Chapter 2: Special Relativity

Verify that the special case ${\mathit{x}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{-}{\mathit{t}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{}\mathit{x}\mathbf{=}\mathbf{0}$leads to equation (2-6) when inserted in linear transformations (2-4) and that special case. ${\mathit{x}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{-}\mathit{c}{\mathit{t}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{}\mathit{x}\mathbf{=}\mathit{c}\mathit{t}$in turn leads to (2-8).

If an object actually occupies less space physically when moving. It cannot depend on the direction we define as positive. As we know, an object aligned with the direction of relative motion is contracted whether it is fixed in frame S and viewed from S'. or the other way around. Use this idea to argue that distances along the y- and y'-axes cannot differ at all. Consider a post of length ${\mathit{L}}_{\mathbf{0}}$ fixed in frame S, jutting up from the origin along the +y-axis. with a saw at the top poised to slice off anything extending any higher in the passing frame S'. Also consider an identical post fixed in frame S'. What happens when the origins cross?

Through a window in Carl's spaceship, passing al 0.5c, you watch Carl doing an important physics calculation. By your watch it takes him 1 min. How much time did Carl spend on his calculation?

According to an observer on Earth, a spacecraft whizzing by at 0.6c is 35 m long. What is the length of the spacecraft according to the passengers onboard?

A pole-vaulter holds a $16\text{ft}$ pole, A barn has doors at both ends, $10\text{ft}$apart. The pole-vaulter on the outside of the barn begins running toward one of the open barn doors, holding the pole level in the direction he's running. When passing through the barn, the pole fits (barely) entirely within the barn all at once. (a) How fast is the pole-vaulter running? (b) According to whom-the pole-vaulter or an observer stationary in the barn--does the pole fit in all at once? (c) According to the other person, which occurs first the front end of the pole leaving the bam or the back end entering, and (d) what is the time interval between these two events?

Bob is watching Anna fly by in her new high-speed plane, which Anna knows to be $\mathbf{60}\mathit{m}$in length. As a greeting, Anna turns on two lights simultaneously, one at the front and one at the tail. According to Bob, the lights come $\mathbf{40}\mathbf{}\mathit{n}\mathit{s}$apart.

(a) Which comes on first?

(b) How fast is the plane moving?

Bob and Bob Jr. stand open doorways. At opposite ends of an aero plane hangar $25\text{m}$ long..Anna owns a spaceship 40m, long as it sits on the .runway. Anna takes off in her spaceship, swoops through the hangar at constant velocity. At precisely time zero onboth Bob's, clock and Anna's, Bob see Anna at. front of her spaceship reach his doorway. At time zero on his clock, Bob Jr. sees the tail of Anna', spaceship .his doorway. (a) How fast is Anna·, spaceship moving? (b) What will Anna's clock read when She sees the tail of spaceship at the doorway where Bob Jr standing her? (c) How far will the Anna say the front of her spaceship is from Bob at this time?

Refer to Figure 2.18. (a) How long is a spaceship? (b) At what speed do the ships move relative to one another? (c) Show that Anna’s times are in accord with the Lorentz transformation equations. (d) Sketch a set of diagrams showing Anna’s complementary view of the passing of the ships. Include times in both frames.

Question: A friend says, "It makes no sense that Anna could turn on lights in her hands simultaneously in her frame but that they don't turn on simultaneously in another-she decides to do it, and its done." By considering the tractable, if somewhat unrealistic, situation of Anna's thought being communicated to her hands by light signals, answer this objection.

You are on a high-speed train, travelling at a decent clip: 0.8C . On the ground are two signal stations 5km apart, each with a status-reporting sign, which always give simultaneous reports. At precisely noon on the train’s clocks, the conductor at the front of the train passes one station and sees a sign reading “All clear,” and another employee at the back passes the other station and sees a sign reading “Severe Electrical Storms Reported! Slow to 0.1c !” (a) How long is the train? (b) Should it slow down? (c) Suppose that both reporting signs display the time very precisely, updated every microsecond. By how much would the two observed time readings differ, if at all?