Chapter 35

Radar-based speed detection works by sending an electromagnetic wave out from a source and examining the Doppler shift of the reflected wave. Suppose a wave of frequency \(10.6 \mathrm{GHz}\) is sent toward a car moving away at a speed of \(32.0 \mathrm{~km} / \mathrm{h}\). What is the difference between the frequency of the wave emitted by the source and the frequency of the wave an observer in the car would detect?

A HeNe laser onboard a spaceship moving toward a remote space station emits a beam of red light toward the space station. The wavelength of the beam, as measured by a wavelength meter on board the spaceship, is \(632.8 \mathrm{nm}\). If the astronauts on the space station see the beam as a blue beam of light with a measured wavelength of \(514.5 \mathrm{nm},\) what is the relative speed of the spaceship with respect to the space station? What is the shift parameter \(z\) in this case?

Sam sees two events as simultaneous: (i) Event \(A\) occurs at the point (0,0,0) at the instant 0: 00: 00 universal time; (ii) Event \(B\) occurs at the point \((500, \mathrm{~m}, 0,0)\) at the same moment. Tim, moving past Sam with a velocity of \(0.999 c \hat{x}\), also observes the two events. a) Which event occurred first in Tim's reference frame? b) How long after the first event does the second event happen in Tim's reference frame?

Use the relativistic velocity addition to reconfirm that the speed of light with respect to any inertial reference frame is \(c\). Assume one-dimensional motion along a common \(x\) -axis.

You are driving down a straight highway at a speed of \(v=50.0 \mathrm{~m} / \mathrm{s}\) relative to the ground. An oncoming car travels with the same speed in the opposite direction. With what relative speed do you observe the oncoming car?

A rocket ship approaching Earth at \(0.90 c\) fires a missile toward Earth with a speed of \(0.50 c,\) relative to the rocket ship. As viewed from Earth, how fast is the missile approaching Earth?

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.65 c\). a) Calculate the total distance Alice traveled during the trip, as measured by Alice. b) With the aforementioned total distance, calculate the total time duration for the trip, as measured by Alice.

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.650 c .\) This can be viewed in terms of Alice's reference frame. a) Show that Alice must travel with a speed of \(0.914 c\) to establish a relative speed of \(0.650 c\) with respect to Earth when Alice is returning back to Earth. b) Calculate the time duration for Alice's return flight toward Earth with the aforementioned speed.

Robert, standing at the rear end of a railroad car of length \(100 . \mathrm{m},\) shoots an arrow toward the front end of the car. He measures the velocity of the arrow as \(0.300 c\). Jenny, who was standing on the platform, saw all of this as the train passed her with a velocity of \(0.750 c .\) Determine the following as observed by Jenny: a) the length of the car b) the velocity of the arrow c) the time taken by arrow to cover the length of the car d) the distance covered by the arrow

Consider motion in one spatial dimension. For any velocity \(v,\) define parameter \(\theta\) via the relation \(v=c \tanh \theta\) where \(c\) is the vacuum speed of light. This quantity is variously called the velocity parameter or the rapidity corresponding to velocity \(v\). a) Prove that for two velocities, which add according to the Lorentzian rule, the corresponding velocity parameters simply add algebraically, that is, like Galilean velocities. b) Consider two reference frames in motion at speed \(v\) in the \(x\) -direction relative to one another, with axes parallel and origins coinciding when clocks at the origin in both frames read zero. Write the Lorentz transformation between the two coordinate systems entirely in terms of the velocity parameter corresponding to \(v\), and the coordinates.