What is an algebraic expression for the Lorentz factor

Problem 1RQ If you walk at 1 km/h down the aisle of a train that moves at 60 km/h, what is your speed relative to the ground?

Problem 1R Electrons are fired at different speeds through a magnetic field and are bent from their straight-line paths to hit the detector at the points shown. Rank the speeds of the electrons from highest to lowest.

Problem 50E Make up four multiple-choice questions, one each that would check a classmate’s understanding of (a) time dilation, (b) length contraction, (c) relativistic momentum, and (d) ?E? = m? c?2.

Problem 2E If you were in a smooth-riding train with no windows, could you sense the difference between uniform motion and rest? Between accelerated motion and rest? Explain how you could make such a distinction with a bowl filled with water.

Problem 1E The idea that force causes acceleration doesn’t seem strange. This and other ideas of Newtonian mechanics are consistent with our everyday experience. But the ideas of relativity do seem odd and more difficult to grasp. Why is this?

To an Earth observer, metersticks on three spaceships are seen to have these lengths. Rank the speeds of the spaceships relative to Earth, from highest to lowest.

Problem 2P Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. A starship passes Earth at 80% of the speed of light and sends a drone ship forward at half the speed of light relative to itself. Show that the drone travels at 93% the speed of light relative to Earth.

If you walk at 1 km/h down the aisle toward the front of a train that moves at 60 km/h, what is your speed relative to the ground? In the preceding question, is your approximate speed relative to the Sun as you walk down the aisle of the train changed slightly or by a lot?

Problem 3E A person riding on the roof of a freight train throws a ball forward. (a) Neglecting air drag and relative to the ground, is the ball moving faster or slower when the train is moving than when it is standing still? (b) Relative to the freight car, is the ball moving faster or slower when the train is moving than when the train is standing still?

Problem 1P Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. Consider a high-speed rocket ship equipped with a flashing light source. If the frequency of flashes seen on an approaching ship is twice what it was when the ship was a fixed distance away, by how much is the period (time interval between flashes) changed? Is this period constant for a constant relative speed? For accelerated motion? Defend your answer.

Problem 3RQ What hypothesis did G. F. FitzGerald make to explain the findings of Michelson and Morley?

Suppose instead that the person riding on top of the freight car shines a searchlight beam in the direction in which the train is traveling. Compare the speed of the light beam relative to the ground when the train is at rest and when it is in motion. How does the behavior of the light beam differ from the behavior of the ball in problem 50? Problem 50 A person riding on the roof of a freight train throws a ball forward. (a) If we ignore air drag and relative to the ground, is the ball moving faster or slower when the train is moving than when it is standing still? (b) Relative to the freight car, is the ball moving faster or slower when the train is moving than when the train is standing still?

Problem 4RQ What classical idea about space and time was rejected by Einstein?

Problem 3P Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. Pretend that the starship in the problem 1 is somehow traveling at ?c with respect to Earth and it fires a drone forward at speed ?c with respect to itself. Use the equation for the relativistic addition of velocities to show that the speed of the drone with respect to Earth is still ?c. Problem 1 Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. A starship passes Earth at 80% of the speed of light and sends a drone ship forward at half the speed of light relative to itself. Show that the drone travels at 93% the speed of light relative to Earth.

Problem 5E Why did Michelson and Morley at first consider their experiment a failure? (Have you ever encountered other examples where failure has to do not with the lack of ability, but with the impossibility of the task?)

Problem 4P Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. A passenger on an interplanetary express bus traveling at ?v = 0.99?c takes a 5-minute catnap, according to her watch. Show that her catnap from the vantage point of a fixed planet lasts 35 minutes.

Problem 5RQ Cite two examples of Einstein’s first postulate.

Problem 5P Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. According to Newtonian mechanics, the momentum of the bus in the problem 1 is ?p? = mv.? According to relativity, it is ?p? = ??mv?. How does the actual momentum of the bus moving at 0.99?c? compare with the momentum it would have if classical mechanics were valid? How does the momentum of an electron traveling at 0.99?c? compare with its classical momentum? Problem 1 Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. A passenger on an interplanetary express bus traveling at ?v? = 0.99?c? takes a 5-minute catnap, according to her watch. Show that her catnap from the vantage point of a fixed planet lasts 35 minutes.

Problem 6E When you drive down the highway, you are moving through space. What else are you moving through?

Problem 7E In Chapter 26, we learned that light travels more slowly in glass than in air. Does this contradict Einstein’s second postulate?

Problem 7RQ Inside the moving compartment of Figure 35.4, light travels a certain distance to the front end and a certain distance to the back end of the compartment. How do these distances compare as seen in the frame of reference of the moving rocket?

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^2}{c^2}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. If the bus in problem 40 were to slow to a “mere” 10% of the speed of light, show that you would measure the passenger’s catnap to last slightly more than 5 minutes. Problem 40 A passenger on an interplanetary express bus traveling at v = 0.99c takes a 5-minute catnap, according to her watch. Show that her catnap from the vantage point of a fixed planet lasts 35 minutes.

Problem 8E Astronomers view light coming from distant galaxies moving away from Earth at speeds greater than 10% the speed of light. How fast does this light meet the telescopes of the astronomers?

Inside the moving compartment of Figure 35.4, light travels a certain distance to the front end and a certain distance to the back end of the compartment. How do these distances compare as seen in the frame of reference of the moving rocket?

Problem 9E Does special relativity allow ?anything? to travel faster than light? Explain.

Problem 9P Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. Assume that rocket taxis of the future move about the solar system at half the speed of light. For a 1-hour trip as measured by a clock in the taxi, a driver is paid 10 stellars. The taxi-driver’s union demands that pay be based on Earth time instead of taxi time. If their demand is met, show that the new payment for the same trip would be 11.5 stellars.

Problem 9RQ How many coordinate axes are usually used to describe three-dimensional space? What does the fourth dimension measure?

Problem 10E When a light beam approaches you, its frequency is greater and its wavelength less. Does this contradict the postulate that the speed of light cannot change? Defend your answer.

Problem 10P Recall, from this chapter, that the factor gamma (?) governs both time dilation and length contraction, where ? When you multiply the time in a moving frame by ?, you get the longer (dilated) time in your fixed fame. When you divide the length in a moving frame by ?, you get the shorter (contracted) length in your fixed frame. The fractional change of reacting mass to energy in a fission reactor is about 0.1%, or 1 part in a thousand. For each kilogram of uranium that undergoes fission, how much energy is released? If energy costs 3 cents per megajoule, how much is this energy worth in dollars?

Problem 10RQ Under what condition will you and a friend share the same realm of spacetime? When will you not share the same realm?

Problem 11E The beam of light from a laser on a rotating turntable casts into space. At some distance, the beam moves across space faster than ?c?. Why does this not contradict relativity?

Problem 11RQ What is special about the ratio of the distance traveled by a flash of light and the time the light takes to travel this distance?

Time is required for light to travel along a path from one point to another. If this path is seen to be longer because of motion, what happens to the time it takes for light to travel this longer path?

Problem 12E Can an electron beam sweep across the face of a cathode- ray tube at a speed greater than the speed of light? Explain.

Problem 13RQ What do we call the “stretching out” of time?

Problem 13E Consider the speed of the point where scissors blades meet when the scissors are closed. The closer the blades are to being closed, the faster the point moves. The point could, in principle, move faster than light. Likewise for the speed of the point where an ax meets wood when the ax blade meets the wood not quite horizontally; the contact point travels faster than the ax. Similarly, a pair of laser beams that are crossed and moved toward being parallel produce a point of intersection that can move faster than light. Why do these examples not contradict special relativity?

Problem 14E If two lightning bolts hit exactly the same place at exactly the same time in one frame of reference, is it possible that observers in other frames will see the bolts hitting at different times or at different places?

Problem 14RQ What is an algebraic expression for the Lorentz factor ?? (gamma)? Why is ?? never less than 1 ?

Problem 15E Event A occurs before event B in a certain frame of reference. How could event B occur before event A in some other frame of reference?

Problem 15RQ How do measurements of time differ for events in a frame of reference that moves at 50% the speed of light relative to us? At 99.5% the speed of light relative to us?

Problem 16RQ What is the evidence for time dilation?

Problem 17E The speed of light is a speed limit in the universe—at least for the four-dimensional universe we comprehend. No material particle can attain or surpass this limit even when a continuous, unremitting force is exerted on it. What evidence supports this?

Problem 17RQ When a flashing light approaches you, each flash that reaches you has a shorter distance to travel. What effect does this have on how frequently you receive the flashes?

Problem 16E Suppose that the lightbulb in the rocket ship in Figures 35.4 and 35.5 is closer to the front than to the rear of the compartment so that the observer in the ship sees the light reaching the front before it reaches the back. Is it still possible that the outside observer will see the light reaching the back first?

Problem 18E Since there is an upper limit on the speed of a particle, does it follow that there is also an upper limit on its momentum, and, therefore, on its kinetic energy? Explain.

Problem 18RQ When a flashing light source approaches you, does the speed of light or the frequency of light—or both—increase?

Problem 19E Light travels a certain distance in, say, 20,000 years. How is it possible that an astronaut, traveling slower than light, could go as far in 20 years of her life as light travels in 20,000 years?

Problem 19RQ If a flashing light source moves toward you fast enough so that the duration between flashes is half as long, how long will be the duration between flashes if the source is moving away from you at the same speed?

Problem 20E Is it possible in principle for a human being who has a life expectancy of 70 years to make a round-trip journey to a part of the universe thousands of light-years distant? Explain.

Problem 21RQ What is the maximum value of ?v?1?v?2/?c?2 in an extreme situation? What is the smallest value?

Problem 21E A twin who makes a long trip at relativistic speeds returns younger than her stay-at-home twin sister. Could she return before her twin sister was born? Defend your answer.

Problem 20RQ How many frames of reference does the stay-at-home twin experience in the twin trip? How many frames of reference does the traveling twin experience?

Problem 22E Is it possible for a son or daughter to be biologically older than his or her parents? Explain,

Is the relativistic rule \(V=\frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}\) consistent with the fact that light can have only one speed in all uniformly moving reference frames?

If you were in a rocket ship traveling away from Earth at a speed close to the speed of light, what changes would you note in your pulse? In your volume? Explain.

Problem 23RQ What two main obstacles prevent us from traveling today throughout the galaxy at relativistic speeds?

Problem 24RQ What is the universal standard of time?

Problem 24E If you were on Earth monitoring a person in a rocket ship traveling away from Earth at a speed close to the speed of light, what changes would you note in his pulse? In his volume? Explain.

Problem 25E Due to length contraction, you see people in a spaceship passing by you as being slightly narrower than they normally appear. How do these people view you?

How long would a meter stick appear to be if it were traveling like a properly thrown spear at 99.5% of the speed of light?

Problem 26E Because of time dilation, you observe the hands of your friend’s watch to be moving slowly. How does your friend view your watch—as running slowly, running rapidly, or neither?

How long would the meter stick in the preceding question appear to be if it were traveling with its length perpendicular to its direction of motion? (Why is your answer different from your answer to the preceding question?)

Problem 27RQ If you were traveling in a high-speed rocket ship, would meter sticks on board appear to you to be contracted? Defend your answer.

Problem 28E If you lived in a world where people regularly traveled at speeds near the speed of light, why would it be risky to make a dental appointment for 10:00 AM next Thursday?

Problem 27E Does the equation for time dilation show dilation occurring for all speeds, whether slow or fast? Explain.

Problem 29E How do the measured densities of a body compare at rest and in motion?

Problem 28RQ What would be the momentum of an object pushed to the speed of light?

When a beam of charged particles moves through a magnetic field, what is the evidence that particles in the beam have momenta greater than the value mv?

Problem 30E If stationary observers measure the shape of a passing object to be exactly circular, what is the shape of the object according to observers on board the object, traveling with it?

Problem 30RQ Compare the amount of mass converted to energy in nuclear reactions and in chemical reactions.

Problem 31E The formula relating speed, frequency, and wavelength of electromagnetic waves, v =f ???, was known before relativity was developed. Relativity has not changed this equation, but it has added a new feature to it. What is that feature?

Problem 32E Light is reflected from a moving mirror. How is the reflected light different from the incident light, and how is it the same?

Problem 31RQ How does the energy from the fissioning of a single uranium nucleus compare with the energy from the combustion of a single carbon atom?

Problem 32RQ Does the equation ?E? = ?mc?2 apply only to nuclear and chemical reactions?

Problem 33E As a meterstick moves past you, your measurements show its momentum to be twice its classical momentum and its length to be 1 m. In what direction is the stick pointing?

Problem 34E In the exercise 1, if the stick is moving in a direction along its length (like a properly thrown spear), how long will you measure its length to be?

Problem 33RQ What is the evidence for ?E? = ?mc?2 in cosmic-ray investigations?

Problem 34RQ How does the correspondence principle relate to special relativity?

Problem 35E If a high-speed spaceship appears shrunken to half its normal length, how does its momentum compare with the classical formula p ? ? = ?mv??

Problem 35RQ Do the relativity equations for time, length, and momentum hold true for everyday speeds? Explain.

Problem 36E How can the momentum of a particle increase by 5% with only a 1% increase in speed?

Problem 37E The 2-mile linear accelerator at Stanford University in California “appears” to be less than a meter long to the electrons that travel in it. Explain.

Problem 39E Two safety pins, identical except that one is latched and one is unlatched, are placed in identical acid baths. After the pins are dissolved, what, if anything, is different about the two acid baths?

Problem 38E Electrons end their trip in the Stanford accelerator with an energy thousands of times greater than their initial rest energy. In theory, if you could travel with them, would you notice an increase in their energy? In their momentum? In your moving frame of reference, what would be the approximate speed of the target they are about to hit?

Problem 40E A chunk of radioactive material encased in an idealized, perfectly insulating blanket gets warmer as its nuclei decay and release energy. Does the mass of the radioactive material and the blanket change? If so, does it increase or decrease?

Problem 41E The electrons that illuminate the screen in the picture tube of yesterday’s TV sets travel at nearly one-fourth the speed of light and possess nearly 3% more energy than hypothetical nonrelativistic electrons traveling at the same speed. Does this relativistic effect tend to increase or decrease the electric bill?

Problem 42E Muons are elementary particles that are formed high in the atmosphere by the interactions of cosmic rays with atomic nuclei up there. Muons are radioactive and have average lifetimes of about two-millionths of a second. Even though they travel at almost the speed of light, very few should be detected at sea level after traveling through the atmosphere—at least according to classical physics. Laboratory measurements, however, show that muons in great number do reach Earth’s surface. What is the explanation?

Problem 43E How might the idea of the correspondence principle be applied outside the field of physics?

Problem 46E Does a fully charged flashlight battery weigh more than the same battery when dead? Defend your answer.

Problem 44E What does the equation ?E? = ?mc?2 mean?

Problem 45E According to ?E?= ?mc?2, how does the amount of energy in a kilogram of leathers compare with the amount of energy in a kilogram of iron?

When we look out into the universe, we see into the past. John Dobson, founder of the San Francisco Sidewalk Astronomers, says that we cannot even see the backs of our own hands now—in fact, we can’t see anything now. Do you agree? Explain.

Problem 48E One of the fads of the future might be “century hopping,” where occupants of high-speed spaceships would depart from Earth for several years and return centuries later. What are the present-day obstacles to such a practice?

Is the statement by the philosopher Soren Kierkegaard that “Life can only be understood backwards; but it must be lived forwards” consistent with the theory of special relativity?

In the preceding question, is your approximate speed relative to the Sun as you walk down the aisle of the train changed slightly or by a lot?

What hypothesis did G. F. FitzGerald make to explain the findings of Michelson and Morley?

What classical idea about space and time did Einstein reject?

What is the same in Einstein’s first postulate?

What is constant in Einstein’s second postulate?

Inside the moving compartment of Figure 35.4, light travels a certain distance to the front end and a certain distance to the back end of the compartment. How do these distances compare as seen in the frame of reference of the moving rocket?

How do the distances in question 7 compare as seen in the frame of reference of an observer on a stationary planet?

How many coordinate axes are usually used to describe three-dimensional space? What does the fourth dimension measure?

Under what condition will you and a friend share the same realm of spacetime? When will you not share the same realm?

What is special about the ratio of the distance traveled by a flash of light to the time the light takes to travel this distance?

Time is required for light to travel along a path from one point to another. If this path is seen to be longer because of motion, what happens to the time it takes for light to travel this longer path?

What do we call the “stretching out” of time?

What is an algebraic expression for the Lorentz factor \(\gamma\) (gamma)? Why is \(\gamma\) never less than 1?

How do measurements of time differ for events in a frame of reference that moves at 50% of the speed of light relative to us? At 99.5% of the speed of light relative to us?

What is the evidence for time dilation?

When a flashing light approaches you, each flash that reaches you has a shorter distance to travel. What effect does this have on how frequently you receive the flashes?

If a flashing light source moves toward you fast enough so that the time interval between flashes is half as long, how long will the time interval between flashes be if the source is moving away from you at the same speed?

When a flashing light source approaches you, does the speed of light or the frequency of light—or both—increase?

How many frames of reference does the stay-at-home twin experience in the twin trip? How many frames of reference does the traveling twin experience?

What is the maximum value of \(v_{1} v_{2} / c^{2}\) in an extreme situation? What is the smallest value?

What two main obstacles prevent us from traveling today throughout the galaxy at relativistic speeds?

Is the relativistic rule \(V=\frac{v_{1}+v_{2}}{1+\frac{v_{1} v_{2}}{c^{2}}}\) consistent with the fact that light can have only one speed in all uniformly moving reference frames?

What is the universal standard of time?

How long would a meter stick appear to be if it were traveling like a properly thrown spear at 99.5% of the speed of light?

How long would the meter stick in the preceding question appear to be if it were traveling with its length perpendicular to its direction of motion? (Why is your answer different from your answer to the preceding question?)

If you were traveling in a high-speed rocket ship, would metersticks on board appear to you to be contracted? Defend your answer.

What would be the momentum of an object if it were moving at the speed of light?

When a beam of charged particles moves through a magnetic field, what is the evidence that particles in the beam have momenta greater than the value mv?

Compare the amounts of mass converted to energy in nuclear reactions and in chemical reactions.

How does the energy from the fissioning of a single uranium nucleus compare with the energy from the combustion of a single carbon atom?

Does the equation \(E=m c^{2}\) apply to chemical reactions?

How does \(E=m c^{2}\) describe the identities of energy and mass?

How does the correspondence principle relate to special relativity?

Do the relativity equations for time, length, and momentum hold true for everyday speeds? Explain.

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. Consider a high-speed rocket ship equipped with a flashing light source. If the frequency of flashes seen on an approaching ship is twice what it was when the ship was a fixed distance away, by how much is the period (time interval between flashes) changed? Is this period constant for a constant relative speed? For accelerated motion? Defend your answer. Equation Transcription: Text Transcription: (gamma) gamma = 1 / sqrt 1-(v^2 / c^2) gamma gamma

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. A starship passes Earth at \(80 \%\) of the speed of light and sends a drone ship forward at half the speed of light relative to itself. Show that the drone travels at \(93 \%\) of the speed of light relative to Earth. Equation Transcription: Text Transcription: (gamma) gamma = 1 / sqrt 1-(v^2 / c^2) gamma gamma 80% 93%

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. Pretend that the starship in the preceding problem is somehow traveling at c with respect to Earth and it fires a drone forward at speed c with respect to itself. Use the equation for the relativistic addition of velocities to show that the speed of the drone with respect to Earth is still c.

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. A passenger on an interplanetary express bus traveling at \(v=0.99c\) takes a \(5\)-minute catnap, according to her watch. Show that her catnap from the vantage point of a fixed planet lasts \(35\) minutes. Equation Transcription: Text Transcription: (gamma) gamma = 1 / sqrt 1-(v^2 / c^2) gamma gamma v = 0.99c 5 35

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. According to Newtonian mechanics, the momentum of the bus in the preceding problem is \(p=mv\). According to relativity, it is \(p=\gamma\). How does the actual momentum of the bus moving at \(0.99c\) compare with the momentum it would have if classical mechanics were valid? How does the momentum of an electron traveling at \(0.99c\) compare with its classical momentum? Equation Transcription: Text Transcription: (gamma) gamma = 1 / sqrt 1-(v^2 / c^2) gamma gamma p = mv p = gamma 0.99c 0.99c

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. The bus in the preceding problems is \(70\) feet long, according to its passengers and driver. Show that its length is seen as slightly less than \(10\) feet from a vantage point on a fixed planet. Equation Transcription: Text Transcription: (gamma) gamma = 1 / sqrt 1-(v^2 / c^2) gamma gamma 70 10

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. If the bus in problem \(40\) were to slow to a “mere” \(10 \%\) of the speed of light, show that you would measure the passenger’s catnap to last slightly more than \(5\) minutes. Equation Transcription: Text Transcription: (gamma) gamma = 1 / sqrt 1-(v^2 / c^2) gamma gamma 40 10% 5

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. If the bus driver in problem \(40\) decided to drive at \(99.99 \%\) of the speed of light in order to gain some time, show that you’d measure the length of the bus to be a little less than \(1\) foot. Equation Transcription: Text Transcription: (gamma) gamma = 1 / sqrt 1-(v^2 / c^2) gamma gamma 40 99.99% 1

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. Assume that rocket taxis of the future move about the solar system at half the speed of light. For a 1-hour trip as measured by a clock in the taxi, a driver is paid 10 stellars. The taxi-driver’s union demands that pay be based on Earth time instead of taxi time. If that demand is met, show that the new payment for the same trip would be 11.5 stellars.

Recall from this chapter that the factor gamma \((\gamma)\) governs both time dilation and length contraction, where \(\gamma=\frac{1}{\sqrt{1-\left(\frac{v^{2}}{c^{2}}\right)}}\) When you multiply the time in a moving frame by \(\gamma\), you get the longer (dilated) time in your fixed frame. When you divide the length in a moving frame by \(\gamma\), you get the shorter (contracted) length in your fixed frame. The fractional change of reacting mass to energy in a fission reactor is about 0.1%, or 1 part in a thousand. For each kilogram of uranium that is totally fissioned, how much energy is released? If energy costs 3 cents per megajoule, how much is this energy worth in dollars?

Electrons are fired at different speeds through a magnetic field and are bent from their straight-line paths to hit the detector at the points shown. Rank the speeds of the electrons, from highest to lowest.

To an Earth observer, meter sticks on three spaceships are seen to have these lengths. Rank the speeds of the spaceships relative to Earth, from highest to lowest.

If you were in a smooth-riding train with no windows, could you sense the difference between uniform motion and rest? Between accelerated motion and rest? Explain how you could make such a distinction with a bowl filled with water.

A person riding on the roof of a freight train throws a ball forward. (a) If we ignore air drag and relative to the ground, is the ball moving faster or slower when the train is moving than when it is standing still? (b) Relative to the freight car, is the ball moving faster or slower when the train is moving than when the train is standing still?

Suppose instead that the person riding on top of the freight car shines a searchlight beam in the direction in which the train is traveling. Compare the speed of the light beam relative to the ground when the train is at rest and when it is in motion. How does the behavior of the light beam differ from the behavior of the ball in problem 50?

When you drive down the highway, you are moving through space. What else are you moving through?

In Chapter 26, we learned that light travels more slowly in glass than in air. Does this contradict Einstein’s second postulate?

Astronomers view light coming from distant galaxies moving away from Earth at speeds greater than 10% of the speed of light. How fast does this light meet the telescopes of the astronomers?

The beam of light from a laser on a rotating turntable casts into space. At some distance, the beam moves across space faster than c. Why doesn’t this contradict relativity?

Can an electron beam sweep across the face of a cathode-ray tube at a speed greater than the speed of light? Explain.

Event A occurs before event B in a certain frame of reference. How could event B occur before event A in some other frame of reference?

If two lightning bolts hit exactly the same place at exactly the same time in one frame of reference, is it possible that observers in other frames will see the bolts hitting at different times or at different places?

Suppose that the lightbulb in the rocket ship in Figures 35.4 and 35.5 is closer to the front than to the rear of the compartment so that the observer in the ship sees the light reaching the front end before it reaches the back end. Is it still possible that the outside observer will see the light reaching the back end first?

Since there is an upper limit on the speed of a particle, does it follow that there is also an upper limit on its momentum and, therefore, on its kinetic energy? Explain.

Light travels a certain distance in, say, 20,000 years. How is it possible that an astronaut, traveling slower than light, could go as far in 20 years of her life as light travels in 20,000 years?

Is it possible in principle for a human being who has a life expectancy of 70 years to make a round-trip journey to a part of the universe thousands of light-years distant? Explain.

A twin who makes a long trip at relativistic speeds returns younger than her stay-at-home twin sister. Could she return before her twin sister was born? Defend your answer.

Is it possible for a son or daughter to be biologically older than his or her parents? Explain.

If you were in a rocket ship traveling away from Earth at a speed close to the speed of light, what changes would you note in your pulse? In your volume? Explain.

If you were on Earth monitoring a person in a rocket ship traveling away from Earth at a speed close to the speed of light, what changes would you note in his pulse? In his volume? Explain.

Due to length contraction, you see people in a spaceship passing by you as being slightly narrower than they normally appear. How do these people view you?

Because of time dilation, you observe the hands of your friend’s watch to be moving slowly. How does your friend view your watch: as running slowly, running rapidly, or neither?

Does the equation for time dilation show dilation occurring for all speeds, whether slow or fast? Explain.

If you lived in a world where people regularly traveled at speeds near the speed of light, why would it be risky to make a dental appointment for 10:00 AM next Thursday?

How do the measured densities of a body compare at rest and in motion?

If stationary observers measure the shape of a passing object to be exactly circular, what is the shape of the object when viewed face-on by observers on board the object, traveling with it?

The formula that relates the speed, frequency, and wavelength of electromagnetic waves, \(v=f \lambda\), was known before relativity was developed. Relativity has not changed this equation, but it has added a new feature to it. What is that feature?

Light is reflected from a moving mirror. How is the reflected light different from the incident light, and how is it the same?

As a meter stick moves past you, your measurements show its momentum to be twice its classical momentum and its length to be 1 m. In what direction is the stick pointing?

In the preceding exercise, if the stick is moving in a direction along its length (like a properly thrown spear), how long will you measure its length to be?

If a high-speed spaceship appears shrunken to half its normal length, how does its momentum compare with the classical formula p=mv?

How can the momentum of a particle increase by 5% with only a 1% increase in speed?

The 2-mile linear accelerator at Stanford University in California “appears” to be less than a meter long to the electrons that travel in it. Explain.

Electrons end their trip in the Stanford accelerator with an energy thousands of times greater than their initial rest energy. In theory, if you could travel with them, would you notice an increase in their energy? In their momentum? In your moving frame of reference, what would be the approximate speed of the target they are about to hit?

The electrons that illuminate the screen in the picture tube of yesterday’s TV sets travel at nearly one-fourth the speed of light and possess nearly 3% more energy than hypothetical non-relativistic electrons traveling at the same speed. Does this relativistic effect tend to increase or decrease the electric bill?

How might the idea of the correspondence principle be applied outside the field of physics?

What does the equation \(E=m c^{2}\) mean? Text Transcription: E=mc^2

According to \(E=m c^{2}\) , how does the amount of energy in a kilogram of feathers compare with the amount of energy in a kilogram of iron?

Does a fully charged flashlight battery weigh more than the same battery when dead? Defend your answer.

When we look out into the universe, we see into the past. John Dobson, founder of the San Francisco Sidewalk Astronomers, says that we cannot even see the backs of our own hands now—in fact, we can’t see anything now. Do you agree? Explain.

Make up four multiple-choice questions, one each that would check a classmate’s understanding of (a) time dilation, (b) length contraction, (c) relativistic momentum, and (d) \(E=m c^{2}\) . Text Transcription: E=mc^2

The idea that force causes acceleration doesn’t seem strange. This and other ideas of Newtonian mechanics are consistent with our everyday experience. But the ideas of relativity do seem odd and more difficult to grasp. Discuss.

Why did Michelson and Morley at first consider their experiment a failure? (Discuss examples you may have encountered where failure has to do not with lack of ability but with the impossibility of the task.)

Does special relativity allow anything to travel faster than light? Discuss.

When a light beam approaches you, its frequency is higher and its wavelength is shorter. Does this contradict the postulate that the speed of light cannot change? Discuss.

The speed of light is a speed limit in the universe—at least for the four-dimensional universe we comprehend. No material particle can attain or surpass this limit even when a continuous, unremitting force is exerted on it. Discuss evidence that supports this.

Two safety pins, identical except that one is latched and one is unlatched, are placed in identical acid baths. After the pins are dissolved, what, if anything, is different about the two acid baths?

A chunk of radioactive material encased in a thick lead container gets warmer as its nuclei decay and release energy. Does the mass of the chunk-container system change? If so, does it increase or decrease?

Muons are elementary particles that are formed high in the atmosphere by the interactions of cosmic rays with atomic nuclei up there. Muons are radioactive and have average lifetimes of about two-millionths of a second. Even though they travel at almost the speed of light, very few should be detected at sea level after traveling through the atmosphere—at least according to classical physics. Laboratory measurements, however, show that muons in great number do reach Earth’s surface. Discuss and explain.

One of the fads of the future might be “century hopping,” where occupants of high-speed spaceships would depart from Earth for several years and return centuries later. Discuss the present-day obstacles to such a practice.

Is the statement by the philosopher Soren Kierkegaard that “Life can only be understood backwards; but it must be lived forwards” consistent with the theory of special relativity?

Your study partner says that matter can be neither created nor destroyed. What do you say to correct this statement?

Discuss with your friends how length contraction occurs for a racing car that travels at 200 miles per hour, but why the decrease can be ignored.