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physics
modern physics

Questions and Answers of Modern Physics

Use the binomial expansion(1 + x)n = 1 + nx + n(n -1)x2/2 + . . . ≈ 1 + nxTo derive the following results for the case when V is much less than c, and use the results when applicable in the
Show that when V << c the transformation equations for x, t, and u reduce to the Galilean equations.
Supersonic jets achieve maximum speeds of about (3 × 10-6)c.(a) By what percentage would you see a jet traveling at this speed contracted in length?(b) During a time of 1 y = 3.15 × 107 s on your
How great must the relative speed of two observers be for the time-interval measurements to differ by 1%? (See Problem 14.)
A spaceship of proper length L’ = 400 m moves past a transmitting station at a speed of 0.76c. At the instant that the nose of the ship passes the transmitter, clocks at the transmitter and in the
A beam of unstable particles emerges from the exit slit of an accelerator with a speed of 0.89c. Particle detectors 3.0 and 6.0 m from the exit slit measure beam intensities of 2 × 108
Show that if and U’x in Equation 39-18a are both less than c, then ux is less than c.
Two events in S are separated by a distance D = x2 - x1 and a time T = t2 - t1.(a) Use the Lorentz transformation to show that in frame S’, which is moving with speed V relative
What is the separation distance between clocks Aï‚¢ï€ and Bï‚¢ï€ according to the observer inS?
As the light pulse from the flashbulb travels toward A¢ with speed c, A¢ travels toward C with speed 0.6c. Show that the clock in S reads 25 min when the flash reaches A¢.
Show that the clock in S reads 100 min when the light flash reaches B¢, which is traveling away from C with speed 0.6c.
The time interval between the reception of the flashes at A’ and B’ in Problems 26 and 27 is 75 min according to the observer in S. How much time does he expect to have elapsed on the clock at
The time interval calculated in Problem 28 is the amount that the clock at A’ leads that at B’ according to the observer in S. Compare this result with LpV/c2.
In frame S, event B occurs 2 ms after event A, which occurs at Dx = 1.5 km from event A. How fast must an observer be moving along the +x axis so that events A and B occur simultaneously? Is it
Observers in reference frame S see an explosion located at x1 = 480 m. A second explosion occurs 5 μs later at x2 = 1200 m. In reference frame S’, which is moving along the +x axis at speed
How fast must you be moving toward a red light (λ = 650 nm) for it to appear green (λ = 525 nm)?
A distant galaxy is moving away from us at a speed of 1.85 × 107 m/s. Calculate the fractional redshift (λ – λ0)/ λ0 in the light from this galaxy.
Sodium light of wavelength 589 nm is emitted by a source that is moving toward the earth with speed V. The wavelength measured in the frame of the earth is 620 nm. Find V.
A student on earth hears a tune on her radio that seems to be coming from a record that is being played too fast. She has a 33-rev/min record of that tune and determines that the tune sounds the same
Derive Equation 39-16a for the frequency received by an observer moving with speed V toward a stationary source of electromagneticwaves.
Herb and Randy are twin jazz musicians who perform as a trombone–saxophone duo. At the age of twenty, however, Randy got an irresistible offer to join a road trip to perform on a star 15
A clock is placed in a satellite that orbits the earth with a period of 90 min. By what time interval will this clock differ from an identical clock on earth after 1 y? (Assume that special
A and B are twins. A travels at 0.6c to Alpha Centauri (which is 4 c · y from earth as measured in the reference frame of the earth) and returns immediately. Each twin sends
A light beam moves along the y’ axis with speed c in frame S’, which is moving to the right with speed V relative to frame S.(a) Find the x and y components of the velocity of the light beam in
A spaceship is moving east at speed 0.90c relative to the earth. A second spaceship is moving west at speed 0.90c relative to the earth. What is the speed of one spaceship relative to the other?
Two spaceships are approaching each other.(a) If the speed of each is 0.6c relative to the earth, what is the speed of one relative to the other?(b) If the speed of each relative to the earth is
A particle moves with speed 0.8c along the x² axis of frame S², which moves with speed 0.8c along the x¢ axis relative to frame S¢. Frame S¢ moves with speed
Find the ratio of the total energy to the rest energy of a particle of rest mass m0 moving with speed(a) 0.1c,(b) 0.5c,(c) 0.8c, and(d) 0.99c.
A proton (rest energy 938 MeV) has a total energy of 1400 MeV.(a) What is its speed?(b) What is its momentum?
How much energy would be required to accelerate a particle of mass m0 from rest to(a) 0.5c,(b) 0.9c, and(c) 0.99c? Express your answers as multiples of the rest energy.
If the kinetic energy of a particle equals its rest energy, what error is made by using p = m0u for its momentum?
A particle with momentum of 6 MeV/c has total energy of 8 MeV.(a) Determine the rest mass of the particle.(b) What is the energy of the particle in a reference frame in which its momentum is 4
Show that
Use Equations 39-21 and 39-25 to derive the equation E2 = p2c2 + (m0c2)2.
Use the binomial expansion (Equation 39-27) and Equation 39-28 to show that when pc
(a) Show that the speed u of a particle of mass m0 and total energy E is given byAnd that when E is much greater than m0c2, this can be approximated byFind the speed of an electron
The rest energy of a proton is about 938 MeV. If its kinetic energy is also 938 MeV, find(a) Its momentum (b) Its speed.
What percentage error is made in using ½m0u2 for the kinetic energy of a particle if its speed is(a) 0.1c (b) 0.9c?
The K0 particle has a rest mass of 497.7 MeV/c2. It decays into a π– and π+, each with rest mass 139.6 MeV/c2. Following the decay of a K0, one of the pions is at rest in the laboratory.
The sun radiates energy at the rate of about 4 × 1026 W. Assume that this energy is produced by a reaction whose net result is the fusion of 4 H nuclei to form 1 He nucleus, with the release of 25
Two protons approach each other head on at 0.5c relative to reference frame S’.(a) Calculate the total kinetic energy of the two protons as seen in frame S’.(b) Calculate the total kinetic energy
An antiproton p0 has the same rest energy as a proton. It is created in the reaction p + p → p + p + p + p . In an experiment, protons at rest in the laboratory are bombarded with protons of
A particle of rest mass 1 MeV/c2 and kinetic energy 2 MeV collides with a stationary particle of rest mass 2 MeV/c2. After the collision, the particles stick together. Find(a) The speed of the first
A horizontal turntable rotates with angular speed ω. There is a clock at the center of the turntable and one at a distance r from the center. In an inertial reference frame, the clock at
The Lorentz transformation for y and z is the same as the classical result: y = y¢ and z = z¢. Yet the relativistic velocity transformation does not give the classical result uy = uy¢ and uz =
A spaceship departs from earth for the star Alpha Centauri, which is 4 light-years away. The spaceship travels at 0.75c. How long does it take to get there(a) As measured on earth and(b) As measured
The total energy of a particle is twice its rest energy.(a) Find u/c for the particle.(b) Show that its momentum is given by p = √3m0c.
How fast must a muon travel so that its mean lifetime is 46 ms if its mean lifetime at rest is 2 μs?
A distant galaxy is moving away from the earth with a speed that results in each wavelength received on earth being shifted such that λ’ = 2λ0. Find the speed of the galaxy relative to the earth.
How fast must a meterstick travel relative to you in the direction parallel to the stick so that its length as measured by you is 50 cm?
Show that if V is much less than c, the doppler shift is given approximately by Δf/f ≈ ±V/c.
If a plane flies at a speed of 2000 km/h, for how long must it fly before its clock loses 1 s because of time dilation?
The radius of the orbit of a charged particle in a magnetic field is related to the momentum of the particle byp = BqR 39-41This equation holds classically for p =
Oblivious to economics and politics, Professor Spenditt proposes building a circular accelerator around the earth’s circumference using bending magnets that provide a magnetic field of magnitude
Frames S and S’ are moving relative to each other along the x and x’ axis. Observers in the two frames set their clocks to t = 0 when the origins coincide. In frame S, event 1
An interstellar spaceship travels from the earth to a distant star system 12 light-years away (as measured in the earth’s frame). The trip takes 15 years as measured on the ship.(a) What is the
The neutral pion π0 has a rest mass of 135 MeV/c2. This particle can be created in a proton–proton collision:p + p → p + p + π0Determine the threshold kinetic energy for the creation of a π0
A rocket with a proper length of 1000 m moves in the +x direction at 0.6c with respect to an observer on the ground. An astronaut stands at the rear of the rocket and fires a bullet toward the front
In a simple thought experiment, Einstein showed that there is mass associated with electromagnetic radiation. Consider a box of length L and mass M resting on a frictionless surface. At the left wall
A rocket with a proper length of 700 m is moving to the right at a speed of 0.9c. It has two clocks, one in the nose and one in the tail, that have been synchronized in the frame of the rocket. A
An observer in frame S standing at the origin observes two flashes of colored light separated spatially by Δx = 2400 m. A blue flash occurs first, followed by a red flash 5 μs later. An observer in
Reference frame S’ is moving along the x’ axis at 0.6c relative to frame S. A particle that is originally at x’ = 10 m at t1’ = 0 is suddenly accelerated and then moves at a
In reference frame S the acceleration of a particle is a = axi + ayj + azk. Derive expressions for the acceleration components ax', ay', and az' of the particle in reference frame
When a projectile particle with kinetic energy greater than the threshold kinetic energy Kth strikes a stationary target particle, one or more particles may be created in the inelastic collision.
A particle of rest mass M0 decays into two identical particles of rest mass m0, where m0 = 0.3M0. Prior to the decay, the particle of rest mass M0 has an energy of 4M0c2 in the laboratory. The
A stick of proper length Lp makes an angle θ with the x axis in frame S. Show that the angle θ‘ made with the x’ axis in frame S’, which is moving along the +x axis with speed V, is given by
Show that if a particle moves at an angle θ with the x axis with speed ?u in frame S, it moves at an angle θ’ with the x’ axis in S’ given by
For the special case of a particle moving with speed u along the y axis in frame S, show that its momentum and energy in frame S’ are related to its momentum and energy in S by the transformation
The equation for the spherical wavefront of a light pulse that begins at the origin at time t = 0 is x2 + y2 + z2 -(ct)2 = 0. Using the Lorentz transformation, show that such a light pulse also
In Problem 90, you showed that the quantity x2 + y2 + z2 -(ct)2 has the same value (0) in both S and S’. Such a quantity is called an invariant. From the results of Problem 89, the
Two identical particles of rest mass m0 are each moving toward the other with speed u in frame S. The particles collide inelastically with a spring that locks shut (Figure) and come to rest in S, and
(Multiple choice)(1)The approximate total energy of a particle of mass m moving at speed u << c is(a) mc2,(b) ½mu2,(c) cmu,(d) ½ mc2,(e) ½cmu.(2)A set of twins work in an office building.
1. You are standing on a corner and a friend is driving past in an automobile. Both of you note the times when the car passes two different intersections and determine from your watch readings the
1. Two events are simultaneous in a frame in which they also occur at the same point in space. Are they simultaneous in other reference frames?2. If event A occurs before event B in some frame, might
Give the symbols for two other isotopes of(a) 14N,(b) 56Fe, and(c) 118Sn
Calculate the binding energy and the binding energy per nucleon from the masses given in Table 40-1 for(a) 12C,(b) 56Fe,(c) 238U.
Repeat Problem 2 for(a) 6Li,(b) 39K,(c) 208Pb.
Use Equation 40-1 to compute the radii of the following nuclei:(a) 16O,(b) 56Fe, and(c) 197Au.
(a) Given that the mass of a nucleus of mass number A is approximately m = CA, where C is a constant, find an expression for the nuclear density in terms of C and the constant R0
Derive Equation 40-2; that is, show that the rest energy of one unified mass unit is 931.5MeV.
Use Equation 40-1 for the radius of a spherical nucleus and the approximation that the mass of a nucleus of mass number A is A u to calculate the density of nuclear matter in grams per
The electrostatic potential energy of two charges q1 and q2 separated by a distance r is U = kq1q2/r, where k is the Coulomb constant.(a) Use Equation 40-1 to calculate the radii of 2H
(a) Calculate the radii of 56141Ba and3692 Kr from Equation 40-1.(b) Assume that after the fission of 235U into 141Ba and 92Kr, the two nuclei are momentarily separated by a distance
Homer enters the visitors’ chambers, and his geiger-beeper goes off. He shuts off the beep, removes the device from his shoulder patch and holds it near the only new object in the room: an orb
A certain source gives 2000 counts/s at time t = 0. Its half-life is 2 min.(a) What is the counting rate after 4 min?(b) After 6 min?(c) After 8 min?
The counting rate from a radioactive source is 8000 counts/s at time t = 0, and 10 min later the rate is 1000 counts/s.(a) What is the half-life?(b) What is the decay constant?(c) What is the
The half-life of radium is 1620 y. Calculate the number of disintegrations per second of 1 g of radium, and show that the disintegration rate is approximately 1 Ci.
A radioactive silver foil (t1/2 = 2.4 min) is placed near a Geiger counter and 1000 counts/s are observed at time t = 0.(a) What is the counting rate at t = 2.4 min and at t = 4.8 min?(b) If the
Using Table 40-1 to calculate the energy in MeV for the Î± decay of(a) 226Ra (b) 242Pu.
Suppose that two billion years ago 10% of the mass of the earth were 14C. Approximately what percent of the mass of the earth today would be 14C, neglecting formation of 14C in the atmosphere?
At the scene of the crime, in the museum’s west wing, Angela found some wood chips, so she slipped them into her purse for future analysis. They were allegedly from an old wooden mask, which the
The thief in problem 21 had been after a valuable carving made from a 10,000 year old bone. The guard said that he chased the thief away, but Angela suspects that the guard is an accomplice, and that
Through a friend in security at the museum, Angela got a sample having 175 g of carbon. The decay rate of 14C was 8.1 Bq.(a) How old is it?(b) Is it from the carving described in problem 22?
A sample of a radioactive isotope is found to have an activity of 115.0 Bq immediately after it is pulled from the reactor that formed it. Its activity 2 h 15 min later is measured to be 85.2 Bq.(a)
Derive the result that the activity of 1 g of natural carbon due to the β decay of 14C is 15 decays/min = 0.25 Bq.
Measurements of the activity of a radioactive sample have yielded the following results. Plot the activity as a function of time, using semilogarithmic paper, and determine the decay constant and
(a) Show that if the decay rate is R0 at time t = 0 and R1 at some later time t1, the decay constant is given by λ = t1-1 ln(R0/R1) and the half-life is given by t1/2
A wooden casket is thought to be 18,000 years old. How much carbon would have to be recovered from this object to yield a 14C counting rate of no less than 5 counts/min?
A 1.00-mg sample of substance of atomic mass 59.934 u emits β particles with an activity of 1.131 Ci. Find the decay constant for this substance in s–1 and its half-life in years.
The counting rate from a radioactive source is measured every minute. The resulting counts per second are 1000, 820, 673, 552, 453, 371, 305, 250. Plot the counting rate versus time on semilog graph

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