Question

Explain time dilation with an example.

Answer 1

Hint:Moving clock ticks slower than rest one.
We can have equation for time dilation \[\Delta t = \gamma \Delta \tau \]
We can substitute \[\gamma = \dfrac{1}{{\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} }}\]

Complete step by step answer:
We can consider two events occurring in the same location, if we measure time interval from an observer such that observer moving inertial frame with a constant high velocity, then there is a dilation of measurement of time interval with respect to the measurement of time interval in rest frame. This is called time dilation.
  Time dilation is given by the equation,
\[\Delta t = \gamma \Delta \tau \]
Where \[\gamma = \dfrac{1}{{\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} }}\]
\[\Delta \tau \]Is the time interval in a stationary frame.
\[v\]is the velocity of the moving frame.
\[c\]Is the velocity of light.
We have a best example for the time dilation is muon decay. The muon is a subatomic particle which has a very short lifetime of \[\tau = 2 \times {10^{ - 6}}s\]and it moves very fast nearly the speed of light around \[0.9999c\] .
By classically it have move by classically distance \[ = 0.9999c \times 2 \times {10^{ - 6}} = 0.999 \times 3 \times {10^8} \times 2 \times {10^{ - 6}} \cong 0.6km\]
 When it moves the decay time much more according to the time dilatation,
It can be
\[\Delta t = \gamma \Delta \tau \]
\[\gamma = \sqrt {\dfrac{1}{{1 - {{.9999}^2}}}} \; = 70.7\]
\[\Delta t = 70.7 \times 2 \times {10^{ - 6}} = 1.414 \times {10^{ - 4}}s\]
Now we can calculate the distance it can move,
\[ = 0.9999c \times 1.414 \times {10^{ - 4}} = 0.999 \times 3 \times {10^8} \times 1.414 \times {10^{ - 4}} \cong 42km\].
The classically we can’t predict the muon at atmosphere of earth, but in real scientist found the muon particle at earth surface,
Approximately only ten kilometer is the thickness of earth's atmosphere, by relativistic idea the muon particle can delay the life time by a factor of seventy. So it can move more than we expected classically, and touch the ground.

Note:We can have another example for time dilation is twin paradox.Like time dilation, we have another phenomena called length contraction.For length contraction, shortening of the length between moving frame and rest frame.The theory of relativity is proposed by the great scientist Albert Einstein. Velocity of light is constant in all frames of reference.