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Chapter 2: 28E (page 63)
The diagram shows Bob's view of the passing of two identical spaceship. Anna's and his own, where . The length of either spaceship in its rest frame is . What are the readings on Anna', two unlabelled clocks?
The unlabelled readings from Anna’s clocks are and .
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Question: A 1 kg object moves at 0.8c relative to Earth.
(a) Calculate the momentum and the energy of the object.
(b) Determine the Lorentz transformation matrix from the earth’s frame to the object’s frame.
(c) Find the momentum and total energy of the object in the new frame via matrix multiplication.
For reasons having to do with quantum mechanics. a given kind of atom can emit only certain wavelengths of light. These spectral lines serve as a " fingerprint." For instance, hydrogen's only visible spectral lines are and . If spectra/ lines were ofabsolutely precise wavelength. they would be very difficult to discern. Fortunately, two factors broaden them: the uncertainty principle (discussed in Chapter 4) and Doppler broadening. Atoms in a gas are in motion, so some light will arrive that was emitted by atoms moving toward the observer and some from atoms moving away. Thus. the light reaching the observer will Cover a range ofwavelengths. (a) Making the assumption that atoms move no foster than their rms speed-given by , where is the Boltzmann constant. Obtain a formula for the range of wavelengths in terms of the wavelength of the spectral line, the atomic mass , and the temperature. (Note: .) (b) Evaluate this range for the hydrogen spectral line, assuming a temperature of .
Verify that the special case leads to equation (2-6) when inserted in linear transformations (2-4) and that special case. in turn leads to (2-8).
In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length-contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about 30g are fatal, and there are serious concerns about the effects of prolonged accelerations greater than 1g. Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity.
(a) Though traveller Anna accelerates, Bob, being on near-inertial Earth, is a reliable observer and will see less time go by on Anna's clock (dt') than on his own (dt). Thus,, whereuis Anna's instantaneous speed relative to Bob. Using the result of Exercise 117(c), withgreplacingF/m, substitute for u, then integrate to show that
(b) How much time goes by for observers on Earth as they “see” Anna age 20 years?
(c) Using the result of Exercise 119, show that when Anna has aged a timet’, she is a distance from Earth (according to Earth observers) of
(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20. How far away from Earth will she end up and how much time will have passed on Earth?
Particle , of mass role="math" localid="1657551290561" , moving at role="math" localid="1657551273841" relative to the lab, collides head-on with particle , of mass , moving at relative to the lab. Afterward, there is a single stationary object. Find in terms of role="math" localid="1657551379188"
(a)
(b) the mass of the final stationary object; and
(c) the change in kinetic energy in this collision.
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