College Physics 2013

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 25

Special Relativity

Educators


Problem 1

Relative motion on an airport walkway A person is walking on a moving walkway in the airport. Her speed with respect to the walkway is 2 $\mathrm{m} / \mathrm{s}$ . The speed of the walkway is 1 $\mathrm{m} / \mathrm{s}$ with respect to the floor. What is her speed with respect to a person walking on the floor in the opposite direction at the speed of 2 $\mathrm{m} / \mathrm{s} ?$

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Problem 2

Running on a treadmill Explain how you can run on a treadmill at 3 $\mathrm{m} / \mathrm{s}$ and remain at the same location.

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Problem 3

Describe the important parts of the Michelson-Morley experimental setup and explain how this setup could help them determine whether Earth moves with respect to ether. Explain how the setup relates to the previous problem.

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Problem 4

Describe what Michelson and Morley would have observed when they rotated their spectrometer if Earth were moving through ether compared to what they would have observed if Earth were not moving through ether.

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Problem 5

Person on a bus A person is sitting on a bus that stops suddenly, causing her head to tilt forward. (a) Explain the acceleration of her head from the point of view of an observer on the ground. (b) Explain the acceleration of her head from the point of view of another person on the bus. (c) Which observer is in an inertial reference frame?

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Problem 6

Turning on a rotating turntable A matchbox is placed on a rotating turntable. The turntable starts turning faster and faster. At some instant the matchbox flies off the turning table. (a) Draw a force diagram for the box when still on the rotating turntable. (b) Draw a force diagram for the box just before it flies off. (c) Explain why the box flies off only when the turntable reaches a certain speed. In what reference frame are you when providing this explanation? (d) How would a bug sitting on the turntable explain the same situation?

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Problem 7

Use your knowledge of electromagnetic waves to give an example illustrating that if the speed of light were different in different inertial reference frames, two inertial frame observers would see the same phenomenon differently. [Hint: Think about radio waves.]

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Problem 8

A particle called $\Sigma^{+}$ lives for $0.80 \times 10^{-10} \mathrm{s}$ in its proper reference frame before transforming into two other particles. How long does the $\Sigma^{+}$ seem to live according to a laboratory observer when the particle moves past the observer at a speed of $2.4 \times 10^{8} \mathrm{m} / \mathrm{s}$ ?

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Problem 9

The $\Sigma^{+}$ particle discussed in the previous problem appears to a laboratory observer to live for $1.0 \times 10^{-10}$ s. How fast is it moving relative to the observer?

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Problem 10

A person on Earth observes 10 flashes of the light on a passing spaceship in 22 s, whereas the same 10 flashes seem to take 12 s to an observer on the ship. What can you determine using this information?

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Problem 11

A spaceship moves away from Earth at a speed of 0.990$c$ . The pilot looks back and measures the time interval for one rotation of Earth on its axis. What time interval does the pilot measure? What assumptions did you make?

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Problem 12

Extending life? A free neutron lives about 1000 s before transforming into an electron and a proton. If a neutron leaves the Sun at a speed of $0.999c$, (a) how long does it live according to an Earth observer? (b) Will such a neutron reach Pluto ( $5.9 \times 10^{12} \mathrm{m}$ from the Sun) before transforming? Explain your answers.

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Problem 13

A $\Sigma^{+}$ particle lives $0.80 \times 10^{-10} \mathrm{s}$ in its proper reference frame. If it is traveling at 0.90$c$ through a bubble chamber, how far will it move before it disintegrates?

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Problem 14

Extending the life of a muon A muon that lives $2.2 \times 10^{-6} \mathrm{s}$ in its proper reference frame is created $10,000$ $\mathrm{m}$ above Earth's surface. At what speed must it move to reach Earth's surface at the instant it disintegrates?

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Problem 15

Effect of light speed on the time interval for a track race Suppose the speed of light were 15 $\mathrm{m} / \mathrm{s}$ . You run a $100-\mathrm{m}$ dace in 10 $\mathrm{s}$ according to the timer's clock. How long did the race last according to your watch?

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Problem 16

Explain why an object moving past you would seem shorter in the direction of motion than when at rest
with respect to you. Draw a sketch to illustrate your reasoning.

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Problem 17

Explain why the length of an object that is oriented perpendicular to the direction of motion would be the same for all observers. Draw a sketch to illustrate your reasoning.

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Problem 18

You sit in a spaceship moving past the Earth at 0.97 c. Your arm, held straight out in front of you, measures 50 $\mathrm{cm} .$ How long is it when measured by an observer on Earth?

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Problem 19

Length of a javelin A javelin hurled by Wonder Woman moves past an Earth observer at 0.90$c .$ Its proper length is 2.7 $\mathrm{m} .$ What is its length according to the Earth observer?

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Problem 20

At what speed must a meter stick move past an observer so that it appears to be 0.50 $\mathrm{m}$ long?

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Problem 21

Changing the shape of a billboard A billboard is 10 $\mathrm{m}$ high and 15 $\mathrm{m}$ long according to a person standing in front of it. At what speed must a person in a fast car drive by parallel to the billboard's surface so that the billboard appears to be square?

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Problem 22

A classmate says that time dilation and length contraction can be remembered in a simple way if you think of a person eating a foot-long “sub” sandwich on a train (the sandwich is oriented parallel to the train’s motion). The person on the train finishes the sandwich in 20 min. You, standing on the platform, observe the person eating a shorter sandwich but for a longer time interval. Do you agree with this example? Explain your answer.

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Problem 23

Give examples of cases in which two observers record the motion of the same object to have different speeds, to have different directions, and to have different velocities. Provide reasonable values for the relevant velocities in your examples. Sketch each example and explain how each observer arrives at the value of the measured speed.

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Problem 24

Now repeat Problem 23, only this time instead of a moving object, use a light flash. Describe what speeds of light different observers should measure according to the second postulate of special relativity.

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Problem 25

Life in a slow-light-speed universe Imagine that you live in a universe where the speed of light is 50 $\mathrm{m} / \mathrm{s}$ . You sit on a train moving west at speed 20 $\mathrm{m} / \mathrm{s}$ relative to the track. Your friend moves on a train in the opposite direction at speed 15 $\mathrm{m} / \mathrm{s}$ What is the speed of his train with respect to yours?

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Problem 26

More slow-light-speed universe In the scenario described in Problem 25, you and your friend listen to music on the same radio station. What is the speed of the radio waves that your antenna is registering compared to the speed of the waves that your friend’s antenna registers if the station is 100 miles to the west of you?

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Problem 27

You are on a spaceship traveling at $0.80c$ with respect to a nearby star sending a laser beam to a spaceship following you, which is moving at $0.50c$ in the same direction. (a) What is the speed of the laser beam registered by the second ship’s personnel according to the classical addition of the velocities? (b) What is the speed of the laser beam registered by the second ship’s personnel according to the relativistic addition of the velocities? (c) What is the speed of the second ship with respect to yours according to the classical addition of the velocities? (d) What is the speed of the second ship with respect to yours according to the relativistic addition of the velocities?

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Problem 28

Your friend says that it is easy to travel faster than the speed of light; you just need to find the right observer. Give physics-based reasons for why your friend would have such an idea. Then explain whether you agree or disagree with him.

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Problem 29

Your friend argues that Einstein’s special theory of relativity says that nothing can move faster than the speed of light. (a) Give physics-based reasons for why your friend would have such an idea. (b) What examples of physical phenomena do you know of that contradict this statement? (c) Restate his idea so it is accurate in terms of physics.

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Problem 30

An electron is moving at a speed of $0.90c$. Compare its momentum as calculated using a non relativistic equation and using a relativistic equation.

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Problem 31

Explain why a relativistic expression is needed for fast-moving particles. Why can’t we use a classical expression?

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Problem 32

If you were to bring an electron from speed zero to $0.95c$ in 10 min, what force would need to be exerted on the electron? What object could possibly exert such a force?

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Problem 33

If a proton has a momentum of $3.00 \times 10^{-19} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s},$ what is its speed?

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Problem 34

Determine the ratio of an electron's total energy to rest energy when moving at the following speeds: (a) 300 $\mathrm{m} / \mathrm{s}$ , (b) $3.0 \times 10^{8} \mathrm{m} / \mathrm{s},$ (c) $3.0 \times 10^{7} \mathrm{m} / \mathrm{s},$ (d) $1.0 \times 10^{8} \mathrm{m} / \mathrm{s}$ (e) $2.0 \times 10^{8} \mathrm{m} / \mathrm{s},$ and ( $\mathrm{f} ) 2.9 \times 10^{8} \mathrm{m} / \mathrm{s} .$

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Problem 35

Solar wind To escape the gravitational pull of the Sun, a proton in the solar wind must have a speed of at least $6.2 \times 10^{5} \mathrm{m} / \mathrm{s}$ . Determine the rest energy, the kinetic energy, and the total energy of the proton.

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Problem 36

At what speed must an object move so that its total energy is 1.0$\%$ greater than its rest energy? 10$\%$ greater? Twice its rest energy?

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Problem 37

Space travel $A 50$ -kg space traveler starts at rest and accelerates at 5$g$ for 30 days. Determine the person's total energy after 30 days. What assumptions did you make?

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Problem 38

A person’s total energy is twice his rest energy when he moves at a certain speed. By what factor must his speed now increase to cause another doubling of his total energy?

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Problem 39

A proton’s energy after passing through the accelerator at Fermilab is 500 times its rest energy. Determine the proton’s speed.

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Problem 40

A rocket of mass $m$ starts at rest and accelerates to a speed of 0.90$c$ . Determine the change in energy needed for this change in speed.

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Problem 41

Determine the total energy, the rest energy, and the kinetic energy of a person with $60-\mathrm{kg}$ mass moving at speed 0.95$c .$

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Problem 42

An electron is accelerated from rest across $50,000 \mathrm{V}$ in a machine used to produce $\mathrm{X}$ -rays. Determine the electron's speed after crossing that potential difference.

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Problem 43

A particle originally moving at a speed 0.90$c$ experiences a 5.0$\%$ increase in speed. By what percent does its kinetic energy increase?

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Problem 44

An electron is accelerated from rest across a potential difference of $9.0 \times 10^{9} \mathrm{V}$ . Determine the electron's speed (a) using the nonrelativistic kinetic energy equation and (b) using
the relativistic kinetic energy equation. Which is the correct answer?

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Problem 45

$A particle of mass $m$ initially moves at speed 0.40$c .(a)$ If the particle's speed is doubled, determine the ratio of its final kinetic energy to its initial kinetic energy. (b) If the particle's kinetic energy increases by a factor of $100,$ by what factor does its speed increase?

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Problem 46

Determine the mass of an object whose rest energy equals the total yearly energy consumption of the world $\left(5 \times 10^{20} \mathrm{J}\right)$

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Problem 47

Mass equivalent of energy to separate a molecule Separating a carbon monoxide molecule $\mathrm{CO}$ into a carbon and an oxygen atom requires $1.76 \times 10^{-18} \mathrm{J}$ of energy. (a) Determine the mass equivalent of this energy. (b) Determine the fraction of the original mass of a CO molecule $4.67 \times 10^{-26} \mathrm{kg}$ that was converted to energy.

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Problem 48

Hydrogen fuel cell A hydrogen-oxygen fuel cell combines 2 $\mathrm{kg}$ of hydrogen with 16 $\mathrm{kg}$ of oxygen to form 18 $\mathrm{kg}$ of water, thus releasing $2.5 \times 10^{8} \mathrm{J}$ of energy. What fraction of the mass has been converted to energy?

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Problem 49

Mass to provide human energy needs Determine the mass that must be converted to energy during a 70 -year lifetime to continually provide electric power for a person at a rate of 1000 $\mathrm{W}$ . The production of the electric power from mass is only about 33$\%$ efficient.

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Problem 50

EST An electric utility company charges a customer about $6-7$ cents for $10^{6} \mathrm{J}$ of electrical energy. At this rate, estimate the cost of 1 $\mathrm{g}$ of mass if converted entirely to energy.

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Problem 51

Mass to produce electric energy in a nuclear power plant A nuclear power plant produces $10^{9} \mathrm{W}$ of electric power and $2 \times 10^{9} \mathrm{W}$ of waste heating. (a) At what rate must mass be converted to energy in the reactor? (b) What is the total mass converted to energy each year?

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Problem 52

BIO EST Metabolic energy Estimate the total metabolic energy you use during a day. (You can find more on metabolic rate in the reading passage in Chapter 6.) Determine the mass equivalent of this energy.

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Problem 53

Energy from the Sun (a) Determine the energy radiated by the Sun each second by its conversion of $4 \times 10^{9} \mathrm{kg}$ of mass to energy. (b) Determine the fraction of this energy intercepted by Earth, which is $1.50 \times 10^{11} \mathrm{m}$ from the Sun and has a radius of $6.38 \times 10^{6} \mathrm{m} .$

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Problem 54

Why no color change? Why don’t the colors of buildings and tree leaves change when we look at them from a flying plane? Shouldn’t the trees ahead look more bluish when you are approaching and reddish when you are receding?

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Problem 55

Change red light to green In a parallel universe the speed of light in a vacuum is 70.000 $\mathrm{m} / \mathrm{s}$ . How fast should a driver's car move so that a red light looks green?

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Problem 56

Effect of the Hubble constant on age and radius of the universe How would the estimated age of the universe change if the new accepted value of the Hubble constant became $(100$ $\mathrm{km} / \mathrm{s} / \mathrm{mpc}$ ? How would the visible radius of the universe change?

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Problem 57

Expanding faster New observations suggest that our universe does not expand at a constant rate but instead is expanding at an increasing rate. How does this finding affect the estimation of the age of the universe using Hubble’s law?

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Problem 58

Baseball Doppler shift In September of 2010 Aroldis Chapman threw what may be the fastest baseball pitch ever recorded at 105 $\mathrm{mi} / \mathrm{h}(47 \mathrm{m} / \mathrm{s}) .$ What would the observed frequency of microwaves reflected from the ball be if the source frequency of were 10.525 GHz? What would be the beat frequency between the source frequency and the observed frequency?

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Problem 59

Were you speeding? A police officer stops you in a 29 $\mathrm{m} / \mathrm{s}$ $(65 \mathrm{mi} / \mathrm{h})$ speed zone and says you were speeding. The officer's radar has source frequency 33.4 $\mathrm{GHz}$ and observed a $3900-\mathrm{Hz}$ beat frequency between the source frequency and waves reflected back to the radar from your car. Were you speeding? Explain.

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Problem 60

Boat trip A boat’s speed is 10 m/s. It makes a round trip between stations A and B and then another between stations A and C. Stations A and B are on the same side of the river 0.5 km apart. Stations A and C are on the opposite sides of the river across from each other and also 0.5 km apart. The river flows at 1.5 m/s. What time interval is the round trip between stations A and B and then between A and C?

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Problem 61

Space travel An explorer travels at speed $2.90 \times 10^{8} \mathrm{m} / \mathrm{s}$ from Earth to a planet of Alpha Centauri, a distance of 4.3 light-years as measured by an Earth observer. (a) How long does the trip laccording to an Earth observer? (b) How long does the trip last for the person on the ship?

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Problem 62

EST Extending life Suppose that the speed of light is 8.0 $\mathrm{m} / \mathrm{s}$ . You walk slowly to all of your classes during one semester while a classmate runs at a speed of 7.5 $\mathrm{m} / \mathrm{s}$ during the time you are walking. Estimate your classmate's change in age, as judged by you, and your change in age according to you during that walking time. Indicate how you chose any numbers used in your estimate.

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Problem 63

Racecar when $c$ is 100 $\mathrm{m} /$ s Suppose that the speed of light is 100 $\mathrm{m} / \mathrm{s}$ and that you are driving a racecar at speed 90 $\mathrm{m} / \mathrm{s}$ . What time interval is required for you to travel 900 $\mathrm{m}$ along a track’s straightaway (a) according to a timer on the track and (b) according to your own clock? (c) How long does the straightaway appear to you? (d) Notice that the speed at which the track moves past is your answer to part (c) divided by your answer to part (b). Does this speed agree with the speed as measured by the stationary timer?

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Problem 64

EST Cherenkov radiation is electromagnetic radiation emitted when a fast-moving particle such as a proton passes through an insulator at a speed faster than the speed of light in that insulator. The Cherenkov radiation looks like a blue glow in the shape of a cone behind the particle. The radiation is named after Soviet physicist Pavel Cherenkov, who received a Nobel Prize in 1958 for describing the radiation. Estimate the smallest speed of a proton moving in oil that will produce Cherenkov radiation behind it.

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Problem 65

A pilot and his spaceship of rest mass 1000 kg wish to travel from Earth to planet Scot ML, 30 light-years from Earth. However, the pilot wishes to be only 10 physiological years older when he reaches the planet. (a) At what constant speed must he travel? (b) What is the total energy of his spaceship and the rest energy, according to an Earth observer, while making the trip?

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Problem 66

Space travel A pilot and her spaceship have a mass of 400$\mathrm {kg}$. The pilot expects to live 50 more Earth years and wishes to travel to a star that requires 100 years to reach even if she were to travel at the speed of light. (a) Determine the average speed she must travel to reach the star during the next 50 Earth years. (b) To attain this speed, a certain mass $m$ of matter is consumed and converted to the spaceship’s kinetic energy. How much mass is needed? (Ignore the energy neededto accelerate the fuel that has not yet been consumed.)

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Problem 67

(a) A container holding 4 $\mathrm{kg}$ of water is heated from $0^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ . Determine the increase in its energy and compare this to the rest energy when at $0^{\circ} \mathrm{C}$ . (b) If the water, initially at $0^{\circ} \mathrm{C},$ is converted to ice at $0^{\circ} \mathrm{C},$ determine the ratio of its energy change to its original rest energy.

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Problem 68

Which principle can we use to determine the frequency $f_{\mathrm{D}}$ “detected” by the ball as it moved toward the source waves from the radar?
(a) The beat frequency equation
(b) The high-speed Doppler effect equation
(c) The low-speed Doppler effect equation
(d) The time dilation equation
(e) The relationship between wave speed, frequency, and wavelength

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Problem 69

Which frequency is closest to the frequency $f_{\mathrm{D}}$ detected by the ball as it moves toward the radar source waves?
(a) 10.525 $\mathrm{GHz}$
(b) $10.525 \mathrm{GHz}+2.0 \times 10^{-6} \mathrm{GHz}$
(c) $10.525 \mathrm{GHz}-2.0 \times 10^{-6} \mathrm{GHz}$
(d) $3 \times 10^{-7} \mathrm{Hz}$

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Problem 70

Which principle can we use to determine the frequency $f_{\mathrm{O}}$ detected by the radar from waves reflected from the ball?
(a) The beat frequency equation
(b) The high-speed Doppler effect equation
(c) The low-speed Doppler effect equation
(d) The time dilation equation
(e) The relationship between wave speed, frequency. and wavelength

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Problem 71

Which answer is closest to the frequency $f_{\mathrm{O}}$ detected by the radar from the waves reflected from the ball?
(a) Exactly 10.525 GHz
(b) $10.525 \mathrm{GHz}+4.0 \times 10^{-6} \mathrm{GHz}$
(c) $10.525 \mathrm{GHz}-4.0 \times 10^{-6} \mathrm{GHz}$
(d) $3 \times 10^{-7} \mathrm{Hz}$

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Problem 72

Which principle is used to determine the frequency that the radar measures of the combined source and observed waves?
(a) The beat frequency equation
(b) The high-speed Doppler effect equation
(c) The low-speed Doppler effect equation
(d) The time dilation equation
(e) The relationship between wave speed, frequency. and wavelength

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Problem 73

Which answer is closest to the frequency that the radar measures of the combined source and observed waves?
(a) $2.0 \times 10^{3} \mathrm{Hz} \quad$ (b) $4.0 \times 10^{3} \mathrm{Hz}$
(c) 10.525 $\mathrm{GHz} \quad$ (d) $3 \times 10^{-7} \mathrm{Hz}$
(e) $6 \times 10^{-7} \mathrm{Hz}$

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Problem 74

What principle would you use to estimate the distance of 3 $\mathrm{C} 273$ from Earth?
(a) The high-speed Doppler effect equation
(b) The low-speed Doppler effect equation
(c) Hubble’s law
(d) The time dilation equation
(e) The relationship between wave speed, frequency, and wavelength

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Problem 75

Which answer below is closest to the distance of 3 $\mathrm{C} 273$ from the Earth in terms of the distance of the Sun from Earth?
(a) $\approx 10^{3}$ Sun distances $\quad$ (b) $\approx 10^{6}$ Sun distances
(c) $\approx 10^{9}$ Sun distances $\quad$ (d) $\approx 10^{12}$ Sun distances
(e) $\approx 10^{14}$ Sun distances $

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Problem 76

Which answer below is closest to the power of light and other forms of radiation emitted by 3 $\mathrm{C} 273 ?$
(a) $\approx 10^{8} \mathrm{J} / \mathrm{s} \quad(\mathrm{b})=10^{18} \mathrm{J} / \mathrm{s}$
(c) $\approx 10^{25} \mathrm{Js} \quad(\mathrm{d}) \approx 10^{32} \mathrm{J} / \mathrm{s}$
(e) $\approx 10^{40} \mathrm{J} / \mathrm{s}$

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Problem 77

Which answer below is closest to the mass of 3 $\mathrm{C} 273$ that is converted to light and other forms of radiation each second?
By comparison, the mass of Earth is $6 \times 10^{24} \mathrm{kg}$ .
(a) $\approx 10^{11} \mathrm{kg} / \mathrm{s} \quad$ (b) $\approx 10^{15} \mathrm{kg} / \mathrm{s}$
(c) $\approx 10^{19} \mathrm{kg} / \mathrm{s} \quad(\mathrm{d}) \approx 10^{23} \mathrm{kg} / \mathrm{s}$
$(e) \approx 10^{29} \mathrm{kg} / \mathrm{s}$

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Problem 78

What is the speed of light emitted by 3 $\mathrm{C} 273$ as detected by an observer on 3 $\mathrm{C} 273 ?$
(a) 1.15$c \quad$ (b) $c$
(c) 0.85$c \quad$ (d) None of these is correct.

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Problem 79

If 3 $\mathrm{C} 273$ is moving away from Earth at $0.16 c,$ what speed below is closest to the light speed we on Earth detect coming from 3 $\mathrm{C} 273 ?$
(a) 1.15$c \quad$ (b) $c$
(c) 0.85$c \quad$ (d) None of these is correct.

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