Chapter 20

Quite apart from effects due to the Earth's rotational and orbital motions, a laboratory frame is not strictly an inertial frame because a particle placed at rest there will not, in general, remain at rest; it will fall under gravity. Often, however, events happen so quickly that we can ignore free fall and treat the frame as inertial. Consider, for example, a $1.0-\mathrm{MeV}$ electron (for which $v=0.941c$ ) projected horizontally into a laboratory test chamber and moving through a distance of $20\mathrm{cm}$. (a) How long would it take, and $(b)$ how far would the electron fall during this interval? What can you conclude about the suitability of the laboratory as an inertial frame in this case?

A $100-\mathrm{MeV}$ electron, for which $v=0.999987c$, moves along the axis of an evacuated tube that has a length of $2.86\mathrm{m}$ as measured by a laboratory observer $S$ with respect to whom the tube is at rest. An observer $S^{\prime }$ moving with the electron, however, would see this tube moving past with speed $v$. What length would this observer measure for the tube?

A rod lies parallel to the $x$ axis of reference frame $S$, moving along this axis at a speed of $0.632c.$ Its rest length is $1.68\mathrm{m}$. What will be its measured length in frame $S?$

The mean lifetime of muons stopped in a lead block in the laboratory is measured to be $2.20\mu \mathrm{s}$. The mean lifetime of high-speed muons in a burst of cosmic rays observed from the Earth is measured to be $16.0\mu \mathrm{s}$. Find the speed of these cosmic ray muons.

An unstable high-energy particle enters a detector and leaves a track $1.05\mathrm{mm}$ long before it decays. Its speed relative to the detector was $0.992c.$ What is its proper lifetime? That is, how long would it have lasted before decay had it been at rest with respect to the detector?

A particle moves along the $x^{\prime }$ axis of frame $S^{\prime }$ with a speed of $0.43c$. Frame $S^{\prime }$ moves with a speed of $0.587c$ with respect to frame $S.$ What is the measured speed of the particle in frame $S?$

A spaceship of rest length $130\mathrm{m}$ drifts past a timing station at a speed of $0.740c.(a)$ What is the length of the spaceship as measured by the timing station? $(b)$ What time interval between the passage of the front and back end of the ship will the station monitor record?

A pion is created in the higher reaches of the Earth's atmosphere when an incoming high-energy cosmic-ray particle collides with an atomic nucleus. A pion so formed descends toward Earth with a speed of $0.99c.$ In a reference frame in which they are at rest, pions have a lifetime of $26\mathrm{ns}$. As measured in a frame fixed with respect to the Earth, how far will such a typical pion move through the atmosphere before it decays?

To circle the Earth in low orbit a satellite must have a speed of about $7.91\mathrm{km}/\mathrm{s}$. Suppose that two such satellites orbit the Earth in opposite directions. (a) What is their relative speed as they pass? Evaluate using the classical Galilean velocity transformation equation. (b) What fractional error was made because the (correct) relativistic transformation equation was not used?

What must be the value of the speed parameter $\beta$ if the Lorentz factor $\gamma$ is to be $(a)1.01?(b)10.0?(c)100?(d)$ $1000?$