Chapter 2: Special Relativity

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$\mathbf{656}\mathbf{,}$ $\mathbf{486}\mathbf{,}$$\mathbf{434}\mathbf{,}$and$\mathbf{410}\mathit{n}\mathit{m}$ . 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 ,${\mathit{v}}_{\mathbf{r}\mathbf{m}\mathbf{s}}\mathbf{=}\sqrt{\mathbf{2}{\mathbf{K}}_{\mathbf{B}}\mathbf{T}\mathbf{/}\mathbf{m}}$ where${\mathit{K}}_{\mathbf{B}}$ is the Boltzmann constant. Obtain a formula for the range of wavelengths in terms of the wavelength$\mathit{\lambda }$ of the spectral line, the atomic mass$\mathit{m}$ , and the temperature$\mathit{T}$. (Note: .${\mathit{v}}_{\mathbf{r}\mathbf{m}\mathbf{s}}\mathbf{<}\mathbf{<}\mathit{c}$) (b) Evaluate this range for the$\mathbf{656}\mathit{n}\mathit{m}$ hydrogen spectral line, assuming a temperature of$\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathit{K}$ .

To catch speeders, a police radar gun detects the beat frequency between the signal it emits and that which reflects off a moving vehicle. What would be the beat frequency for an emitted signal of 900 Mhz reflected from a car moving at 30 m/s ?

Bob is on Earth. Anna is on a spacecraft moving away from Earth at 0.6c . At some point in Anna's outward travel, Bob fires a projectile loaded with supplies out to Anna's ship. Relative to Bob, the projectile moves at 0.8c . (a) How fast does the projectile move relative to Anna? (b) Bob also sends a light signal, " Greetings from Earth:' out to Anna's ship. How fast does the light signal move relative to Anna?

According Anna, on Earth, Bob is on a spaceship moving at 0.8c toward Earth, and Carl, a little farther out. is on a spaceship moving at 0.9c toward Earth. (a) According to Bob, how fast and in what direction is Carl moving relative to himself (Bob)? (b) According to Bob, how fast is Carl moving relative to Earth?

Prove that if v and u' are less than c, it is impossible for a speed u greater than c to result from equation (2-l9b). [Hint: The product (c-u')(c-v) is positive.]

In a particle collider experiment, particle I is moving to the right at 0.99c and particle 2 to the left at 0.99c, both relative to the laboratory. What is the relative velocity of the two panicles according to (an observer moving with) particle 2?

Question: A light beam moves in the xy plane and has an x component of velocity of ${\mathit{u}}_{\mathbf{x}}$ (a) In terms of ${\mathit{u}}_{\mathbf{x}}$, and c, what is its y component? (b) Using equations (2-20a) and (2-20b). Calculate its velocity components in a frame moving in the x direction at speed$\mathit{v}\mathbf{=}{\mathit{u}}_{\mathbf{x}}$ and comment on your result.

A light beam moves at an angle$\mathit{\theta }$ with the x-axis as seen from frame S. Using the relativistic velocity transformation, find the components of its velocity when viewed from frame $\mathit{S}\mathbf{\text{'}}$. From these, verify explicitly that its speed is c.

You fire a light signal at $\mathbf{60}\mathbf{°}$north of west (a) Find the velocity component of this, signal according to an observer moving eastward relative to you at half the speed of light. From them. determine the magnitude and direction of the light signal's velocity according to this other observer. (b) Find the component according to a different observer, moving westward relative to you at half the speed of light.

At t=0, a bright beacon at the origin flashes, sending light uniformly in all directions. Anna is moving at speed v in the +x direction relative to the beacon and passes through the origin at t=0. (a) Show that according to Anna, the only light with a positive${\mathbf{x}}^{\mathbf{\text{'}}}$-component is that which in the beacon’s reference frame is within an angle $\mathit{\theta }\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\left(v/c\right)$of the +x -axis. (b) What are the limits of $\mathit{\theta }$ as v approaches 0 and as it approaches c? (c) The phenomenon is called the head-light effect. Why?