Chapter 2: Special Relativity

A famous experiment detected 527 muons per hour at the top of Mt. Washington, New Hemisphere, elevation 1910 m . At sea level, the same equipment detected 395 muons per hour. A discriminator selected for muons whose speed was between 0.9950 c and 0.9954c . Given that the mean lifetime T of a muon in a frame in which it is at rest is $\mathbf{2}\mathbf{.}\mathbf{2}\mathbf{}\mathit{\mu }\mathit{s}$and that in this frame the number of muons decays exponentially with time according to $\mathbf{N}\mathbf{=}{\mathbf{N}}_{\mathbf{0}}{\mathbf{e}}^{\mathbf{-}\mathbf{i}\mathbf{/}\mathbf{T}}$, show that the results obtained in the experiment are sensible.

In the frame in which they are at rest, the number of muons at time t is given by

$\mathbf{N}\mathbf{=}{\mathbf{N}}_{\mathbf{0}}{\mathbf{e}}^{\mathbf{-}\mathbf{l}\mathbf{/}\mathbf{T}}$

Where ${\mathbf{N}}_{\mathbf{0}}$ is the number at t = 0 and T is the mean life-time 2.2 $\mathit{\mu }\mathit{s}$. (a) If muons are produced at a height of 4.0 km , heading toward the ground at 0.93 c, what fraction will survive to reach the ground? (b) What fraction would reach the ground if classical mechanics were valid?

Question: The Lorentz transformation equations have x and t and x^' and t^'. Why no v and v^' ?

An experimenter determines that a particle created at one end of the laboratory apparatus moved at 0.94c and survived for $\mathbf{0}\mathbf{.}\mathbf{032}\mathbf{}\mathit{\mu }\mathit{s}$decaying just as it reached the other end. (a) According to the experimenter. How far did the particle move? (b) In its own frame of reference, how long did the particle survive? (c) According to the particle, what was the length of the laboratory apparatus?

A muon has a mean lifetime of 2.2$\mathbf{\mu s}$in its rest frame. Suppose muons are travelling at 0.92c relative to Earth. What is the mean distance a muon will travel as measured by an observer on Earth?

A pion is an elementary particle that on average disintegrates $\mathbf{2}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathbf{}\mathbf{s}$after creation in a frame at rest relative to the pion. An experimenter finds that pions created in the laboratory travel 13m on average before disintegrating. How fast are the pions traveling through the lab?

Anna and Bob have identical spaceship 60 m long. The diagram shows Bob’s observations of Anna’s ship, which passes at a speed of $\mathit{c}\mathbf{/}\sqrt{\mathbf{2}}$. Clocks at the back of both ships read 0 just as they pass. Bob is at the center of his ship and at t = 0 on his wrist watch peers at a second clock on Anna’s ship.

(a) What does this clock read?

(b) Later, the back of Anna’s ship passes Bob. At what time does this occur according to Bob?

(c) What will observers in Bob’s frame see on Anna’s two clocks at this time?

(d) Identify two events that show time dilation and two that show length contraction according to Anna.

Demonstrate that equations (2 - 12) and (2 - 13) become the classical transformation equations (2 - 1) when v << c, except when applied to events very far away, in which case time is still not absolute.

Planet W is 12ly from Earth. Anna and Bob are both 20 yr old. Anna travels to Planet W at 0.6c, quickly turns around, and returns to Earth at . How old will Anna and Bob be when Anna gets back?

Anna and Bob are both born just as Anna's spaceship passes Earth at $0.9c$. According to Bob on Earth planet Z is a fixed away. As Anna passes planet Z on her continuing onward journey, what will be

(a) Bob's age according to Bob

(b) Bob's age according to Anna

(c) Anna's age according to Anna

(d) Anna's age according to Bob