Chapter 35

Suppose you are explaining the theory of relativity to a friend, and you have told him that nothing can go faster than $300,000\mathrm{km}/\mathrm{s}$. He says that is obviously false: Suppose a spaceship traveling past you at $200,000\mathrm{km}/\mathrm{s}$, which is perfectly possible according to what you are saying, fires a torpedo straight ahead whose speed is $200,000\mathrm{km}/\mathrm{s}$ relative to the spaceship, which is also perfectly possible; then, he says, the torpedo's speed is $400,000\mathrm{km}/\mathrm{s}$. How would you answer him?

Consider a positively charged particle moving at constant speed parallel to a current-carrying wire, in the direction of the current. As you know (after studying Chapters 27 and 28), the particle is attracted to the wire by the magnetic force due to the current. Now suppose another observer moves along with the particle, so according to him the particle is at rest. Of course, a particle at rest feels no magnetic force. Does that observer see the particle attracted to the wire or not? How can that be? (Either answer seems to lead to a contradiction: If the particle is attracted, it must be by an electric force because there is no magnetic force, but there is no electric field from a neutral wire; if the particle is not attracted, you see that the particle is, in fact, moving toward the wire.)

At rest, a rocket has an overall length of $L.$ A garage at rest (built for the rocket by the lowest bidder) is only $L/2$ in length. Luckily, the garage has both a front door and a back door, so that when the rocket flies at a speed of $v=0.866c$, the rocket fits entirely into the garage. However, according to the rocket pilot, the rocket has length $L$ and the garage has length $L/4$. How does the rocket pilot observe that the rocket does not fit into the garage?

A rod at rest on Earth makes an angle of ${10}^{\circ }$ with the $x$ -axis. If the rod is moved along the $x$ -axis, what happens to this angle, as viewed by an observer on the ground?

An astronaut in a spaceship flying toward Earth's Equator at half the speed of light observes Earth to be an oblong solid, wider and taller than it appears deep, rotating around its long axis. A second astronaut flying toward Earth's North Pole at half the speed of light observes Earth to be a similar shape but rotating about its short axis. Why does this not present a contradiction?

Consider two clocks carried by observers in a reference frame moving at speed $v$ in the positive $x$ -direction relative to ours. Assume that the two reference frames have parallel axes, and that their origins coincide when clocks at that point in both frames read zero. Suppose the clocks are separated by distance $l$ in the $x^{\prime }-$ direction in their own reference frame; for instance, $x^{\prime }=0$ for one clock and $x^{\prime }=I$ for the other, with $y^{\prime }=z^{\prime }=0$ for both. Determine the readings $t^{\prime }$ on both clocks as functions of the time coordinate $t$ in our reference frame.

Prove that in all cases, two sub-light-speed velocities "added" relativistically will always yield a sub-light-speed velocity. Consider motion in one spatial dimension only.

A famous result in Newtonian dynamics is that if a particle in motion collides elastically with an identical particle at rest, the two particles emerge from the collision on perpendicular trajectories. Does the same hold in the special theory of relativity? Suppose a particle of rest mass $m$ and total energy $E$ collides with an identical particle at rest, the same two particles emerging from the collision with new velocities. Are those velocities necessarily perpendicular? Explain.

Find the speed of light in feet per nanosecond, to three significant figures.

Find the value of $g$, the gravitational acceleration at Earth's surface, in light-years per year, to three significant figures.