Solved: Calculate the kinetic energy and momentum of a

A spaceship passes you at a speed of 0.850c. You measure its length to be 44.2 m. How long would it be when at rest?

A certain type of elementary particle travels at a speed of At this speed, the average lifetime is measured to be What is the particles lifetime at rest?

You travel to a star 135 light-years from Earth at a speed of What do you measure this distance to be?

What is the speed of a pion if its average lifetime is measured to be At rest, its average lifetime

In an Earth reference frame, a star is 49 light-years away. How fast would you have to travel so that to you the distance would be only 35 light-years?

(II) At what speed \(v\) will the length of a 1.00-m stick look 10.0% shorter (90.0 cm)? Equation Transcription: Text Transcription: v

At what speed do the relativistic formulas for (a) length and (b) time intervals differ from classical values by 1.00%? (This is a reasonable way to estimate when to use relativistic calculations rather than classical.)

You decide to travel to a star 62 light-years from Earth at a speed that tells you the distance is only 25 lightyears. How many years would it take you to make the trip?

A friend speeds by you in her spacecraft at a speed of 0.720c. It is measured in your frame to be 4.80 m long and 1.35 m high. (a) What will be its length and height at rest? (b) How many seconds elapsed on your friends watch when 20.0 s passed on yours? (c) How fast did you appear to be traveling according to your friend? (d) How many seconds elapsed on your watch when she saw 20.0 s pass on hers?

A star is 21.6 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers: (a) on Earth, (b) on the spacecraft? (c) What is the distance traveled according to observers on the spacecraft? (d) What will the spacecraft occupants compute their speed to be from the results of (b) and (c)?

A fictional news report stated that starship Enterprise had just returned from a 5-year voyage while traveling at 0.70c. (a) If the report meant 5.0 years of Earth time, how much time elapsed on the ship? (b) If the report meant 5.0 years of ship time, how much time passed on Earth?

A box at rest has the shape of a cube 2.6 m on a side. This box is loaded onto the flat floor of a spaceship and the spaceship then flies past us with a horizontal speed of 0.80c. What is the volume of the box as we observe it?

Escape velocity from the Earth is What would be the percent decrease in length of a 68.2-m-long spacecraft traveling at that speed as seen from Earth?

An unstable particle produced in an accelerator experiment travels at constant velocity, covering 1.00 m in 3.40 ns in the lab frame before changing (decaying) into other particles. In the rest frame of the particle, determine (a) how long it lived before decaying, (b) how far it moved before decaying

How fast must a pion be moving on average to travel 32 m before it decays? The average lifetime, at rest, is 2.6 * 108 s.

What is the momentum of a proton traveling at v = 0.68c?

(a) A particle travels at By what percentage will a calculation of its momentum be wrong if you use the classical formula?

A particle of mass m travels at a speed At what speed will its momentum be doubled?

An unstable particle is at rest and suddenly decays into two fragments. No external forces act on the particle or its fragments. One of the fragments has a speed of 0.60c and a mass of while the other has a mass of What is the speed of the less massive fragment?

What is the percent change in momentum of a proton that accelerates from (a) 0.45c to 0.85c, (b) 0.85c to 0.98c?

Calculate the rest energy of an electron in joules and in 1 MeV = 1.60 * 1013 JB

When a uranium nucleus at rest breaks apart in the process known as fission in a nuclear reactor, the resulting fragments have a total kinetic energy of about 200 MeV. How much mass was lost in the process?

The total annual energy consumption in the United States is about How much mass would have to be converted to energy to fuel this need?

Calculate the mass of a proton in MeVc2

A certain chemical reaction requires of energy input for it to go. What is the increase in mass of the products over the reactants?

Calculate the kinetic energy and momentum of a proton traveling 2.90 * 108 ms.

What is the momentum of a 950-MeV proton (that is, its kinetic energy is 950 MeV)?

What is the speed of an electron whose kinetic energy is 1.12 MeV?

(a) How much work is required to accelerate a proton from rest up to a speed of 0.985c? (b) What would be the momentum of this proton?

At what speed will an objects kinetic energy be 33% of its rest energy?

(II) Determine the speed and the momentum of an electron \(\left(m=9.11 \times 10^{-31} \mathrm{~kg}\right)\) whose \(\mathrm{KE}\) equals its rest energy. Equation Transcription: ????? Text Transcription: (m = 9.11 x 10^-31 kg) KE

(II) A proton is traveling in an accelerator with a speed of \(1.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\). By what factor does the proton's kinetic energy increase if its speed is doubled? Equation Transcription: Text Transcription: 1.0 x 10^8 m/s

How much energy can be obtained from conversion of 1.0 gram of mass? How much mass could this energy raise to a height of 1.0 km above the Earths surface?

(II) To accelerate a particle of mass \(m\) from rest to speed 0.90c requires work \(W_{1}\). To accelerate the particle from speed 0.90c to 0.99c requires work \(W_{2}\). Determine the ratio \(W_{2} / W_{1}\). Equation Transcription: Text Transcription: m W_1 W_2 W_2/W_1

(II) Suppose there was a process by which two photons, each with momentum 0.65 MeV/c, could collide and make a single particle. What is the maximum mass that the particle could possess?

What is the speed of a proton accelerated by a potential difference of 165 MV?

What is the speed of an electron after being accelerated from rest by 31,000 V?

(II) The kinetic energy of a particle is 45 MeV. If the momentum is 121 MeV/c, what is the particle’s mass?

(II) Calculate the speed of a proton \(\left(m=1.67 \times 10^{-27} \mathrm{~kg}\right)\) whose kinetic energy is exactly half (a) its total energy, (b) its rest energy. Equation Transcription: Text Transcription: (m = 1.67 x 10^-27 kg)

Calculate the kinetic energy and momentum of a proton \(\left(m=1.67 \times 10^{-27} \mathrm{~kg}\right)\) traveling \(8.65 \times 10^{7} \mathrm{~m} / \mathrm{s}\). By what percentages would your calculations have been in error if you had used classical formulas? Equation Transcription: Text Transcription: (m = 1.67 x 10^-27 kg) 8.65 x 10^7 m/s

Suppose a spacecraft of mass 17,000 kg is accelerated to 0.15c. (a) How much kinetic energy would it have? (b) If you used the classical formula for kinetic energy, by what percentage would you be in error?

(II) A negative muon traveling at 53 % the speed of light collides head-on with a positive muon traveling at 65 % the speed of light. The two muons (each of mass \(105.7 \mathrm{MeV} / \mathrm{c}^{2}\)) annihilate, and produce how much electromagnetic energy? Equation Transcription: Text Transcription: 105.7 MeV/c^2

(II) Two identical particles of mass \(m\) approach each other at equal and opposite speeds, \(v\). The collision is completely inelastic and results in a single particle at rest. What is the mass of the new particle? How much energy was lost in the collision? How much kinetic energy was lost in this collision? Equation Transcription: Text Transcription: m v

(III) The americium nucleus, \({ }_{95}^{241} \mathrm{Am}\), decays to a neptunium nucleus, \({ }_{93}^{237} \mathrm{~Np}\), by emitting an alpha particle of mass 4.00260 u and kinetic energy 5.5 MeV.. Estimate the mass of the neptunium nucleus, ignoring its recoil, given that the americium mass is 241.05682 u. Equation Transcription: Text Transcription: { }_{95}^{241} {Am} { }_{93}^{237} {Np}

(III) Show that the kinetic energy KE of a particle of mass \(m\) is related to its momentum \(p\) by the equation \(p=\sqrt{\mathrm{KE}^{2}+2 \mathrm{KE} m c^{2}} / c.\) Equation Transcription: Text Transcription: m p p=sqrt{{KE}^{2} + 2 {KE} m c^{2}} / c.

(III) What magnetic field \(B\) is needed to keep 998-GeV protons revolving in a circle of radius 1.0 km? Use the relativistic mass. The proton's "rest mass" is \(0.938 \mathrm{GeV} / c^{2}\). \(\left(1 \mathrm{GeV}=10^{9} \mathrm{eV} .\right)\) [Hint: In relativity, \(m_{\mathrm{rel}} v^{2} / r=q v B\) is still valid in a magnetic field, where \(m_{\mathrm{rel}}=\gamma_{m .}\) ] Equation Transcription: Text Transcription: B 0.938 GeV/c^2 (1 GeV = 10^9 eV) m_{rel}v^2/r = qvB m_{rel} = gamma m

A person on a rocket traveling at 0.40c (with respect to the Earth) observes a meteor come from behind and pass her at a speed she measures as 0.40c. How fast is the meteor moving with respect to the Earth?

Two spaceships leave Earth in opposite directions, each with a speed of 0.60c with respect to Earth. (a) What is the velocity of spaceship 1 relative to spaceship 2? (b) What is the velocity of spaceship 2 relative to spaceship 1?

A spaceship leaves Earth traveling at 0.65c. A second spaceship leaves the first at a speed of 0.82c with respect to the first. Calculate the speed of the second ship with respect to Earth if it is fired (a) in the same direction the first spaceship is already moving, (b) directly backward toward Earth.

An observer on Earth sees an alien vessel approach at a speed of 0.60c. The fictional starship Enterprise comes to the rescue (Fig. 26–13), overtaking the aliens while moving directly toward Earth at a speed of 0.90c relative to Earth. What is the relative speed of one vessel as seen by the other?

(II) A spaceship in distress sends out two escape pods in opposite directions. One travels at a speed \(v_{1}=+0.70 c\) in one direction, and the other travels at a speed \(v_{2}=-0.80 c\) in the other direction, as observed from the spaceship. What speed does the first escape pod measure for the second escape pod? Equation Transcription: Text Transcription: v_1 = + 0.70 c v_2 = - 0.80 c

Rocket A passes Earth at a speed of 0.65c. At the same time, rocket B passes Earth moving 0.95c relative to Earth in the same direction as A. How fast is B moving relative to A when it passes A?

Your spaceship, traveling at 0.90c, needs to launch a probe out the forward hatch so that its speed relative to the planet that you are approaching is 0.95c. With what speed must it leave your ship?

What is the speed of a particle when its kinetic energy equals its rest energy? Does the mass of the particle affect the result?

The nearest star to Earth is Proxima Centauri, 4.3 lightyears away. (a) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 4.9 years, as measured by travelers on the spacecraft? (b) How long does the trip take according to Earth observers?

According to the special theory of relativity, the factor \(\gamma\) that determines the length contraction and the time dilation is given by \(\gamma=1 / \sqrt{1-v^{2} / c^{2}}\). Determine the numerical values of \(\gamma\) for an object moving at speed \(v=0.01 c, 0.05 c, 0.10 c, 0.20 c, 0.30 c, 0.40 c, 0.50 c, 0.60 c, 0.70 c, 0.80 c, 0.90 c, 0.95 c\), and \(0.99 c\). Make a graph of \(\gamma\) versus \(v\). Equation Transcription: Text Transcription: gamma gamma=1 / sqrt{1-v^{2} / c^{2} gamma v = 0.01 c, 0.05 c, 0.10 c, 0.20 c, 0.30 c, 0.40 c, 0.50 c, 0.60 c, 0.70 c, 0.80 c, 0.90 c, 0.95 c 0.99 c gamma v

A healthy astronaut’s heart rate is 60 beats/min. Flight doctors on Earth can monitor an astronaut’s vital signs remotely while in flight. How fast would an astronaut be flying away from Earth if the doctor measured her having a heart rate of 25 beats/min?

(a) What is the speed \(v\) of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference \(c - v\). Such speeds are reached in the Stan- ford Linear Accelerator, SLAC. (b) If the electrons travel in the lab through a tube 3.0 km long (as at SLAC), how long is this tube in the electrons’ reference frame? [Hint: Use the binomial expansion.] Equation Transcription: Text Transcription: v c - v

What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 276.)

How many grams of matter would have to be totally destroyed to run a 75-W lightbulb for 1.0 year?

A free neutron can decay into a proton, an electron, and a neutrino. Assume the neutrinos mass is zero; the other masses can be found in the Table inside the front cover. Determine the total kinetic energy shared among the three particles when a neutron decays at rest.

An electron \(\left(m=9.11 \times 10^{-31} \mathrm{~kg}\right)\) is accelerated from rest to speed \(v\) by a conservative force. In this process, its potential energy decreases by \(6.20 \times 10^{-14} \mathrm{~J}\). Determine the electron's speed, \(v\). Equation Transcription: Text Transcription: (m = 9.11 x 10^-31 kg) v 6.20 x 10^14 J v

The Sun radiates energy at a rate of about \(4 \times 10^{26} \mathrm{~W}\). (a) At what rate is the Sun's mass decreasing? (b) How long does it take for the Sun to lose a mass equal to that of Earth? (c) Estimate how long the Sun could last if it radiated constantly at this rate. Equation Transcription: Text Transcription: 4 x 10^26 W

How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 \mathrmu, and 4.00260 u. (This energy difference is called the total binding energy of the \({ }_{2}^{4} \mathrm{He}\) nucleus.) Equation Transcription: Text Transcription: { }_{2}^{4} He

Show analytically that a particle with momentum \(p\) and energy \(E\) has a speed given by \(v=\frac{p c^{2}}{E}=\frac{p c}{\sqrt{m^{2} c^{2}+p^{2}}}\) Equation Transcription: Text Transcription: p E v = frac{p c^{2}}{E} = frac{p c}{\sqrt{m^{2} c^{2}+p^{2}

Two protons, each having a speed of 0.990c in the laboratory, are moving toward each other. Determine (a) the momentum of each proton in the laboratory, (b) the total momentum of the two protons in the laboratory, and (c) the momentum of one proton as seen by the other proton

When two moles of hydrogen molecules \(\left(\mathrm{H}_{2}\right)\) and one mole of oxygen molecules \(\left(\mathrm{O}_{2}\right)\) react to form two moles of water \(\left(\mathrm{H}_{2} \mathrm{O}\right)\), the energy released is 484 kJ. How much does the mass decrease in this reaction? What \% of the total original mass is this? Equation Transcription: Text Transcription: (H_2) (O_2) (H_2{O}}

The fictional starship Enterprise obtains its power by combining matter and antimatter, achieving complete conversion of mass into energy. If the mass of the Enterprise is approximately \(6 \times 10^{9} \mathrm{~kg}\), how much mass must be converted into kinetic energy to accelerate it from rest to one-tenth the speed of light? Equation Transcription Text Transcription: 6 x 10^9 kg

Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

A spaceship and its occupants have a total mass of 160,000 kg. The occupants would like to travel to a star that is 35 light-years away at a speed of 0.70c. To accelerate, the engine of the spaceship changes mass directly to energy. (a) Estimate how much mass will be converted to energy to accelerate the spaceship to this speed. (b) Assuming the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70c, determine how long the trip will take according to the astronauts on board.

In a nuclear reaction two identical particles are created, traveling in opposite directions. If the speed of each particle is 0.82c, relative to the laboratory frame of reference, what is one particles speed relative to the other particle?

A 36,000-kg spaceship is to travel to the vicinity of a star 6.6 light-years from Earth. Passengers on the ship want the (one-way) trip to take no more than 1.0 year. How much work must be done on the spaceship to bring it to the speed necessary for this trip?

Suppose a 14,500-kg spaceship left Earth at a speed of 0.90c. What is the spaceship's kinetic energy? Compare with the total U.S. annual energy consumption (about \(10^{20}J\)). Equation Transcription: Text Transcription: 10^20 J

A pi meson of mass \(m_{\pi}\) decays at rest into a muon (mass \(m_{\mu}\)) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is \(\mathrm{KE}_{\mu}=\left(m_{\pi}-m_{\mu}\right)^{2} c^{2} /\left(2 m_{\pi}\right)\). Equation Transcription: Text Transcription: m_pi m_mu KE_mu = (m_pi - m_mu)^2 c^2/(2m_pi)

An astronaut on a spaceship traveling at 0.75c relative to Earth measures his ship to be 23 m long. On the ship, he eats his lunch in 28 min. (a) What length is the spaceship according to observers on Earth? (b) How long does the astronauts lunch take to eat according to observers on Earth?

Astronomers measure the distance to a particular star to be 6.0 light-years (1 ly = distance light travels in 1 year). A spaceship travels from Earth to the vicinity of this star at steady speed, arriving in 3.50 years as measured by clocks on the spaceship. (a) How long does the trip take as measured by clocks in Earth’s reference frame? (b) What distance does the spaceship travel as measured in its own reference frame?

An electron is accelerated so that its kinetic energy is greater than its rest energy \(mc^{2}\) by a factor of (a) 5.00, (b) 999. What is the speed of the electron in each case? Equation Transcription: Text Transcription: mc^2

You are traveling in a spaceship at a speed of 0.70c away from Earth. You send a laser beam toward the Earth traveling at velocity c relative to you. What do observers on the Earth measure for the speed of the laser beam?

A farm boy studying physics believes that he can fit a 13.0-m-long pole into a 10.0-m-long barn if he runs fast enough, carrying the pole. Can he do it? Explain in detail. How does this fit with the idea that when he is running the barn looks even shorter than 10.0 m?

An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 2.0 years has elapsed? [Hint: Use the binomial expansion, Appendix A.]

An airplane travels 1300 km/h around the Earth in a circle of radius essentially equal to that of the Earth, returning to the same place. Using special relativity, estimate the difference in time to make the trip as seen by Earth and by airplane observers. [Hint: Use the binomial expansion, Appendix A.]

Problem 1COQ A rocket is headed away from Earth at a speed of 0.80c. The rocket fires a small payload at a speed of 0.70c (relative to the rocket) aimed away from Earth. How fast is the payload moving relative to Earth? (a) 1.50c; (b) a little less than 1.50c; (c) a little over c; (d) a little under c; (e) 0.75c.

Problem 1MCQ The fictional rocket ship Adventure is measured to be 50 m long by the ship’s captain inside the rocket. When the rocket moves past a space dock at 0.5c, space-dock personnel measure the rocket ship to be 43.3 m long. What is its proper length? (a) 50m. (b) 43.3 m. (c) 93.3 m. (d) 13.3 m.

Problem 1P (I) A spaceship passes you at a speed of 0.850c. You measure its length to be 44.2 m. How long would it be when at rest?

Problem 1Q You are in a windowless car in an exceptionally smooth train moving at constant velocity. Is there any physical experiment you can do in the train car to determine whether you are moving? Explain.

Problem 2MCQ As rocket ship Adventure (MisConceptual Question 1) passes by the space dock, the ship’s captain flashes a flashlight at 1.00-s intervals as measured by space-dock personnel. How often does the flashlight flash relative to the captain? (a) Every 1.15 s. (b) Every 1.00 s. (c) Every 0.87 s. (d)We need to know the distance between the ship and the space dock.

Problem 2P (I) A certain type of elementary particle travels at a speed Of 2.70 X 108 m/s. At this speed, the average lifetime is measured to be 4.76 X 10-6 s. What is the particle’s lifetime at rest?

Problem 2Q You might have had the experience of being at a red light when, out of the corner of your eye, you see the car beside you creep forward. Instinctively you stomp on the brake pedal, thinking that you are rolling backward. What does this say about absolute and relative motion?

Problem 3MCQ For the flashing of the flashlight in MisConceptual Question 2, what time interval is the proper time interval? (a) 1.15 s. (b) 1.00 s. (c) 0.87 s. (d) 0.13 s.

Problem 3P (II) You travel to a star 135 light-years from Earth at a speed of 2.90 X 108 m/s. What do you measure this distance to be?

Problem 3Q A worker stands on top of a railroad car moving at constant velocity and throws a heavy ball straight up (from his point of view). Ignoring air resistance, explain whether the ball will land back in his hand or behind him.

Problem 4MCQ The rocket ship of MisConceptual Question 1 travels to a star many light-years away, then turns around and returns at the same speed. When it returns to the space dock, who would have aged less: the space-dock personnel or ship’s captain? (a) The space-dock personnel. (b) The ship’s captain. (c) Both the same amount, because both sets of people were moving relative to each other. (d)We need to know how far away the star is.

Using Example as a guide, show that for objects that move slowly in comparison to \(c\) , the length contraction formula is roughly \(l \approx l_{0}\left(1-\frac{1}{2} v^{2} / c^{2}\right)\). Use this approximation to find the "length shortening" \(\Delta l=l_{0}-l\) of the train in Example if the train travels at \(100 \mathrm{~km} / \mathrm{h}\) (rather than \(0.92 c\) ). Equation Transcription: Text Transcription: c l \approx l0(1-12v2/c2) \Delta l=l0-l 100 km/h 0.92c

Problem 4P 4. (II) What is the speed of a pion if its average lifetime is measured to be 4.40 X 10-8 s ? At rest, its average lifetime is 2.60 X 10-8 s.

Problem 4Q Does the Earth really go around the Sun? Or is it also valid to say that the Sun goes around the Earth? Discuss in view of the relativity principle (that there is no best reference frame). Explain. See Section 5–8.

Problem 4SL In Example 26–5, the spaceship is moving at 0.90c in the horizontal direction relative to an observer on the Earth. If instead the spaceship moved at 0.90c directed at 30° above the horizontal, what would be the painting’s dimensions as seen by the observer on Earth?

Problem 5MCQ An Earth observer notes that clocks on a passing spacecraft run slowly. The person on the spacecraft (a) agrees her clocks move slower than those on Earth. (b) feels normal, and her heartbeat and eating habits are normal. (c) observes that Earth clocks are moving slowly. (d) The real time is in between the times measured by the two observers. (e) Both (a) and (b). (f) Both (b) and (c).

Problem 5P (II) In an Earth reference frame, a star is 49 light-years away. How fast would you have to travel so that to you the distance would be only 35 light-years?

Problem 5Q If you were on a spaceship traveling at 0.6c away from a star, at what speed would the starlight pass you?

Problem 5SL Protons from outer space crash into the Earth’s atmosphere at a high rate. These protons create particles that eventually decay into other particles called muons. This cosmic debris travels through the atmosphere. Every second, dozens of muons pass through your body. If a muon is created 30 km above the Earth’s surface, what minimum speed and kinetic energy must the muon have in order to hit Earth’s surface? A muon’s mean lifetime (at rest) is 2.30 ?s and its mass is 105.7 MeV/c2.

Problem 6MCQ Spaceships A and B are traveling directly toward each other at a speed 0.5c relative to the Earth, and each has a headlight aimed toward the other ship. What value do technicians on ship B get by measuring the speed of the light emitted by ship A’s headlight? (a) 0.5c. (b) 0.75c. (c) 1.0c. (d) 1.5c.

Problem 6P (II) Calculate the kinetic energy and momentum of a proton traveling 2.90 X 108 m/s.

Problem 6Q The time dilation effect is sometimes expressed as “moving clocks run slowly.” Actually, this effect has nothing to do with motion affecting the functioning of clocks. What then does it deal with?

Problem 6SL As a rough rule, anything traveling faster than about 0.1c is called relativistic—that is, special relativity is a significant effect. Determine the speed of an electron in a hydrogen atom (radius 0.53 X 10-10 m) and state whether or not it is relativistic. (Treat the electron as though it were in a circular orbit around the proton. See hint for Problem 46.)

Problem 7MCQ Relativistic formulas for time dilation, length contraction, and mass are valid (a) only for speeds less than 0.10c. (b) only for speeds greater than 0.10c. (c) only for speeds very close to c. (d) for all speeds.

Problem 7P (II) At what speed do the relativistic formulas for (a) length and (b) time intervals differ from classical values by 1.00%? (This is a reasonable way to estimate when to use relativistic calculations rather than classical.)

Problem 7Q Does time dilation mean that time actually passes more slowly in moving reference frames or that it only seems to pass more slowly?

Problem 8MCQ Which of the following will two observers in inertial reference frames always agree on? (Choose all that apply.) (a) The time an event occurred. (b) The distance between two events. (c) The time interval between the occurence of two events. (d) The speed of light. (e) The validity of the laws of physics. (f) The simultaneity of two events.

Problem 8P (II) You decide to travel to a star 62 light-years from Earth at a speed that tells you the distance is only 25 light-years. How many years would it take you to make the trip?

Problem 8Q A young-looking woman astronaut has just arrived home from a long trip. She rushes up to an old gray-haired man and in the ensuing conversation refers to him as her son. How might this be possible?

Problem 9MCQ Two observers in different inertial reference frames moving relative to each other at nearly the speed of light see the same two events but, using precise equipment, record different time intervals between the two events. Which of the following is true of their measurements? (a) One observer is incorrect, but it is impossible to tell which one. (b) One observer is incorrect, and it is possible to tell which one. (c) Both observers are incorrect. (d) Both observers are correct.

Problem 9P (II) A friend speeds by you in her spacecraft at a speed of 0.720c. It is measured in your frame to be 4.80 m long and 1.35 m high. (a) What will be its length and height at rest? (b) How many seconds elapsed on your friend’s watch when 20.0 s passed on yours? (c) How fast did you appear to be traveling according to your friend? (d) How many seconds elapsed on your watch when she saw 20.0 s pass on hers?

Problem 9Q If you were traveling away from Earth at speed 0.6c, would you notice a change in your heartbeat? Would your mass, height, or waistline change? What would observers on Earth using telescopes say about you?

Problem 10MCQ You are in a rocket ship going faster and faster. As your speed increases and your velocity gets closer to the speed of light, which of the following do you observe in your frame of reference? (a) Your mass increases. (b) Your length shortens in the direction of motion. (c) Your wristwatch slows down. (d) All of the above. (e) None of the above.

Problem 10P (II) A star is 21.6 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers: (a) on Earth, (b) on the spacecraft? (c) What is the distance traveled according to observers on the spacecraft? (d) What will the spacecraft occupants compute their speed to be from the results of (b) and (c)?

Problem 10Q Do time dilation and length contraction occur at ordinary speeds, say 90 km/h?

Problem 11MCQ You are in a spaceship with no windows, radios, or other means to check outside. How could you determine whether your spaceship is at rest or moving at constant velocity? (a) By determining the apparent velocity of light in the spaceship. (b) By checking your precision watch. If it’s running slow, then the ship is moving. (c) By measuring the lengths of objects in the spaceship. If they are shortened, then the ship is moving. (d) Give up, because you can’t tell.

(II) A fictional news report stated that starship Enterprise had just returned from a 5-year voyage while traveling at 0.70c. (a) If the report meant 5.0 years of Earth time, how much time elapsed on the ship? (b) If the report meant 5.0 years of ship time, how much time passed on Earth?

Problem 11Q Suppose the speed of light were infinite. What would happen to the relativistic predictions of length contraction and time dilation?

The period of a pendulum attached in a spaceship is 2 s while the spaceship is parked on Earth. What is the period to an observer on Earth when the spaceship moves at 0.6c with respect to the Earth? (a) Less than 2 s. (b)More than 2 s. (c) 2 s.

Problem 12P (II) A box at rest has the shape of a cube 2.6 m on a side. This box is loaded onto the flat floor of a spaceship and the spaceship then flies past us with a horizontal speed of 0.80c. What is the volume of the box as we observe it?

Problem 12Q Explain how the length contraction and time dilation formulas might be used to indicate that c is the limiting speed in the universe.

Problem 13MCQ Two spaceships, each moving at a speed 0.75c relative to the Earth, are headed directly toward each other. What do occupants of one ship measure the speed of other ship to be? (a) 0.96c. (b) 1.0c. (c) 1.5c. (d) 1.75c. (e) 0.75c.

Problem 13P (III) Escape velocity from the Earth is 11.2 km/s. What would be the percent decrease in length of a 68.2-m-long spacecraft traveling at that speed as seen from Earth?

Problem 13Q Discuss how our everyday lives would be different if the speed of light were only 25 m/s.

Problem 14P (III) An unstable particle produced in an accelerator experiment travels at constant velocity, covering 1.00 m in 3.40 ns in the lab frame before changing (“decaying”) into other particles. In the rest frame of the particle, determine (a) how long it lived before decaying, (b) how far it moved before decaying.

Problem 14Q The drawing at the start of this Chapter shows the street as seen by Mr Tompkins, for whom the speed of light is c = 20 mi/h. What does Mr Tompkins look like to the people standing on the street (Fig. 26–12)? Explain.

Problem 15P (III) How fast must a pion be moving on average to travel 32 m before it decays? The average lifetime, at rest, is 2.6 X 10-8 s.

Problem 15Q An electron is limited to travel at speeds less than c. Does this put an upper limit on the momentum of an electron? If so, what is this upper limit? If not, explain.

Problem 16P (I) What is the momentum of a proton traveling at v =0.68c?

Problem 16Q Can a particle of nonzero mass attain the speed of light? Explain.

Problem 17P (II) (a) A particle travels at v =0.15c. By what percentage will a calculation of its momentum be wrong if you use the classical formula? (b) Repeat for v =0.75c.

Does the equation \(E=m c^{2}\) conflict with the conservation of energy principle? Explain. Equation Transcription: Text Transcription: E = mc^2

Problem 18P (II) A particle of mass m travels at a speed v = 0.22c. At what speed will its momentum be doubled?

If mass is a form of energy, does this mean that a spring has more mass when compressed than when relaxed? Explain.

Problem 19P (II) An unstable particle is at rest and suddenly decays into two fragments. No external forces act on the particle or its fragments. One of the fragments has a speed of 0.60c and a mass of 6.68 X 10-27 kg, while the other has a mass of 1.67 X 10-27 kg. What is the speed of the less massive fragment?

Problem 19Q It is not correct to say that “matter can neither be created nor destroyed.” What must we say instead?

Problem 20P (II) What is the percent change in momentum of a proton that accelerates from (a) 0.45c to 0.85c, (b) 0.85c to 0.98c?

Problem 20Q Is our intuitive notion that velocities simply add, as in Section 3–8, completely wrong?

Problem 21P (I) Calculate the rest energy of an electron in joules and in MeV (1 MeV = 1.60 X 10-13 J)

Problem 22P (I) When a uranium nucleus at rest breaks apart in the process known as fission in a nuclear reactor, the resulting fragments have a total kinetic energy of about 200 MeV. How much mass was lost in the process?

Problem 23P (I) The total annual energy consumption in the United States is about 1X1020 J. How much mass would have to be converted to energy to fuel this need?

(I) Calculate the mass of a proton \((1.67 \times 10^{-27}\ kg)\) in \(MeV/c^2\).

Problem 25P (I) A certain chemical reaction requires 4.82 X 104 J of energy input for it to go. What is the increase in mass of the products over the reactants?

Problem 26EA What is the muon’s mean lifetime (Example 26–1) if it is traveling at EXAMPLE 26-1 Lifetime of a moving muon. What will be the mean lifetime of a muon as measured in the laboratory if it is traveling at = 0.60c = 1.80 x 108 m/s with respect to the laboratory? A muon’s mean lifetime at rest is 2.20 = 2.20 x 10-6 s. How far does a muon travel in the laboratory, on average, before decaying?

A certain atomic clock keeps precise time on Earth. If the clock is taken on a spaceship traveling at a speed \(v=0.60 c\), does this clock now run slow according to the people (a) on the spaceship, (b) on Earth? Equation Transcription: Text Transcription: v=0.60c

What is the length of the tunnel as measured by observers on the train in Example 26–6?

Problem 26ED For 1% accuracy, does an electron with KE = 100eV need to be treated relativistically? [Hint: The mass of an electron is 0.511 MeV.]

Problem 6EE Use Eq. 26–10 to calculate the speed of rocket 2 in Fig. 26–11 relative to Earth if it was shot from rocket 1 at a speed Assume rocket 1 had a speed

Problem 6EF Return to the Chapter-Opening Question, page 744, and answer it again now. Try to explain why you may have answered differently the first time. CHAPTER-OPENING QUESTION---Guess now! A rocket is headed away from Earth at a speed of 0.80c. The rocket fires a small payload at a speed of 0.70c (relative to the rocket) aimed away from Earth. How fast is the payload moving relative to Earth? 1. 1.50c; 2. A little less than 1.50c; 3. A little over c; 4. A little under c; 5. 0.75c.

Problem 27P (II) What is the momentum of a 950-MeV proton (that is, its kinetic energy is 950 MeV)?

Problem 28P (II) What is the speed of an electron whose kinetic energy is 1.12 MeV?

Problem 29P (II) (a) How much work is required to accelerate a proton from rest up to a speed of 0.985c? (b) What would be the momentum of this proton?

Problem 30P (II) At what speed will an object’s kinetic energy be 33% of its rest energy?

Problem 31P (II) Determine the speed and the momentum of an electron (m =9.11 X 10-31 kg) whose KE equals its rest energy.

Problem 32P (II) A proton is traveling in an accelerator with a speed of 1.0 X 108 m/s. By what factor does the proton’s kinetic energy increase if its speed is doubled?

Problem 33P (II) How much energy can be obtained from conversion of 1.0 gram of mass? How much mass could this energy raise to a height of 1.0 km above the Earth’s surface?

Problem 34P (II) To accelerate a particle of mass m from rest to speed 0.90c requires work W1. To accelerate the particle from speed 0.90c to 0.99c requires work W2 Determine the ratio W2/W1.

Problem 35P (II) Suppose there was a process by which two photons, each with momentum 0.65 MeV/c, could collide and make a single particle. What is the maximum mass that the particle could possess?

Problem 36P (II) What is the speed of a proton accelerated by a potential difference of 165 MV?

Problem 37P (II) What is the speed of an electron after being accelerated from rest by 31,000 V?

Problem 38P (II) The kinetic energy of a particle is 45 MeV. If the Momentum is 121 Me V/c, what is the particle’s mass?

Problem 39P (II) Calculate the speed of a proton (m =1.67 X 10-27 kg) whose kinetic energy is exactly half (a) its total energy, (b) its rest energy.

Problem 40P (II) Calculate the kinetic energy and momentum of a Proton (m =1.67 X 10-27 kg) traveling 8.65 X 107 m/s. By what percentages would your calculations have been in error if you had used classical formulas?

Problem 41P (II) Suppose a spacecraft of mass 17,000 kg is accelerated to 0.15c. (a) How much kinetic energy would it have? (b) If you used the classical formula for kinetic energy, by what percentage would you be in error?

Problem 42P (II) A negative muon traveling at 53% the speed of light collides head on with a positive muon traveling at 65% the speed of light. The two muons (each of mass 105 .7 Me V/c2 ) annihilate, and produce how much electromagnetic energy?

Problem 43P (II) Two identical particles of mass m approach each other at equal and opposite speeds, v. The collision is completely inelastic and results in a single particle at rest. What is the mass of the new particle? How much energy was lost in the collision? How much kinetic energy was lost in this collision?

\(\text { (III) }\)The americium nucleus, \({ }_{95}^{241} \mathrm{Am}\), decays to a neptunium nucleus, \({ }_{93}^{237} \mathrm{~Np}\), by emitting an alpha particle of mass \(4.00260 u\) and kinetic energy \(5.5 \mathrm{MeV}\). Estimate the mass of the neptunium nucleus, ignoring its recoil, given that the americium mass is \(241.05682 u\). Equation Transcription: Text Transcription: (III) 95 241Am 93 237Np 4.00260u 5.5MeV 241.05682u

(III) Show that the kinetic energy ke of a particle of mass m is related to its momentum p by the equation \(p=\sqrt{\mathrm{KE}^{2}+2 \mathrm{KE}\ m c^{2}} / c\).

Problem 46P (III) What magnetic field B is needed to keep 998-GeV protons revolving in a circle of radius 1.0 km? Use the relativistic mass. The proton’s “rest mass” is 0.938 Ge V/c2. (1 Ge V = 109 eV.) [Hint: In relativity, M rel V2/r =qvB is still valid in a magnetic field, where M ret =? m.]

Problem 47P (I) A person on a rocket traveling at 0.40c (with respect to the Earth) observes a meteor come from behind and pass her at a speed she measures as 0.40c. How fast is the meteor moving with respect to the Earth?

Problem 48P (II) Two spaceships leave Earth in opposite directions, each with a speed of 0.60c with respect to Earth. (a) What is the velocity of spaceship 1 relative to spaceship 2? (b) What is the velocity of spaceship 2 relative to spaceship 1?

Problem 49P (II) A spaceship leaves Earth traveling at 0.65c. A second spaceship leaves the first at a speed of 0.82c with respect to the first. Calculate the speed of the second ship with respect to Earth if it is fired (a) in the same direction the first spaceship is already moving, (b) directly backward toward Earth.

(II) An observer on Earth sees an alien vessel approach at a speed of \(0.60 \mathrm{c}\). The fictional starship Enterprise comes to the rescue (Fig. 26–13), overtaking the aliens while moving directly toward Earth at a speed of \(0.90 \mathrm{c}\) relative to Earth. What is the relative speed of one vessel as seen by the other? Equation Transcription: Text Transcription: 0.60c 0.90c v=0.60c v=0.90c

Problem 51P (II) A spaceship in distress sends out two escape pods in opposite directions. One travels at a speed v1 = +0.70c in one direction, and the other travels at a speed v2 = -0.80c in the other direction, as observed from the spaceship. What speed does the first escape pod measure for the second escape pod?

Problem 52P (II) Rocket A passes Earth at a speed of 0.65c. At the same time, rocket B passes Earth moving 0.95c relative to Earth in the same direction as A. How fast is B moving relative to A when it passes A?

Problem 53P (II) Your spaceship, traveling at 0.90c, needs to launch a probe out the forward hatch so that its speed relative to the planet that you are approaching is 0.95c. With what speed must it leave your ship?

Problem 54GP What is the speed of a particle when its kinetic energy equals its rest energy? Does the mass of the particle affect the result?

Problem 55GP The nearest star to Earth is Proxima Centauri, 4.3 lightyears away. (a) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 4.9 years, as measured by travelers on the spacecraft? (b) How long does the trip take according to Earth observers?

According to the special theory of relativity, the factor \(\gamma\) that determines the length contraction and the time dilation is given by Determine the numerical values of \(\gamma\) for an object moving at speed \(v=0.01 c, 0.05 c, 0.10 c, 0.20 c, 0.30 c, 0.40 c, 0.50 c, 0.60 c 0.70 c, 0.80 c, 0.90 c, 0.95 c \text { and } 0.99 c\). Make a graph of \(\gamma \text { versus } v\) Equation Transcription: Text Transcription: \gamma \gamma=1/1-v2/c2 \gamma v=0.01c, 0.05c,0.10c,0.20c,0.30c,0.40c, 0.50c, 0.60c, 0.70c, 0.80c, 0.90c, 0.95c and 0.99c \gamma versus v

Problem 57GP A healthy astronaut’s heart rate is 60 beats/min. Flight doctors on Earth can monitor an astronaut’s vital signs remotely while in flight. How fast would an astronaut be flying away from Earth if the doctor measured her having a heart rate of 25 beats/min?

Problem 58GP What is the speed v of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference c-v Such speeds are reached in the Stanford Linear Accelerator, SLAC. (b) If the electrons travel in the lab through a tube 3.0 km long (as at SLAC), how long is this tube in the electrons’ reference frame? [Hint: Use the binomial expansion.]

Problem 59GP What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 27–6.)

Problem 60GP How many grams of matter would have to be totally destroyed to run a 75-W lightbulb for 1.0 year?

Problem 61GP A free neutron can decay into a proton, an electron, and a neutrino. Assume the neutrino’s mass is zero; the other masses can be found in the Table inside the front cover. Determine the total kinetic energy shared among the three particles when a neutron decays at rest.

An electron \(\left(m=9.11 \times 10^{-31} \mathrm{~kg}\right)\) is accelerated from rest to speed \(v\). by a conservative force. In this process, its potential energy decreases by \(6.20 \times 10^{-14} j\). Determine the electron's speed, \(v\). Equation Transcription: Text Transcription: (m=9.11 x 10-31kg) v 6.20 x 10-14 J v

Problem 63GP The Sun radiates energy at a rate of about 4x1026 W. (a) At what rate is the Sun’s mass decreasing? (b) How long does it take for the Sun to lose a mass equal to that of Earth? (c) Estimate how long the Sun could last if it radiated constantly at this rate.

Problem 64GP How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 u, and 4.00260 u. (This energy difference is called the total binding energy of the 42He nucleus.)

Show analytically that a particle with momentum \(p\) and energy \(E\) has a speed given by \(v=\frac{p c^{2}}{E}=\frac{p c}{\sqrt{m^{2} c^{2}+p^{2}}}\) Equation Transcription: Text Transcription: p E v=\frac p c^2 E=\fracp c\sqrt m^2 c^2+p^2

Problem 66GP Two protons, each having a speed of 0.990c in the laboratory, are moving toward each other. Determine (a) the momentum of each proton in the laboratory, (b) the total momentum of the two protons in the laboratory, and (c) the momentum of one proton as seen by the other proton.

Problem 67GP When two moles of hydrogen molecules and one mole of oxygen molecules (O2) react to form two moles of water (H2O) the energy released is 484 kJ. How much does the mass decrease in this reaction? What % of the total original mass is this?

Problem 68GP The fictional starship Enterprise obtains its power by combining matter and antimatter, achieving complete conversion of mass into energy. If the mass of the Enterprise is approximately 6 x 109 kg. how much mass must be converted into kinetic energy to accelerate it from rest to one-tenth the speed of light?

Problem 69GP Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

A spaceship and its occupants have a total mass of 160,000 kg. The occupants would like to travel to a star that is 35 light-years away at a speed of 0.70c. To accelerate, the engine of the spaceship changes mass directly to energy. (a) Estimate how much mass will be converted to energy to accelerate the spaceship to this speed. (b) Assuming the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70c, determine how long the trip will take according to the astronauts on board.

Problem 71GP In a nuclear reaction two identical particles are created, traveling in opposite directions. If the speed of each particle is 0.82c, relative to the laboratory frame of reference, what is one particle’s speed relative to the other particle?

Problem 72GP A 36,000-kg spaceship is to travel to the vicinity of a star 6.6 light-years from Earth. Passengers on the ship want the (one-way) trip to take no more than 1.0 year. How much work must be done on the spaceship to bring it to the speed necessary for this trip?

Problem 73GP Suppose a 14,500-kg spaceship left Earth at a speed of 0.90c. What is the spaceship’s kinetic energy? Compare with the total U.S. annual energy consumption (about 1020 J).

A pi meson of mass \(m_{\pi}\) decays at rest into a muon (mass \(m_{\pi}\) ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is \(K E_{\mu}=\left(m_{\pi}-m_{\mu}\right)^{2} c^{2} /\left(2 m_{\pi}\right)\). Equation Transcription: Text Transcription: m_\pi m_\pi K E_\mu=(m_\pi-m_\mu)^2 c^2 (2 m_\pi)

Problem 75GP An astronaut on a spaceship traveling at 0.75c relative to Earth measures his ship to be 23 m long. On the ship, he eats his lunch in 28 min. (a) What length is the spaceship according to observers on Earth? (b) How long does the astronaut’s lunch take to eat according to observers on Earth?

Problem 76GP Astronomers measure the distance to a particular star to be 6.0 light-years (1 ly=distance light travels in 1 year). A spaceship travels from Earth to the vicinity of this star at steady speed, arriving in 3.50 years as measured by clocks on the spaceship. (a) How long does the trip take as measured by clocks in Earth’s reference frame? (b) What distance does the spaceship travel as measured in its own reference frame?

Problem 77GP An electron is accelerated so that its kinetic energy is greater than its rest energy mc2 by a factor of (a) 5.00, (b) 999. What is the speed of the electron in each case?

Problem 78GP You are traveling in a spaceship at a speed of 0.70c away from Earth. You send a laser beam toward the Earth traveling at velocity c relative to you. What do observers on the Earth measure for the speed of the laser beam?

Problem 79GP A farm boy studying physics believes that he can fit a 13.0-m-long pole into a 10.0-m-long barn if he runs fast enough, carrying the pole. Can he do it? Explain in detail. How does this fit with the idea that when he is running the barn looks even shorter than 10.0 m?

Problem 80GP An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 2.0 years has elapsed? [Hint: Use the binomial expansion, Appendix A.]

Problem 81GP An airplane travels 1300 km/h around the Earth in a circle of radius essentially equal to that of the Earth, returning to the same place. Using special relativity, estimate the difference in time to make the trip as seen by Earth and by airplane observers. [Hint: Use the binomial expansion, Appendix A.]

A spaceship passes you at a speed of 0.850c. You measure its length to be 44.2 m. How long would it be when at rest?

A certain type of elementary particle travels at a speed of At this speed, the average lifetime is measured to be What is the particles lifetime at rest?

You travel to a star 135 light-years from Earth at a speed of What do you measure this distance to be?

What is the speed of a pion if its average lifetime is measured to be At rest, its average lifetime

In an Earth reference frame, a star is 49 light-years away. How fast would you have to travel so that to you the distance would be only 35 light-years?

At what speed v will the length of a 1.00-m stick look 10.0% shorter (90.0 cm)?

At what speed do the relativistic formulas for (a) length and (b) time intervals differ from classical values by 1.00%? (This is a reasonable way to estimate when to use relativistic calculations rather than classical.)

You decide to travel to a star 62 light-years from Earth at a speed that tells you the distance is only 25 lightyears. How many years would it take you to make the trip?

A friend speeds by you in her spacecraft at a speed of 0.720c. It is measured in your frame to be 4.80 m long and 1.35 m high. (a) What will be its length and height at rest? (b) How many seconds elapsed on your friends watch when 20.0 s passed on yours? (c) How fast did you appear to be traveling according to your friend? (d) How many seconds elapsed on your watch when she saw 20.0 s pass on hers?

A star is 21.6 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers: (a) on Earth, (b) on the spacecraft? (c) What is the distance traveled according to observers on the spacecraft? (d) What will the spacecraft occupants compute their speed to be from the results of (b) and (c)?

A fictional news report stated that starship Enterprise had just returned from a 5-year voyage while traveling at 0.70c. (a) If the report meant 5.0 years of Earth time, how much time elapsed on the ship? (b) If the report meant 5.0 years of ship time, how much time passed on Earth?

A box at rest has the shape of a cube 2.6 m on a side. This box is loaded onto the flat floor of a spaceship and the spaceship then flies past us with a horizontal speed of 0.80c. What is the volume of the box as we observe it?

Escape velocity from the Earth is What would be the percent decrease in length of a 68.2-m-long spacecraft traveling at that speed as seen from Earth?

An unstable particle produced in an accelerator experiment travels at constant velocity, covering 1.00 m in 3.40 ns in the lab frame before changing (decaying) into other particles. In the rest frame of the particle, determine (a) how long it lived before decaying, (b) how far it moved before decaying

How fast must a pion be moving on average to travel 32 m before it decays? The average lifetime, at rest, is 2.6 * 108 s.

What is the momentum of a proton traveling at v = 0.68c?

(a) A particle travels at By what percentage will a calculation of its momentum be wrong if you use the classical formula?

A particle of mass m travels at a speed At what speed will its momentum be doubled?

An unstable particle is at rest and suddenly decays into two fragments. No external forces act on the particle or its fragments. One of the fragments has a speed of 0.60c and a mass of while the other has a mass of What is the speed of the less massive fragment?

What is the percent change in momentum of a proton that accelerates from (a) 0.45c to 0.85c, (b) 0.85c to 0.98c?

Calculate the rest energy of an electron in joules and in 1 MeV = 1.60 * 1013 JB

When a uranium nucleus at rest breaks apart in the process known as fission in a nuclear reactor, the resulting fragments have a total kinetic energy of about 200 MeV. How much mass was lost in the process?

The total annual energy consumption in the United States is about How much mass would have to be converted to energy to fuel this need?

Calculate the mass of a proton in MeVc2

A certain chemical reaction requires of energy input for it to go. What is the increase in mass of the products over the reactants?

Calculate the kinetic energy and momentum of a proton traveling 2.90 * 108 ms.

What is the momentum of a 950-MeV proton (that is, its kinetic energy is 950 MeV)?

What is the speed of an electron whose kinetic energy is 1.12 MeV?

(a) How much work is required to accelerate a proton from rest up to a speed of 0.985c? (b) What would be the momentum of this proton?

At what speed will an objects kinetic energy be 33% of its rest energy?

Determine the speed and the momentum of an electron whose equals its rest energy

A proton is traveling in an accelerator with a speed of By what factor does the protons kinetic energy increase if its speed is doubled?

How much energy can be obtained from conversion of 1.0 gram of mass? How much mass could this energy raise to a height of 1.0 km above the Earths surface?

To accelerate a particle of mass m from rest to speed 0.90c requires work To accelerate the particle from speed 0.90c to 0.99c requires work Determine the ratio

Suppose there was a process by which two photons, each with momentum could collide and make a single particle. What is the maximum mass that the particle could possess?

What is the speed of a proton accelerated by a potential difference of 165 MV?

(II) What is the speed of an electron after being accelerated from rest by 31,000 V?

The kinetic energy of a particle is 45 MeV. If the momentum is what is the particles mass?

Calculate the speed of a proton whose kinetic energy is exactly half (a) its total energy, (b) its rest energy

Calculate the kinetic energy and momentum of a proton traveling By what percentages would your calculations have been in error if you had used classical formulas?

Suppose a spacecraft of mass 17,000 kg is accelerated to 0.15c. (a) How much kinetic energy would it have? (b) If you used the classical formula for kinetic energy, by what percentage would you be in error?

(II) A negative muon traveling at 53% the speed of light collides head on with a positive muon traveling at 65% the speed of light. The two muons (each of mass \(105.7\ MeV/c^2\)) annihilate, and produce how much electromagnetic energy?

Two identical particles of mass m approach each other at equal and opposite speeds, v. The collision is completely inelastic and results in a single particle at rest. What is the mass of the new particle? How much energy was lost in the collision? How much kinetic energy was lost in this collision?

The americium nucleus, decays to a neptunium nucleus, by emitting an alpha particle of mass 4.00260 u and kinetic energy 5.5 MeV. Estimate the mass of the neptunium nucleus, ignoring its recoil, given that the americium mass is 241.05682 u.

Show that the kinetic energy ke of a particle of mass m is related to its momentum p by the equation

What magnetic field B is needed to keep 998-GeV protons revolving in a circle of radius 1.0 km? Use the relativistic mass. The protons rest mass is [Hint: In relativity, is still valid in a magnetic field, where ]

A person on a rocket traveling at 0.40c (with respect to the Earth) observes a meteor come from behind and pass her at a speed she measures as 0.40c. How fast is the meteor moving with respect to the Earth?

Two spaceships leave Earth in opposite directions, each with a speed of 0.60c with respect to Earth. (a) What is the velocity of spaceship 1 relative to spaceship 2? (b) What is the velocity of spaceship 2 relative to spaceship 1?

A spaceship leaves Earth traveling at 0.65c. A second spaceship leaves the first at a speed of 0.82c with respect to the first. Calculate the speed of the second ship with respect to Earth if it is fired (a) in the same direction the first spaceship is already moving, (b) directly backward toward Earth.

An observer on Earth sees an alien vessel approach at a speed of 0.60c. The fictional starship Enterprise comes to the rescue (Fig. 2613), overtaking the aliens while moving directly toward Earth at a speed of 0.90c relative to Earth. What is the relative speed of one vessel as seen by the other?

A spaceship in distress sends out two escape pods in opposite directions. One travels at a speed in one direction, and the other travels at a speed in the other direction, as observed from the spaceship. What speed does the first escape pod measure for the second escape pod?

Rocket A passes Earth at a speed of 0.65c. At the same time, rocket B passes Earth moving 0.95c relative to Earth in the same direction as A. How fast is B moving relative to A when it passes A?

Your spaceship, traveling at 0.90c, needs to launch a probe out the forward hatch so that its speed relative to the planet that you are approaching is 0.95c. With what speed must it leave your ship?

What is the speed of a particle when its kinetic energy equals its rest energy? Does the mass of the particle affect the result?

The nearest star to Earth is Proxima Centauri, 4.3 lightyears away. (a) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 4.9 years, as measured by travelers on the spacecraft? (b) How long does the trip take according to Earth observers?

According to the special theory of relativity, the factor that determines the length contraction and the time dilation is given by Determine the numerical values of for an object moving at speed 0.05c, 0.10c, 0.20c, 0.30c, 0.40c, 0.50c, 0.60c, 0.70c, 0.80c, 0.90c, 0.95c, and 0.99c. Make a graph of versus v.

A healthy astronauts heart rate is Flight doctors on Earth can monitor an astronauts vital signs remotely while in flight. How fast would an astronaut be flying away from Earth if the doctor measured her having a heart rate of

What is the speed v of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference Such speeds are reached in the Stanford Linear Accelerator, SLAC. (b) If the electrons travel in the lab through a tube 3.0 km long (as at SLAC), how long is this tube in the electrons reference frame? [Hint: Use the binomial expansion.]

What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 276.)

How many grams of matter would have to be totally destroyed to run a 75-W lightbulb for 1.0 year?

A free neutron can decay into a proton, an electron, and a neutrino. Assume the neutrinos mass is zero; the other masses can be found in the Table inside the front cover. Determine the total kinetic energy shared among the three particles when a neutron decays at rest.

An electron is accelerated from rest to speed v by a conservative force. In this process, its potential energy decreases by Determine the electrons speed, v

The Sun radiates energy at a rate of about \(4 \times 10^{26}\ W\). (a) At what rate is the Sun’s mass decreasing? (b) How long does it take for the Sun to lose a mass equal to that of Earth? (c) Estimate how long the Sun could last if it radiated constantly at this rate.

How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The masses of a proton (including an electron), a neutron, and neutral helium are, respectively, 1.00783 u, 1.00867 u, and 4.00260 u. (This energy difference is called the total binding energy of the nucleus.)

Show analytically that a particle with momentum p and energy E has a speed given by

Two protons, each having a speed of 0.990c in the laboratory, are moving toward each other. Determine (a) the momentum of each proton in the laboratory, (b) the total momentum of the two protons in the laboratory, and (c) the momentum of one proton as seen by the other proton

When two moles of hydrogen molecules and one mole of oxygen molecules react to form two moles of water the energy released is 484 kJ. How much does the mass decrease in this reaction? What % of the total original mass is this?

The fictional starship Enterprise obtains its power by combining matter and antimatter, achieving complete conversion of mass into energy. If the mass of the Enterprise is approximately how much mass must be converted into kinetic energy to accelerate it from rest to one-tenth the speed of light?

Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (b) a particle with zero mass.

A spaceship and its occupants have a total mass of 160,000 kg. The occupants would like to travel to a star that is 35 light-years away at a speed of 0.70c. To accelerate, the engine of the spaceship changes mass directly to energy. (a) Estimate how much mass will be converted to energy to accelerate the spaceship to this speed. (b) Assuming the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70c, determine how long the trip will take according to the astronauts on board.

In a nuclear reaction two identical particles are created, traveling in opposite directions. If the speed of each particle is 0.82c, relative to the laboratory frame of reference, what is one particles speed relative to the other particle?

A 36,000-kg spaceship is to travel to the vicinity of a star 6.6 light-years from Earth. Passengers on the ship want the (one-way) trip to take no more than 1.0 year. How much work must be done on the spaceship to bring it to the speed necessary for this trip?

Suppose a 14,500-kg spaceship left Earth at a speed of 0.90c. What is the spaceships kinetic energy? Compare with the total U.S. annual energy consumption (about )

A pi meson of mass decays at rest into a muon (mass ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is kem = Amp - mmB2 c2 A2mpB

An astronaut on a spaceship traveling at 0.75c relative to Earth measures his ship to be 23 m long. On the ship, he eats his lunch in 28 min. (a) What length is the spaceship according to observers on Earth? (b) How long does the astronauts lunch take to eat according to observers on Earth?

Astronomers measure the distance to a particular star to be 6.0 light-years ( light travels in 1 year). A spaceship travels from Earth to the vicinity of this star at steady speed, arriving in 3.50 years as measured by clocks on the spaceship. (a) How long does the trip take as measured by clocks in Earths reference frame? (b) What distance does the spaceship travel as measured in its own reference frame?

An electron is accelerated so that its kinetic energy is greater than its rest energy by a factor of (a) 5.00, (b) 999. What is the speed of the electron in each case?

You are traveling in a spaceship at a speed of 0.70c away from Earth. You send a laser beam toward the Earth traveling at velocity c relative to you. What do observers on the Earth measure for the speed of the laser beam?

A farm boy studying physics believes that he can fit a 13.0-m-long pole into a 10.0-m-long barn if he runs fast enough, carrying the pole. Can he do it? Explain in detail. How does this fit with the idea that when he is running the barn looks even shorter than 10.0 m?

An atomic clock is taken to the North Pole, while another stays at the Equator. How far will they be out of synchronization after 2.0 years has elapsed? [Hint: Use the binomial expansion, Appendix A.]

An airplane travels around the Earth in a circle of radius essentially equal to that of the Earth, returning to the same place. Using special relativity, estimate the difference in time to make the trip as seen by Earth and by airplane observers. [Hint: Use the binomial expansion, Appendix A.]