Show that if is much less than the Doppler shift is given

The approximate total energy of a particle of mass moving at speed is (a) (b) (c) (d) (e)

Aset of twins work in an office building. One twin works on the top floor and the other twin works in the basement. Considering general relativity, which twin will age more quickly? (a) They will age at the same rate. (b) The twin who works on the top floor will age more quickly. (c) The twin who works in the basement will age more quickly. (d) It depends on the speed of the office building. (e) None of the above.

True or false: (a) The speed of light is the same in all reference frames. (b) The time interval between two events is never shorter than the proper time interval between the two events. (c) Absolute motion can be determined by means of length contraction. (d) The light-year is a unit of distance. (e) Simultaneous events must occur at the same place. (f) If two events are not simultaneous in one frame, they cannot be simultaneous in any other frame. (g) The mass of a system that consists of two particles tightly bound together by attractive forces is less than the sum of the masses of the individual particles when separated.

An observer sees a system moving past her that consists of a mass oscillating on the end of a spring and measures the period of the oscillations. A second observer, who is moving with the massspring system, also measures its period. The second observer will find a period that is (a) equal to (b) less than (c) greater than (d) either (a) or (b) depending on whether the system was approaching or receding from the first observer, (e) Not enough information is given to answer the question.

The Lorentz transformation for and is the same as the classical result: and Yet the relativistic velocity transformation does not give the classical result and Explain why this result occurs.

The Sun radiates energy at the rate of approximately Assume that this energy is produced by a reaction whose net result is the fusion of four protons to form a single 4He nucleus and the release of of energy that is radiated into space. Calculate the Suns loss of mass per day.

The most distant galaxies that can be seen by the Hubble telescope are moving away from us and have a redshift parameter of about [The redshift parameter is defined as where is the frequency measured in the rest frame of the emitter and is the frequency measured in the rest frame of the receiver.] (a) What is the speed of the galaxies relative to us (expressed as a fraction of the speed of light)? (b) Hubbles law states that the recession speed is given by the expression where is the speed of recession, is the distance, and , the Hubble constant, is equal to where (The abbreviation for parsec is ) Estimate the distance of such a galaxy from us using the information given. SSM

The proper mean lifetime of a muon is Muons in a beam are traveling through a laboratory at (a) What is their mean lifetime as measured in the laboratory? (b) How far do they travel, on average, before they decay?

In the Stanford linear collider, small bundles of electrons and positrons are fired at each other. In the laboratorys frame of reference, each bundle is approximately long and in diameter. In the collision region, each particle has an energy of and the electrons and the positrons are moving in opposite directions. (a) How long and how wide is each bundle in its own reference frame? (b) What must be the minimum proper length of the accelerator for a bundle to have both its ends simultaneously in the accelerator in its own reference frame? (The actual proper length of the accelerator is less than ) (c) What is the length of a positron bundle in a reference frame that moves with the electron bundle?

Use the binomial expansion equation to derive the following results for the case when is much less than (a) (b) (c)

Star A and Star B are at rest relative to Earth. Star A is 27 from Earth, and as viewed from Earth, Star B is located beyond (behind) Star A. (a) A spaceship is making a trip from Earth to Star A at a speed such that the trip from Earth to Star Atakes according to clocks on the spaceship. At what speed, relative to Earth, must the spaceship travel? (Assume that the times for the accelerations are very short compared to the overall trip time.) (b) Upon reaching Star A, the spaceship speeds up and departs for Star B at a speed such that the gamma factor, is twice that of Part (a). The trip from Star A to Star B takes (spaceships time). How far, in is Star B from Star Ain the rest frame of Earth and the two stars? (c) Upon reaching Star B, the spaceship departs for Earth at the same speed as in Part (b). It takes it (spaceships time) to return to Earth. If you were born on Earth the day the ship left Earth and you remain on Earth, how old are you on the day the ship returns to Earth?

A spaceship travels to a star 35 away at a speed of How long does the spaceship take to get to the star (a) as measured on Earth and (b) as measured by a passenger on the spaceship?

Unobtainium (Un) is an unstable particle that decays into normalium (Nr) and standardium (St) particles. (a) An accelerator produces a beam of Un that travels to a detector located 100 m away from the accelerator. The particles travel with a velocity of How long do the particles take (in the laboratory frame) to get to the detector? (b) By the time the particles get to the detector, half of the particles have decayed. What is the half-life of Un? (Note: half-life as it would be measured in a frame moving with the particles) (c) A new detector is going to be used, which is located away from the accelerator. How fast should the particles be moving if half of the particles are to make it to the new detector? SSM

A clock on Spaceship A measures the time interval between two events, both of which occur at the location of the clock. You are on Spaceship B. According to your careful measurements, the time interval between the two events is 1.00 percent longer than that measured by the two clocks on Spaceship A. How fast is Spaceship A moving relative to Spaceship B. (Hint: Use one or more of the results of Problem 10.)

If a plane flies at a speed of how long must the plane fly before its clock loses because of time dilation? (Hint: Use one or more of the results of Problem 10.)

Show that when the relativistic transformation equations for and reduce to the classical transformation equations.

A spaceship of proper length moves past a transmitting station at a speed of (The transmitting station broadcasts signals that travel at the speed of light.) A clock is attached to the nose of the spaceship and a second clock is attached to the transmitting station. The instant that the nose of the spaceship passes the transmitter, the clock attached to the transmitter and the clock attached to the nose of the spaceship are set equal to zero. The instant that the tail of the spaceship passes the transmitter a signal is sent by the transmitter that is subsequently detected by a receiver in the nose of the spaceship. (a) When, according to the clock attached to the nose of spaceship, is the signal sent? (b) When, according to the clocks attached to the nose of spaceship, is the signal received? (c) When, according to the clock attached to the transmitter, is the signal received by the spaceship? (d) According to an observer that works at the transmitting station, how far from the transmitter is the nose of the spaceship when the signal is received?

In frame event occurs after event and event occurs at the origin whereas event occurs on the axis at How fast and in what direction must an observer be traveling along the axis so that events and occur simultaneously? Is it possible for event to precede event for some observer?

Observers in reference frame see an explosion located on the axis at A second explosion occurs, later, at In reference frame which is moving along the axis in the direction at speed the two explosions occur at the same point in space. What is the separation in time between the two explosions as measured in

In reference frame events 1 and 2 are separated by a distance and a time (a) Use the Lorentz transformation to show that in frame which is moving along the axis with speed relative to the time separation is (b) Show that the events can be simultaneous in frame only if is greater than (c) If one of the events is the cause of the other, the separation must be less than because is the smallest time that a signal can take to travel from to in frame Show that if is less than is greater than in all reference frames. This shows that if the cause precedes the effect in one frame, it must precede it in all reference frames. (d) Suppose that a signal could be sent with speed so that in frame the cause precedes the effect by the time Show that there is then a reference frame moving with speed less than in which the effect precedes the cause.

A rocket that has a proper length of is moving to the right at a speed of It has two clocksone in the nose and one in the tailthat have been synchronized in the frame of the rocket. Aclock on the ground and the clock in the nose of the rocket both read zero as they pass by each other. (a) At the instant the clock on the ground reads zero, what does the clock in the tail of the rocket read according to observers on the ground? When the clock in the tail of the rocket passes the clock on the ground, (b) what does the clock in the tail read according to observers on the ground, and (c) what does the clock in the nose read according to observers on the ground, and (d) what does the clock in the nose read according to observers on the rocket? (e) At the instant the clock in the nose of the rocket reads a light signal is sent from the nose of the rocket to an observer standing by the clock on the ground. What does the clock on the ground read when the observer on the ground receives the signal? (f) When the observer on the ground receives the signal, he immediately sends a return signal to the nose of the rocket. What is the reading of the clock in the nose of the rocket when that signal is received at the nose of the rocket?

SPREADSHEET A spaceship, at rest in a certain reference frame is given a speed increase of (call this increase boost 1). Relative to its new rest frame, the spaceship is given a further increase 10 seconds later (as measured in its new rest frame; call this increase boost 2). This process is continued indefinitely, at intervals, as measured in the rest frame of the spaceship. (Assume that the boosts take a very short time compared to ) (a) Using a spreadsheet program, calculate and graph the speed of the spaceship in reference frame as a function of the boost number for boost 1 to boost 10. (b) Graph the gamma factor in the same manner. (c) How many boosts does it take until the speed of the ship in is greater than (d) How far does the spaceship move between boost 1 and boost 6, as measured in reference frame What is the average speed of the spaceship between boost 1 and boost 6, as measured in

Light is emitted by a sodium sample that is moving toward Earth with speed The wavelength of the light is 589 nm in the rest frame of the sample. The wavelength measured in the frame of Earth is Find

A distant galaxy is moving away from us at a speed of \(1.85 \times 10^7\) m/s. Calculate the fractional redshift \((\lambda^\prime - \lambda_0)/\lambda_0\) that we observe the light from the galaxy to have.

Derive (Equation 39-16a) for the frequency received by an observer moving with speed toward a stationary source of electromagnetic waves.

Show that if is much less than the Doppler shift is given approximately by

A clock is placed in a satellite that orbits Earth with an orbital period of By what time interval will this clock differ from an identical clock on Earth after (Assume that special relativity applies and neglect general relativity.)

For light that is Doppler-shifted with respect to an observer, we define the redshift parameter where is the frequency of the light measured in the rest frame of the emitter and is the frequency measured in the rest frame of the receiver. If the emitter is moving directly away from the receiver, show that the relative velocity between the emitter and the receiver is where u _ z _ 1.

A light beam moves along the axis with speed in frame which is moving in the direction with speed relative to frame (a) Find the and components of the velocity of the light beam in frame (b) Show that, according to the velocity transformation equations, the magnitude of the velocity of the light beam in is

A spaceship is moving east at speed relative to Earth. A second spaceship is moving west at speed relative to Earth. What is the speed of one spaceship relative to the other spaceship?

A particle moves with speed in the direction along the axis of frame which moves with the same speed and in the same direction along the axis relative to frame Frame moves with the same speed and in the same direction along the axis relative to frame (a) Find the speed of the particle relative to frame (b) Find the speed of the particle relative to frame

A proton that has a rest energy equal to has a total energy of (a) What is its speed? (b) What is its momentum?

If the kinetic energy of a particle equals twice its rest energy, what percentage error is made by using for the magnitude of its momentum?

In a certain reference frame, a particle has momentum of and total energy of (a) Determine the mass of the particle. (b) What is the total energy of the particle in a reference frame in which its momentum is (c) What is the relative speed of the two reference frames?

Show that Note: This relation was used to derive the relativistically correct expression for kinetic energy (Equation 39-22).

The particle has a mass of It decays into a and each having mass Following the decay of a one of the pions is at rest in the laboratory. Determine the kinetic energy of the other pion after the decay and of the prior to the decay.

In reference frame two protons, each moving at approach each other head-on. (a) Calculate the total kinetic energy of the two protons in frame (b) Calculate the total kinetic energy of the protons as seen in reference frame which is moving with one of the protons.

An antiproton has the same mass as a proton p. The antiproton is created during the reaction During an experiment, protons at rest in the laboratory are bombarded with protons of kinetic energy which must be great enough so that an amount of kinetic energy equal to can be converted into the rest energy of the two particles. In the frame of the laboratory, the total kinetic energy cannot be converted into rest energy because of conservation of momentum. However, in the zero-momentum reference frame in which the two initial protons are moving toward each other with equal speed the total kinetic energy can be converted into rest energy. (a) Find the speed of each proton so that the total kinetic energy in the zero-momentum frame is (b) Transform to the laboratorys frame in which one proton is at rest, and find the speed of the other proton. (c) Show that the kinetic energy of the moving proton in the laboratorys frame is KL _ 6mc2.

A particle of mass and kinetic energy collides with a stationary particle of mass After the collision, the particles stick together. Find (a) the speed of the first particle before the collision, (b) the total energy of the first particle before the collision, (c) the initial total momentum of the system, (d) the total kinetic energy after the collision, and (e) the mass of the system after the collision.

Light traveling in the direction of increasing gravitational potential undergoes a frequency redshift. Calculate the shift in wavelength if a beam of light of wavelength \(\lambda=632.8\) is sent up a vertical shaft of height L = 100 m.

Let us revisit a problem from Chapter 3: Two cannons are pointed directly toward each other, as shown in Figure 39-17. When fired, the cannonballs will follow the trajectories shown. Point is the point where the trajectories cross each other. Ignore any effects due to air resistance. Using the principle of equivalence, show that if the cannons are fired simultaneously (in the rest frame of the cannons), the cannonballs will hit each other at point P.

A horizontal turntable rotates with angular speed There is a clock at the center of the turntable and an identical clock mounted on the turntable a distance from the center. In an inertial reference frame, in which the clock at the center is at rest, the clock at distance is moving with speed (a) Show that from time dilation according to special relativity, the time between ticks, for the clock at rest and for the moving clock, are related by (b) In a reference frame rotating with the table, both clocks are at rest. Show that the clock at distance experiences a pseudoforce in the rotating frame and that this is equivalent to a difference in gravitational potential between and the origin of (c) Use the difference in gravitational potential given in Part (b) to show that in this frame the difference in time intervals is the same as in the inertial frame.

How fast must a muon travel so that its mean lifetime is if its mean lifetime at rest is

Adistant galaxy is moving away from Earth with a speed that results in each wavelength received on Earth being shifted so that Find the speed of the galaxy relative to Earth.

Frames and are moving relative to each other along the and axes (which superpose). Observers at rest in the two frames set their clocks to when the two origins coincide. In frame event 1 occurs at and and event 2 occurs at and The events occur simultaneously in frame (a) Find the magnitude and direction of the velocity of relative to (b) At what time do both events occur as measured in S_? SSM

An interstellar spaceship travels from Earth to a star system 12 light-years away (as measured in Earths frame). The trip takes as measured by clocks on the spaceship. (a) What is the speed of the spaceship relative to Earth? (b) When the spaceship arrives, it sends an electromagnetic signal to Earth. How long after the spaceship leaves Earth will observers on Earth receive the signal?

The neutral pion has a mass of This particle can be created in a protonproton collision: Determine the threshold kinetic energy for the creation of a in a collision of a moving proton and a stationary proton. (See Problem 38.)

A rocket that has a proper length of moves away from a space station and in the direction at relative to an observer on the station. An astronaut stands at the rear of the rocket and fires a dart toward the front of the rocket at relative to the rocket. How long does it take the dart to reach the front of the rocket (a) as measured in the frame of the rocket, (b) as measured in the frame of the space station, and (c) as measured in the frame of the dart?

Using a simple thought experiment, Einstein showed that there is mass associated with electromagnetic radiation. Consider a box of length and mass resting on a frictionless surface. Attached to the left wall of the box is a light source that emits a directed pulse of radiation of energy which is completely absorbed at the right wall of the box. According to classical electromagnetic theory, the radiation carries momentum of magnitude (Equation 30-24). The box recoils when the pulse is emitted by the light source. (a) Find the recoil velocity of the box so that momentum is conserved when the light is emitted. (Because is small and is large, you may use classical mechanics.) (b) When the light is absorbed at the right wall of the box the box stops, so the total momentum of the system remains zero. If we neglect the very small velocity of the box, the time it takes for the radiation to travel across the box is Find the distance moved by the box in that time. (c) Show that if the center of mass of the system is to remain at the same place, the radiation must carry mass

Using the relativistic conservation of momentum and energy and the relation between energy and momentum for a photon prove that a free electron (an electron not bound to an atomic nucleus) cannot absorb or emit a photon.

When a moving particle that has a kinetic energy greater than the threshold kinetic energy strikes a stationary target particle, one or more particles may be created in the inelastic collision. Show that the threshold kinetic energy of the moving particle is given by Here is the sum of the masses of the particles prior to the collision, is the sum of the masses of the particles following the collision, and is the mass of the target particle. Use this expression to determine the threshold kinetic energy of protons incident on a stationary proton target for the production of a protonantiproton pair; compare your result with the result of Problem 38. SSM

A particle of mass decays into two identical particles, each of mass where Prior to the decay, the particle of mass has a total energy of in the laboratory reference frame. The velocities of the decay products are along the direction of motion of Find the velocities of the decay products in the laboratory reference frame.

A rod of proper length makes an angle with the axis in frame Show that the angle made with the axis in frame which is moving in the direction with speed is given by and that the length of the stick in is

Show that if a particle moves at an angle with the axis with speed in frame it moves at an angle with the axis in given by

For the special case of a particle moving with speed along the axis in frame show that its momentum and energy in frame a frame that is moving along the axis with velocity are related to its momentum and energy in by the transformation equations Compare these equations with the Lorentz transformation equations for and Notice that the quantities and transform in the same way as do and

The equation for the spherical wavefront of a light pulse that begins at the origin at time is Frame moves with velocity along the axis. Using the Lorentz transformation, show that such a light pulse also has a spherical wavefront in frame by showing that

In Problem 56, you showed that the quantity has the same value (zero) in both and A quantity that has the same value in all inertial frames is called a Lorentz invariant. From the results of Problem 55, the quantity must also be a Lorentz invariant. Show that this quantity has the value in both the and reference frames.

A long rod that is parallel to the axis is released from rest. Subsequently, it is in free fall with an acceleration of magnitude in the direction. An observer in a rocket ship moving with speed parallel to the axis passes by. Using the Lorentz transformations, show that the observer on the rocket ship will measure the rod to be bent into a parabolic shape. Is the parabola concave upward or concave downward?

Show that if and in (Equation 39-18a) are both positive and less than then is positive and less than (Hint: Let and where and are positive numbers that are less than 1.)

In reference frame the acceleration of a particle is Derive expressions for the acceleration components and of the particle in reference frame that is moving relative to S in the x direction with velocity v.