5.3 Time Dilation

What is $\gamma$ if $v = 0.650 c ?$

a. A particle travels at $1.90 \times 10^{8} \text{m}/\text{s}$ and lives $2.10 \times 10^{−8} \text{s}$ when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

b. Spacecraft A and B pass in opposite directions at a relative speed of $4.00 \times 10^{7} \text{m}/\text{s} .$ An internal clock in spacecraft A causes it to emit a radio signal for 1.00 s. The computer in spacecraft B corrects for the beginning and end of the signal having traveled different distances, to calculate the time interval during which ship A was emitting the signal. What is the time interval that the computer in spacecra…

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 the interval of proper time?

(a) What is $\gamma$ if $v = 0.250 c ?$ (b) If $v = 0.500 c ?$

Particles called $\pi$-mesons are produced by accelerator beams. If these particles travel at $2.70 \times 10^{8} \text{m}/\text{s}$ and live $2.60 \times 10^{−8} \text{s}$ 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 $1.40 \times 10^{−16} \text{s}$ as measured in the laboratory, and $0.840 \times 10^{−16} \text{s}$ 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 $\gamma = 1.01 ?$