleration with a comma. increasing decreasing not changing 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 18/29 The acceleration is defined as the ratio of the change in velocity to the interval of time, and its direction is given by the quantity , which represents the change in velocity that occurs in the interval of time . ANSWER: Correct Part E The power ball has now reached its highest point above the ground and starts to descend again. The motion diagram representing the velocity vectors is the same as that after the initial release, as shown in the figure of Part B. Indicate whether the velocity and acceleration of the ball at its highest point are positive (upward), negative, or zero. acceleration with a comma. In Part D you found that the velocity of the ball is positive during the upward motion. Once the ball starts its descent, its velocity is negative, as you found in Part B. Since velocity changes continuously in time, it has to be zero at some point along the path of the ball. In Part D, you found that the acceleration of the ball is negative and constant during the upward motion, as well as once the ball has started its descent, which you found in Part B. Since acceleration is a continuous function of time, it has to be negative at the highest point along the path as well. P,N 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 19/29 ANSWER: Correct These examples should show you that the velocity and acceleration can have opposite or similar signs or that physical quantities when working with kinematics. Analyzing Position versus Time Graphs: Conceptual Question Two cars travel on the parallel lanes of a twolane 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? Two objects can pass each other only if they have the same position at the same time. ANSWER: Z,N 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 20/29 Correct Part B Are the two cars traveling in the same direction when they pass each other? ANSWER: Correct Part C At which of the lettered times, if any, does car #1 momentarily stop? 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. ANSWER: A B C D E None Cannot be determined yes no A B C D E none cannot be determined 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 21/29 Correct Part D At which of the lettered times, if any, does car #2 momentarily stop? 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. ANSWER: Correct Part E At which of the lettered times are the cars moving with nearly identical velocity? 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. ANSWER: A B C D E none cannot be determined 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 22/29 Correct Conceptual Question 2.05 Part A Suppose that an object is moving with constant nonzero acceleration. Which of the following is an accurate statement concerning its motion? ANSWER: Correct Conceptual Question 2.09 Part A The graph in the figure shows the position of an object as a function of time. The letters HL represent particular moments of time. At which moments shown (H, I, etc.) is the speed of the object A B C D E None Cannot be determined In equal times its speed changes by equal amounts. In equal times its velocity changes by equal amounts. A graph of its position as a function of time has a constant slope. In equal times it moves equal distances. A graph of its velocity as a function of time is a horizontal line. 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 23/29 (a) the greatest? ANSWER: Correct Part B (b) the smallest? ANSWER: Correct Conceptual Question 2.04 Part A H I J K L H I J K L 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 24/29 When can we be certain that the average velocity of an object is always equal to its instantaneous velocity? ANSWER: Correct Conceptual Question 2.18 Part A Two objects are thrown from the top of a tall building and experience no appreciable air resistance. One is thrown up, and the other is thrown down, both with the same initial speed. are their speeds when they hit the street? ANSWER: Correct Conceptual Question 2.16 Part A instantaneous acceleration equal to zero? always only when the acceleration is constant never only when the acceleration is changing at a constant rate only when the velocity is constant The one thrown down is traveling faster. The one thrown up is traveling faster. They are traveling at the same speed. 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 25/29 ANSWER: Correct Problem 2.27 Part A A package is dropped from a helicopter moving upward at 15 m/s. If it takes 8 s before the package strikes the ground, how high above the ground was the package when it was released if air resistance is negligible? ANSWER: Correct Problem 2.10 Part A 190 m 114 m 152 m 228 m 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 26/29 ANSWER: Correct Conceptual Question 2.02 Part A If the graph of the position as a function of time for an object is a horizontal line, that object cannot be accelerating. ANSWER: Correct Prelecture Concept Question 2.01 Part A Which of the following best describes how to calculate the average acceleration of any object? ANSWER: 2.0 m/s 3.5 m/s 2.5 m/s 3.0 m/s True False Average acceleration is always equal to the change in velocity of an object divided by the time interval. Average acceleration is always halfway between the initial acceleration of an object and its final acceleration. Average acceleration is always equal to the displacement of an object divided by the time interval. Average acceleration is always equal to the change in speed of an object divided by the time interval. 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 27/29 Correct Conceptual Question 2.13 Part A as a function of time is as a function of time? ANSWER: 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 28/29 11/10/2014 HW_Week2 http://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=2992507 29/29 Correct Score Summary: Your score on this assignment is 97.1%. You received 17.48 out of a possible total of 18 points.

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