MP_Assignment#9_soln

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Unformatted text preview: MasteringPhysics: Course Home Student View 5/2/11 12:30 PM Summary View Diagnostics View Print View withEdit Assignment Answers Settings per Student MP_Assignment#9 Due: 12:33pm on Monday, May 2, 2011 Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy ± Understanding Lorentz Transformations Description: ± Includes Math Remediation. A simple numerical problem using Lorentz transformations, after which the length contraction formula is derived from the Lorentz transformations. Learning Goal: To be able to perform Lorentz transformations between inertial reference frames. Suppose that an inertial reference frame S' moves in the positive x direction at speed with respect to another inertial reference frame S. In classical physics, the Galilean transformations relate the coordinates measured for an event in frame S to the coordinates measured for the same event in frame S' . Assuming that both frames have the same origin (i.e., at , ), the Galilean transformations take the following simple form: , . The Galilean transformations are not valid at very large speeds. To transform between inertial frames when is close to the speed of light , we need to use the Lorentz transformations of special relativity. Again, assuming that both frames have the same origin, the Lorentz transformations take the following form: . These equations become more manageable with the introduction of the quantity , so that the Lorentz transformations become , . Often, the space- time coordinates for an event will be given in the form , or just when the y and z coordinates are not important. Part A Consider an event with space- time coordinates in an inertial frame of reference S. Let S' be a second inertial frame of reference moving, in the positive x direction, with speed relative to frame S. Find the value of that will be needed to transform coordinates http://session.masteringphysics.com/myct/courseHome?start=1 Page 1 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM between frames S and S' . Use for the speed of light in vacuum. Express your answer to three significant figures. ANSWER: = Part B Suppose that S and S' share the same origin; that is, at calculated in Part A, find , . Using the you , the x coordinate of the event in frame S' . Express your answer in meters to three significant figures. ANSWER: = Part C Now find , the t coordinate of the event in frame S' . Express your answer in seconds to three significant figures. ANSWER: = Suppose that you are stationary with respect to an inertial reference frame Z. A spaceship flies by you in the positive x direction with speed . Let Z' be the frame of reference associated with the spaceship; that is, the ship is stationary with respect to Z' . The frames Z and Z' have the same origin at proper length of the ship (the length of the ship as measured in the ship's frame, Z' ) is passenger on the ship measures the back of the ship to be at . The . In other words, a and the front to be at . Part D Find the factor that should be used to transform between frames Z and Z' . Express your answer in terms of and . ANSWER: = http://session.masteringphysics.com/myct/courseHome?start=1 Page 2 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Part E At time , in your frame of reference Z, you measure the back of the spaceship to be at front of the ship to be at proper length Hint E.1 . Find an equation relating the length that you measure and the to the ship's . Transforming between Z and Z' According to the Lorentz transformation, this equation simplifies to . Since you make your measurements at , . To find the length, you must apply this transformation to the ship's front and back coordinates: . In your frame of reference Z, Express your answer in terms of and and . . ANSWER: = You should recognize this as the equation for length contraction. The time dilation equation can also be found from the Lorentz transformations. A Moving Light Clock Appears Shorter Description: The relativity postulates make a light clock moving longitudinally appear shorter to keep same time as a transverse light clock. Assign after "Moving Light Clock Ticks Slower." (uses applet) Learning Goal: To understand length transformation starting from relativity postulates. In this problem we calculate the length of a light clock that moves at speed parallel to its axis in the moving system. A light clock ticks every time light makes a round trip between two mirrors separated by a distance . The key point is that we can deduce the length of this clock directly from the postulates of relativity: The laws of physics are the same in any coordinate system that moves at constant velocity (i.e., any inertial reference frame). The speed of light is when measured with respect to any coordinate system moving at a constant velocity. http://session.masteringphysics.com/myct/courseHome?start=1 Page 3 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM As shown in the figure , the light clock is based on the propagation time of light between mirrors spaced a distance apart (in the rest frame of the clock). As the light pulse bounces back and forth between the mirrors, a small part of it is allowed to escape through the partially silvered mirror on the right. These small transmitted pulses of light hit a detector, which therefore emits "ticks" evenly spaced (in time) by one round - trip time of the pulse. (New pulses are injected in phase with the detected pulses to keep the clock going.) In this problem, we show that the clock's length appears shortened when viewed from a reference frame S in which the clock is moving with relative speed to the right, parallel to its long axis. The key is to calculate explicitly the length of time that an observer in S will measure between "ticks" of the moving clock in terms of , the length of the clock in the frame S. Comparing this with the known period of the clock in the S frame allows us to compute the length of the clock in the S frame. Einstein used gedanken (thought) experiments such as this to clarify his thinking as he strove to remove inconsistencies from the basic formulations of physics. Part A What is the period between successive ticks of the clock in its rest frame ? Express your answer in terms of variables given in the introduction. Use for the speed of light. ANSWER: = Part B What is the time Hint B.1 between the ticks of the light clock as viewed from reference frame ? Determine the time dilation factor The time interval measured by a stationary observer for a moving clock is a measurement that is made at the same position in the moving system against clocks synchronized in the stationary one. This is the standard time dilation situation where an interval of on the moving clock appears longer by the amount (i.e., Express ). in terms of the velocity of relative motion and the speed of light . ANSWER: http://session.masteringphysics.com/myct/courseHome?start=1 Page 4 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM ANSWER: = Express the time between these ticks in terms of , the relative speed , and the speed of light . ANSWER: = Any clock in the moving frame must keep time at this slower pace. It is straightforward to show explicitly that a light clock traveling perpendicular to its long axis appears slowed by this factor when viewed from the stationary frame. If reorienting the light clock so that it travels parallel to its long axis were to cause it to change its rate, then the laws of physics would depend on the velocity of the moving frame - - a clear violation of the relativity postulate. The moving light clock has to keep the same time despite its orientation (although it can appear to run at different rates to relatively moving observers). We now come to the key step: calculating the round - trip travel time (the clock tick time) from the perspective of an observer in S. This is not trivial, because we know that the separation of the moving mirrors is in the moving frame, but it is not necessarily in the stationary reference frame. In fact, the point of this problem is to calculate the length of light will travel at speed of the clock in the system S. We do know that the pulse relative to the coordinate system S. Therefore our job is to find the time takes to catch the moving mirror (which appears to be it ahead of the detector mirror) and return to the mirror at the end of the clock with the detector mirror. Part C What is the round - trip time Hint C.1 ? Where is the mirror when the light pulse hits it? Find the location at which the light pulse from the source hits the right - hand mirror. The stationary observer sees the light travel at , but it has to catch up to the right - hand mirror, which moves at speed . Assume that the source is at position http://session.masteringphysics.com/myct/courseHome?start=1 at the time when the pulse is emitted. Page 5 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Express the x position of the mirror at time terms of , , and when the pulse of light emitted at hits it in . ANSWER: = Hint C.2 Find the travel time to the mirror Since the light pulse is at when it strikes the mirror, it is easy to find the time light pulse hits the mirror. What is Express in terms of , when the first ? , and . ANSWER: = Hint C.3 Find the return - trip time The time to the mirror can be calculated in one line if you realize that is the speed of the light pulse relative to the right - hand mirror as seen by the observer in S. Find the return- trip time . Hint C.3.1 Relative speed on return trip Notice that the time for the light pulse to reach the right - hand mirror, in the S frame, is just the length of the mirror divided by the speed of the light pulse relative to the right - hand mirror in the S frame. Similarly, the time for the return trip is just the length divided by the speed of the light pulse relative to the source in the S frame. When the light pulse is "catching up" with the right - hand mirror, its relative speed is . Since the source is heading toward the light pulse, on the return trip the relative speed is Express . in terms of , , and . ANSWER: = Use kinematics to express in terms of , the velocity , and the speed of light . ANSWER: = http://session.masteringphysics.com/myct/courseHome?start=1 Page 6 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Part D What is the length Hint D.1 of the clock according to a stationary observer in S? Expressing the length of the clock in S You have two expressions for the clock tick time between Express and as viewed by S. Together they give a relationship . in terms of the length of the clock in its rest frame and the velocities and . ANSWER: = The length of the moving clock appears shorter to the stationary observer. This is called the Lorentz contraction, but it is a reflection of the fundamental fact that the length of a moving object has lost its conventional meaning in relativity. Length is the spatial separation of two events: The (moving) ends of the clock must be measured at the same instant. Since simultaneity is relative (as a consequence of the relativity postulates), the length of a moving object depends on which observer defines what simultaneous means. Length and time have become relative quantities, as was realized first by Einstein. This property of nature has significant philosophical implications. Now, look at this applet , which shows the situation you've been studying in motion. The top shows how things look in the rest frame of the clock. On the bottom, you see the same situation from a frame of reference moving relative to the light clock. The clock's motion can be set by changing the value of with the slider. Run the applet a few times to be sure that you have a good understanding of how everything works. There is much that can be learned from studying the various clocks and how their rates change with varying . Part E Notice that the clocks at the front and rear mirrors of the light clock do not show the same time when viewed from the moving frame. Why is this ? ANSWER: The front clock experiences a greater time dilation because it moves ahead of the rear clock. The clocks start simultaneously in the rest frame, so they cannot start simultaneously in the moving frame. Since the light clock is passing the observer, the signal from the front clock takes longer to reach the observer, making the time appear earlier. http://session.masteringphysics.com/myct/courseHome?start=1 Page 7 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM One of the most important concepts derived from special relativity is that simultaneity is a relative concept (i.e., events that are simultaneous for observers in one reference frame are not simultaneous for observers in another reference frame). Recall that this is not just an optical illusion or the result of signals taking different amounts of time to reach the observer. An intelligent observer, who corrects for the time signals take to reach him ot her, will find that events that are simultaneous in another reference frame take place at different times in the observer's reference frame. Energy of a Speeding Proton Description: Find energy, momentum, and speed of a relativistically moving proton. A proton (with a rest mass ) has a total energy that is 4.00 times its rest energy. Part A What is the kinetic energy Hint A.1 of the proton? Converting to Often in particle physics, masses of particles are given in units of electron - volts and , where are billions of is the speed of light. These units are useful, because the energy of a beam in a particle accelerator is measured in electron - volts. Knowing the mass in makes it easy to compare the energy needed to create a particle to the energy of the beam, or to compute the kinetic energy of a particle if you know the beam energy. The mass of a proton in such units is . Hint A.2 Total energy components The total energy of the particle can be broken down into its components as follows: , where is the rest energy of the particle, is the kinetic energy, and energy. In this problem we are dealing with a free particle so Hint A.3 is the potential . Rest energy Recall that the rest energy of a particle is defined by , where is the mass and is the speed of light. Express your answer in billions of electron volts to three significant figures. ANSWER: = Part B http://session.masteringphysics.com/myct/courseHome?start=1 Page 8 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM What is the magnitude of the momentum Hint B.1 of the proton? Find the relativistic momentum Derive the magnitude of the relativistic momentum using the relativistic energy- momentum relation . Hint B.1.1 Definition of total particle energy Recall that the total energy of the particle is given by , where is the Lorentz factor. Express your answer in terms of (the Lorentz factor), , and the speed of light . ANSWER: = Hint B.2 Calculate the Lorentz factor Using the information given, calculate the Lorentz factor . Hint B.2.1 Definition of total particle energy Recall that the total energy of the particle is given by , where is the Lorentz factor. ANSWER: = Express your answer in kilogram - meters per second to three significant figures. ANSWER: = Part C What is the speed Hint C.1 of the proton? Rewrite the equation for speed http://session.masteringphysics.com/myct/courseHome?start=1 Page 9 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM What is the equation for speed in terms of the Lorentz factor and the speed of light ? ANSWER: = Hint C.2 Definition of Recall that is defined as , where is the speed of the particle and is the speed of light. Express your answer as a fraction of the speed of light to four significant figures. ANSWER: = Rest Length and Length Contraction Ranking Task Description: Ranking task on rest length and observed length in different frame of references. (ranking task) The graph below shows the value of the term as a function of velocity. Six spaceships with rest lengths speed of each ship, zoom past an intergalactic speed trap. The officer on duty records the . (No ship is going in excess of the stated speed limit of , so she doesn’t have to pull anyone over for a ticket.) http://session.masteringphysics.com/myct/courseHome?start=1 Page 10 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Part A Rank these spaceships on the basis of their length measured by the police officer. Hint A.1 Rest length versus "moving" length The rest length of each spaceship is given. The length measured by the police officer will be less than this amount because, in the police officer’s frame of reference, the ships are moving and will contract as the result of length contraction. Hint A.2 The length contraction relation The mathematical relationship between the rest length of the spaceship ( ) and "moving" length ( ) is . Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER: View Part B Rank these spaceships on the basis of their length as measured by their respective captains. Hint B.1 The captain’s frame of reference Relative to each ship's respective captain, sitting in the captain’s chair, each spaceship is at rest. This is true regardless of the speed of the ship relative to the police officer or anyone else. Rank from largest to smallest. To rank items as equivalent, overlap them. ANSWER: View http://session.masteringphysics.com/myct/courseHome?start=1 Page 11 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Time is Relative Description: Consider two events occurring at the same space point in a frame of reference, find their space distance as measured in a moving frame of reference, given the time intervals between the two events measured in both frames of reference. Part A Two events are observed in a frame of reference S to occur at the same space point, with the second event occurring after a time of 1.70 . In a second frame S' moving relative to S, the second event is observed to occur after a time of 2.25 . What is the difference between the positions of the two events as measured in S' ? Hint A.1 How to approach the problem Although the two events are observed to occur at the same location in S, from the point of view of an observer at rest in S' they occur at a space distance that depends on the relative speed of the two frames of reference and the time elapsed between the two events as measured by the observer in S' . Note that the observer at rest in S' will see the source of the event moving with speed . Therefore, to find the difference between the positions of the two events as measured in S' , you need to determine the relative speed . To do that, use the relativistic relationship between the time interval measured in S and the one measured in S' . Hint A.2 Find the expression for the space distance between the two events If the speed of S' relative to S is , which of the following expressions for the space distance between the two events as measured in S' is correct? Here, and are the time intervals between the two events as measured in S and S' , respectively. ANSWER: Hint A.3 Find the relative speed What is the speed of the frame of reference S ' with respect to the frame S? Hint A.3.1 Time dilation Since the speed of light interval is the same in all frames of reference (Einstein's postulate), the time between two events that are observed to occur at the same space point in a given frame of http://session.masteringphysics.com/myct/courseHome?start=1 Page 12 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM reference S is not the same in all reference frames. An observer in a frame S' moving with speed relative to S would measure a longer time interval between those two same events because of the relativistic effect of time dilation. The time interval measured in the moving frame S' is . If you know the time interval relative speed , also called proper time , and , you can derive an expression for the . Hint A.3.2 Find the proper time The time interval between two events occurring at the same space point as measured in a particular frame of reference is called the proper time . What is the value of the proper time in this case ? Express your answer numerically in seconds. ANSWER: = Express your answer numerically in meters per second. ANSWER: = Use 3.00×10 8 for the speed of light in a vacuum. ANSWER: = Do not be tempted to interpret the distance that you have just calculated as the distance traveled by the moving frame S' in the time elapsed between the two events as measured by an observer in S. An observer in S would measure a distance equal to , where is the relative speed of the two reference frames and is the time between events in S. http://session.masteringphysics.com/myct/courseHome?start=1 Page 13 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM ± The Space - Time Interval Description: ± Includes Math Remediation. The space- time interval is explored through Lorentz transformations. For this problem, assume that any two frames have the same origin. Consider two events with coordinates and in an inertial frame of reference S. The coordinates of these two events in an inertial frame S' moving to the right with speed are and relative to frame S . If we assume that the y and z coordinates are constant in the two frames, then is the space interval, more commonly known as distance, between the events as measured in frame S' , and interval in the frame S. These two intervals are more commonly written and is called the time interval between the events in frame S' , and frame S. These two intervals are more commonly written and is the space , respectively. Similarly, is called the time interval in , respectively. Part A Find the value of Hint A.1 . Find Find the space interval between these two events. Use . Hint A.1.1 How to approach the problem Recall that the Lorentz transformations have the following form: , , where . Observe that the transformations are linear, so will transform in the same manner that Express your answer in terms of some or all of the variables , , , does. , and . ANSWER: = http://session.masteringphysics.com/myct/courseHome?start=1 Page 14 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM [P If you used the hint, you may have already read this, but it is worth reiterating. If you did not use rint ] the hint, note that the only reason that transforms in the same manner that does is because the Lorentz transformations are linear. Find Hint A.2 Now that you have a value for ? Be certain to use , simplify it using and . Then square. What is the value of to simplify your expression wherever possible. If you don't, you will have a very complicated - looking equation. ANSWER: Hint A.3 Find Find the time interval in the second frame. Hint A.3.1 How to approach the problem Recall that the Lorentz transformations have the following form: , , where . Observe that the transformations are linear, so will transform in the same manner that Express your answer in terms of some or all of the variables , , , does. , and . ANSWER: = http://session.masteringphysics.com/myct/courseHome?start=1 Page 15 of 20 MasteringPhysics: Course Home Hint A.4 5/2/11 12:30 PM Find Now that you have a value for is the value of , simplify it using ? Be certain to use and . Then square and multiply by . What to simplify your expression wherever possible. ANSWER: Express your answer in terms of any of the following , , , and . ANSWER: = In classical physics, the space interval and time interval are each separately conserved. In special relativity, however, the conserved quantity is a mixture of space and time called the space- time interval . To avoid possibly having to take the square root of a negative number, we usually talk about the square of the space- time interval, which is what you actually showed to be conserved in this part: . Part B A pair of events are observed to have coordinates What is the proper time interval Hint B.1 and in a frame S. between the two events ? How to approach the problem Since the square of the space- time interval is conserved, find for one frame, then use that value of to find the time interval in the other frame. Hint B.2 Which frame has the proper time? Recall that the proper time interval between two events is defined as the time interval between the two events in a frame of reference in which the two events occur at the same point. Think about what http://session.masteringphysics.com/myct/courseHome?start=1 Page 16 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM this means for the value of the space interval Hint B.3 in the frame where proper time is measured. Calculate Calculate the value of events in frame S are for the two events in frame S. The coordinates for the two and . Express your answer in meters to three significant figures. ANSWER: = Express your answer in seconds to three significant figures. ANSWER: = Just as conservation of energy allowed you to solve problems where Newton's laws would have been difficult to work with, conservation of the space- time interval can simplify many relativity problems. In any situation, the proper time between two events will equal the space- time interval divided by , since the space interval is, by definition, equal to zero in the frame where the proper time is measured. You may have heard that in the general theory of relativity (the relativistic description of gravity), gravity is described by a "curvature of space- time" or that matter curves space- time. This essentially means that, in the presence of matter, the space- time interval is no longer conserved. The difficult mathematics required to describe how the space- time interval changes, as well as the conceptual leaps needed to understand how the shape of space and time can be changed, led to the widely quoted figure that five years after the theory had been put forward, only three people on earth understood it. Conceptual Question 37.1 Description: The figure shows two balls. (a) What is the speed of the ball 1 in a reference frame that moves with ball 1 ? (b) What is the direction of motion of the ball 1 in a reference frame that moves with ball 1 ? (c) What is the speed of the ball 2 in a... The figure shows two balls. http://session.masteringphysics.com/myct/courseHome?start=1 Page 17 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Part A What is the speed of the ball 1 in a reference frame that moves with ball 1 ? Express your answer using one significant figure. ANSWER: = Part B What is the direction of motion of the ball 1 in a reference frame that moves with ball 1 ? ANSWER: To the left To the right Zero Part C What is the speed of the ball 2 in a reference frame that moves with ball 1 ? Express your answer using one significant figure. ANSWER: = Part D What is the direction of motion of the ball 2 in a reference frame that moves with ball 1 ? ANSWER: To the left To the right Zero Part E What is the speed of the ball 1 in a reference frame that moves with ball 2 ? http://session.masteringphysics.com/myct/courseHome?start=1 Page 18 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Express your answer using one significant figure. ANSWER: = Part F What is the direction of motion of the ball 1 in a reference frame that moves with ball 2 ? ANSWER: Zero To the right To the left Part G What is the speed of the ball 2 in a reference frame that moves with ball 2 ? Express your answer using one significant figure. ANSWER: = Part H What is the direction of motion of the ball 2 in a reference frame that moves with ball 2 ? ANSWER: To the left Zero To the right Problem 37.15 Description: You are flying your personal rocketcraft at 0.9c from Star A toward Star B. The distance between the stars, in the stars' reference frame, is 1.0 ly. Both stars happen to explode simultaneously in your reference frame at the instant you are exactly... You are flying your personal rocketcraft at 0.9 from Star A toward Star B. The distance between the stars, in the stars' reference frame, is 1.0 . Both stars happen to explode simultaneously in your reference frame at the instant you are exactly halfway between them. Part A http://session.masteringphysics.com/myct/courseHome?start=1 Page 19 of 20 MasteringPhysics: Course Home 5/2/11 12:30 PM Do you see the flashes simultaneously? ANSWER: Yes No Score Summary: Your score on this assignment is 0%. You received 0 out of a possible total of 8 points. http://session.masteringphysics.com/myct/courseHome?start=1 Page 20 of 20 ...
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This note was uploaded on 12/28/2011 for the course PHYSICS 270 taught by Professor Drake during the Fall '08 term at Maryland.

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