28.2 Simultaneity And Time Dilation

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the Earth on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the pre…

(a) What is $\gamma$ if $v = 0 . \text{250} c$? (b) If $v = 0 . \text{500} c$?

Particles called $\pi$-mesons are produced by accelerator beams. If these particles travel at $2 . \text{70} \times \text{10}^{8} \text{m}/\text{s}$ and live $2 . \text{60} \times \text{10}^{- 8} \text{s}$ when at rest relative to an observer, how long do they live as viewed in the laboratory?

A neutral $\pi$-meson is a particle that can be created by accelerator beams. If one such particle lives $1 . \text{40} \times \text{10}^{- \text{16}} \text{s}$ as measured in the laboratory, and $0 . \text{840} \times \text{10}^{- \text{16}} \text{s}$ when at rest relative to an observer, what is its velocity relative to the laboratory?

If relativistic effects are to be less than 1%, then $\gamma$ must be less than 1.01. At what relative velocity is $\gamma = 1 . \text{01}$?

(a) At what relative velocity is $\gamma = 1 . \text{50}$? (b) At what relative velocity is $\gamma = \text{100}$?

(a) Find the value of $\gamma$ for the following situation. An Earth-bound observer measures 23.9 h to have passed while signals from a high-velocity space probe indicate that $\text{24}.\text{0 h}$ have passed on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?