Explain how the length contraction and time dilation

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.

(I) A certain type of elementary particle travels at a speed of \(2.70 \times 10^8 \mathrm{~m} / \mathrm{s}\). At this speed, the average lifetime is measured to be \(4.76 \times 10^{-6} \mathrm{~s}\). What is the particle's lifetime at rest?

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 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 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 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 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 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?

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

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

Problem 12Q Discuss how our everyday lives would be different if the speed of light were only 25 m/s.

Problem 13Q Explain how the length contraction and time dilation formulas might be used to indicate that c is the limiting speed in the universe.

The drawing at the start of this Chapter shows the street as seen by Mr Tompkins, for whom the speed of light is \(c=20 \mathrm{mi} / \mathrm{h}\). What does Mr Tompkins look like to the people standing on the street (Fig. 26-11)? Explain.

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

Problem 17Q Does the equation E = mc2 conflict with the conservation of energy principle? Explain.

Problem 18Q If mass is a form of energy, does this mean that a spring has more mass when compressed than when relaxed? Explain.

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

Problem 20P (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 20Q Is our intuitive notion that velocities simply add. as in Section 3-8. completely wrong?

Problem 21P (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 22P (I) Calculate the rest energy of an electron in joules and in MeV (1 MeV = 1.60 X 10-13 J)

Problem 41P Make a graph of the kinetic energy versus momentum for (a) a particle of nonzero mass, and (0) a particle with zero mass.

Problem 59GP 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.)

An observer on Earth sees an alien vessel approach at a speed of 0.60c. The Enterprise comes to the rescue (Fig. 26-12), overtaking the aliens while moving directly toward Earth at a speed of relative to Earth. What is the relative speed of one vessel as seen by the other?

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.

Problem 70GP 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.

A 42,000-kg spaceship is to travel to the vicinity of a star 6.0 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?

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

Problem 28P (a) How much work is required to accelerate a proton from rest up to a speed of 0.997c? (b) What would be the momentum of this proton?

Problem 1P A spaceship passes you at a speed of 0.750c. You measure its length to be 28.2 m. How long would it be when at rest?

(I) Lengths and time intervals depend on the factor \(\sqrt{1-v^2/c^2}\) according to the theory of relativity (Eqs. and . Evaluate this factor for speeds of: \((a)\ v=20,000\ \mathrm{m/s}\) (typical speed of a satellite); \((b)\ v=0.020c\); \((c)\ v=0.200c\); \((d)\ v=0.95c\); \((e)\ v=0.98c\); \((f)\ v=0.999c\). ________________ Equation Transcription: Text Transcription: sqrt{1-v^2/c^2} (a) v=20,000 m/s (b) v=0.020c (c) v=0.200c (d) v=0.95c (e) v=0.98c (f) v=0.999c

A worker stands on top of a moving railroad car, and throws a heavy ball straight up (from his point of view). Ignoring air resistance, will the ball land on the car or behind it?

(II) If you were to travel to a star 125 light-years from Earth at a speed of \(2.50 \times 10^8 \mathrm{~m} / \mathrm{s}\), what would you measure this distance to be?

Problem 5P What is the speed of a pion if its average lifetime is measured to be 4.10 × 10?8 s? At rest, its average lifetime is 2.60 × 10?8 s.

Problem 5Q If you were on a spaceship traveling at 0.5c away from a star, at what speed would the starlight pass you?

Problem 6P In an Earth reference frame, a star is 82 light-years away. How fast would you have to travel so that to you the distance would be only 35 light-years?

Problem 7P Suppose you decide to travel to a star 85 light-years away 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 8P At what speed v will the length of a 1.00-m stick look 10.0% shorter (90.0 cm)?

Problem 9P Escape velocity from the Earth is 40,000 km/h. What would be the percent decrease in length of a 95.2-m-long spacecraft traveling at that speed?

Problem 11P Suppose a news report stated that starship Enterprise had just returned from a 5-year voyage while traveling at 0.84c. (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 12P A certain star is 10.6 light-years away. How long would it take a spacecraft traveling 0.960c to reach that star from Earth, 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 13P A friend speeds by you in her “Ferrari” spacecraft at a speed of 0.660c. It is measured in your frame to be 4.80 m long and 1.25 m high. (a) What will be its length and height at rest? (b) How many seconds would you say 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 would she say elapsed on your watch when she saw 20.0 s pass on hers?

Problem 14P How fast must an average pion be moving to travel 15 m before it decays? The average lifetime, at rest, is 2.6 × 10?8 s.

(I) What is the momentum of a proton traveling at v=0.85c?

Problem 16P At what speed will an object’s relativistic mass be twice its rest mass?

Problem 17P A particle of rest mass m0 travels at a speed v = 0.20c. At what speed will its momentum be doubled?

Problem 18P (a) A particle travels at v = 0.10c. By what percentage will a calculation of its momentum be wrong if you use the classical formula? (b) Repeat for v = 0.50c.

Problem 19P What is the percent change in momentum of a proton that accelerates (a) from 0.45c to 0.90c, (b) from 0.90c to 0.98c?

(I) Calculate the rest mass of a proton in \(\mathrm{MeV} / c^2\).

Problem 24P The total annual energy consumption in the United States is about 8 × 104 J. How much mass would have to be converted to energy to fuel this need?

Problem 25P 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 0.25 km above the Earth’s surface?

Problem 26P What is the speed of a particle when its kinetic energy equals its rest energy?

(II) At what speed will an object’s kinetic energy be 25% of its rest energy?

Calculate the kinetic energy and momentum of a proton traveling \(2.60 \times 10^8 \mathrm{~m} / \mathrm{s}\).

(I) What is the momentum of a 750-MeV proton (that is, its kinetic energy is 750 MeV)?

Problem 31P What is the speed of a proton accelerated by a potential difference of 105 MV?

Problem 32P What is the speed of an electron whose kinetic energy is 1.00 MeV?

Problem 33P What is the speed of an electron just before it hits a television screen after being accelerated from rest by the 25,000 V of the picture tube?

Problem 34P Two identical particles of rest mass m0 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 rest mass of the new particle? How much energy was lost in the collision? How much kinetic energy is lost in this collision?

Problem 35P Calculate the speed of a proton (m0 = 1.67 × 10?27 kg) whose kinetic energy is exactly half (a) its total energy, (b) its rest energy.

Problem 36P What is the speed and the momentum of an electron (m = 9.11 × 10?31 kg) whose kinetic energy equals its rest energy?

Problem 37P Suppose a spacecraft of mass 27,000 kg is accelerated to 0.21c. (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 38P Calculate the kinetic energy and momentum of a proton (m0 = 1.67 × 10?27kg) traveling 7.35 × 107m/s. By what percentages would your calculations have been in error if you had used classical formulas?

(II) The americium nucleus, \({ }_{95}^{241} \mathrm{Am}\), decays to a neptunium nucleus, \({ }_{93}^{237} \mathrm{N p}\), 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

Problem 40P An electron (m0 = 9.11 × 10?31 kg) is accelerated from rest to speed v by a conservative force. In this process, its potential energy decreases by 6.60 × 10?14 J. Determine the electron’s speed, v.

Problem 42P What magnetic field intensity is needed to keep 998-GeV protons revolving in a circle of radius 1.0 km (at, say, the Fermilab synchrotron)? Use the relativistic mass. The proton’s rest mass is 0.938 GeV/c2. (l GeV = 109 eV.) [Hint: in relativity, it is still true that mv2/r = qvB in a magnetic field.]

Problem 43P A person on a rocket traveling at 0.50c (with respect to the Earth) observes a meteor come from behind and pass her at a speed she measures as 0.50c. How fast is the meteor moving with respect to the Earth?

Problem 44P Two spaceships leave Earth in opposite directions, each with a speed of 0.50c 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 0.71c. A second spaceship leaves the first at a speed of 0.87c 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.

Problem 47P A spaceship in distress sends out two escape pods in opposite directions. One travels at a speed v1 = ?0.60c in one direction, and the other travels at a speed v2 = +0.70c 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 0.75c. At the same time, rocket B passes Earth moving 0.95c relative to Earth in the same direction. How fast is B moving relative to A when it passes A?

Problem 49GP The nearest star to Earth is Proxima Centauri, 4.3 light-years away. (a) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 4.0 years, as measured by travelers on the spacecraft? (b) How long does the trip take according to Earth observers?

Problem 50GP As a rule of thumb, anything traveling faster than about 0.1c is called relativistic—i.e., for which the correction using special relativity is a significant effect. Determine the speed of an electron in a hydrogen atom (radius 0.5 × 10?10 m) and state if it is relativistic. (Treat the electron as though it were in a circular orbit around the proton.)

Problem 51GP (a) What is the speed v of an electron whose kinetic energy is 14,000 times its rest energy? You can state 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.]

How many grams of matter would have to be totally destroyed to run a 100-W lightbulb for 1 year?

Problem 53GP What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same rest mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 27-6.)

An electron \((m = 9.11 \times 10^{?31} \mathrm {kg})\) enters a uniform magnetic field B = 1.8 T, and moves perpendicular to the field lines with a speed v = 0.92c. What is the radius of curvature of its path? See hint for Problem 42.

Problem 55GP A negative muon traveling at 33% the speed of light collides head on with a positive muon traveling at 50% the speed of light. The two muons (each of rest mass 105.7 MeV/c2) annihilate, and produce how much electromagnetic energy?

A free neutron can decay into a proton, an electron, and a neutrino. Assume the neutrino’s rest mass is zero, and 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.

Problem 58GP An unknown particle is measured to have a negative charge and a speed of 2.24 × 108m/s. Its momentum is determined to be 3.07 × 10?22kg·m/s. Identify the particle by finding its rest mass.

Problem 60GP What is the percentage increase in the (relativistic) mass of a car traveling 110 km/h as compared to at rest? [Hint: use the binomial expansion.]

Two protons, each having a speed of \(0.935 c\) 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.

Show analytically that a particle with momentum and energy has a speed given by \(v=\frac{p c^{2}}{E}=\frac{p c}{\sqrt{m_{0}^{2} c^{2}+p^{2}}}\) ________________ Equation Transcription: Text Transcription: v={{pc^2} over E}={pc}over sqrt{m_0^2c^2+p^2}

Problem 63GP 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 5 × 109kg, how much mass must be converted into kinetic energy to accelerate it from rest to one-tenth the speed of light?

An electron is accelerated so that its kinetic energy is greater than its rest energy \(m_{0} c^2\) by a factor of (a) 5.00, (b) 999. What is the speed of the electron in each case?

Problem 65GP A farm boy studying physics believes that he can fit a 15.0-m-long pole into a 12.0-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 to him than 12.0 m?

When two moles of hydrogen and one mole of oxygen react to form two moles of water, the energy released is \(484 \mathrm{~kJ}\). How much does the mass of the elements decrease in this reaction? What % of the total original mass of the system does this mass change represent?

In a nuclear reaction two identical particles are created, traveling in opposite directions. If the speed of each particle is 0.75c, relative to the laboratory frame of reference, what is one particle’s speed relative to the other particle?

Problem 68GP An astronaut on a spaceship traveling at 0.75c relative to Earth measures his ship to be 25 m long. On the ship, he eats his lunch in 23 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 69GP You are traveling in a spaceship at a speed of 0.85c 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?

Suppose a \(12,500-\mathrm{kg}\) spaceship left Earth at a speed of \(0.99 \mathrm{c}\). What is the spaceship's kinetic energy? Compare with the total U.S. annual energy consumption (about \(10^{20} \mathrm{~J}\) ).

Problem 73GP A 1.68-kg mass oscillates on the end of a spring whose spring stiffness constant is k = 48.7 N/m. If this system is in a spaceship moving past Earth at 0.900c, what is its period of oscillation according to (a) observers on the ship, and (b) observers on Earth?