28.2 Simultaneity and time dilation  (Page 5/8)

1. What is $\gamma$ if $v=0\text{.650}c$ ?

Solution

2. A particle travels at $1\text{.}\text{90}\times {\text{10}}^8\text{m/s}$ and lives $2\text{.}\text{10}\times {\text{10}}^{-8}\mathrm{s}$ when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

Does motion affect the rate of a clock as measured by an observer moving with it? Does motion affect how an observer moving relative to a clock measures its rate?

To whom does the elapsed time for a process seem to be longer, an observer moving relative to the process or an observer moving with the process? Which observer measures proper time?

How could you travel far into the future without aging significantly? Could this method also allow you to travel into the past?

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

(a) What is $\gamma$ if $v=0\text{.}\text{100}c$ ? (b) If $v=0\text{.}\text{900}c$ ?

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

Suppose a particle called a kaon is created by cosmic radiation striking the atmosphere. It moves by you at $0\text{.}\text{980}c$ , and it lives $1\text{.}\text{24}\times {\text{10}}^{-8}\text{s}$ when at rest relative to an observer. How long does it live as you observe it?

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

A neutron lives 900 s when at rest relative to an observer. How fast is the neutron moving relative to an observer who measures its life span to be 2065 s?

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

If relativistic effects are to be less than 3%, then $\gamma$ must be less than 1.03. At what relative velocity is $\gamma =1\text{.}\text{03}$ ?

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

(a) At what relative velocity is $\gamma =2\text{.}\text{00}$ ? (b) At what relative velocity is $\gamma =\text{10}\text{.}0$ ?

Unreasonable Results

(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 $\text{24.0 h}$ have passed on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?