At what speed $v$ will the length of a $1.00~\mathrm{m}$ stick look $10.0 \%$ shorter $(90.0 \mathrm{~cm})$ ?

What is the speed v of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference $c-v$. Such speeds are reached in the Stanford Linear Accelerator, SLAC.

Suppose a spacecraft of mass $17,000 \mathrm{~kg}$ was accelerated to $0.22 c$. If you used the classical formula for kinetic energy, by what percentage would you be in error?

A spacecraft (reference frame $\mathrm{S}^{\prime}$ ) moves past Earth (reference frame S) at velocity $\overrightarrow{\mathbf{v}}$, which points along the $x$ and $x^{\prime}$ axes. The spacecraft emits a light beam (speed $c$ ) along its $y^{\prime}$ axis as shown in Fig. 36-17. (a) What angle $\theta$ does this light beam make with the $x$ axis in the Earth's reference frame?

Suppose a spacecraft of mass $17,000 \mathrm{~kg}$ was accelerated to $0.22 c$. How much kinetic energy would it have?

A radar "speed gun" emits microwaves of frequency $f_0=36.0 \mathrm{GHz}$. When the gun is pointed at an object moving toward it at speed $v$, the object senses the microwaves at the Doppler-shifted frequency $f$. The moving object reflects these microwaves at this same frequency $f$. The stationary radar apparatus detects these reflected waves at a (double) Doppler-shifted frequency $f^{\prime}$. The gun combines its emitted wave at $f_0$ and its detected wave at $f^{\prime}$. These waves interfere, creating a beat pattern whose beat frequency is $f_{\text {beat }}=f^{\prime}-f_0$. (a) Show that

$$v \approx \frac{c f_{\text {beat }}}{2 f_0} \text {, }$$

if $f_{\text {beat }} \ll f_0$. If $f_{\text {beat }}=6670 \mathrm{~Hz}$, what is $v(\mathrm{~km} / \mathrm{h})$ ?

A healthy astronaut's heart rate is 60 beats/min. Flight doctors on Earth can monitor an astronaut's vital signs remotely while in flight. How fast would an astronaut have to be flying away from Earth in order for the doctor to measure her having a heart rate of 30 beats/min?

Calculate the kinetic energy and momentum of a proton $\left(\mathrm{m}=1.67 \times 10^{-27} \mathrm{~kg}\right)$ traveling $7.85 \times 10^7 \mathrm{~m} / \mathrm{s}$. By what percentages would your calculations have been in error if you had used classical formulas?

Starting from the said equation, show that the Doppler shift in wavelength is

$$\frac{\Delta \lambda}{\lambda}=\frac{v}{c}$$

if $v \ll c$.

A quasar emits familiar hydrogen lines whose wave-lengths are $2.5$ times longer than what we measure in the laboratory.$(b)$ What result would you obtain if you used the "classical" Doppler shift discussed in Chapter 16 ?

A quasar emits familiar hydrogen lines whose wave-lengths are $2.5$ times longer than what we measure in the laboratory.(a) What is the speed of this quasar?

The nearest star to Earth is Proxima Centauri, $4.3$ lightyears away.(b) How long does the trip take according to Earth observers?

Show that $p_x, p_y, p_z$, E/c transform according to the Lorentz transformation in the same way as x, y, z, ct.

The nearest star to Earth is Proxima Centauri, $4.3$ lightyears away.(a) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in $4.6$ years, as measured by travelers on the spacecraft?