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Calculating Δ t For a relativistic event: how long does a speedy muon live?

Suppose a cosmic ray colliding with a nucleus in the Earth’s upper atmosphere produces a muon that has a velocity v = 0 . 950 c size 12{v=0 "." "950"c} {} . The muon then travels at constant velocity and lives 1 . 52 μ s size 12{1 "." "52"` ital "μs"} {} as measured in the muon’s frame of reference. (You can imagine this as the muon’s internal clock.) How long does the muon live as measured by an Earth-bound observer? (See [link] .)

A muon is moving far above the earth. A teenage boy is looking towards the muon. A velocity vector arrow V starting from Muon is pointing toward the boy. A clock depicting time delta-t-zero is shown near the muon, and another time clock depicting the time delta-t is shown near the boy.
A muon in the Earth’s atmosphere lives longer as measured by an Earth-bound observer than measured by the muon’s internal clock.

Strategy

A clock moving with the system being measured observes the proper time, so the time we are given is Δ t 0 = 1 . 52 μ s . The Earth-bound observer measures Δ t as given by the equation Δ t = γ Δ t 0 . Since we know the velocity, the calculation is straightforward.

Solution

1) Identify the knowns. v = 0 . 950 c , Δ t 0 = 1 . 52 μ s

2) Identify the unknown. Δ t

3) Choose the appropriate equation.

Use,

Δ t = γ Δ t 0 ,

where

γ = 1 1 v 2 c 2 .

4) Plug the knowns into the equation.

First find γ .

γ = 1 1 v 2 c 2 = 1 1 ( 0.950 c ) 2 c 2 = 1 1 ( 0.950 ) 2 = 3.20.

Use the calculated value of γ size 12{γ} {} to determine Δ t .

Δ t = γ Δ t 0 = ( 3.20 ) ( 1.52 μ s ) = 4.87 μ s

Discussion

One implication of this example is that since γ = 3 . 20 size 12{γ=3 "." "20"} {} at 95 . 0% size 12{"95" "." 0%} {} of the speed of light ( v = 0 . 950 c size 12{v=0 "." "950"c} {} ), the relativistic effects are significant. The two time intervals differ by this factor of 3.20, where classically they would be the same. Something moving at 0 . 950 c size 12{0 "." "950"c} {} is said to be highly relativistic.

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Another implication of the preceding example is that everything an astronaut does when moving at 95 . 0% size 12{"95" "." 0%} {} of the speed of light relative to the Earth takes 3.20 times longer when observed from the Earth. Does the astronaut sense this? Only if she looks outside her spaceship. All methods of measuring time in her frame will be affected by the same factor of 3.20. This includes her wristwatch, heart rate, cell metabolism rate, nerve impulse rate, and so on. She will have no way of telling, since all of her clocks will agree with one another because their relative velocities are zero. Motion is relative, not absolute. But what if she does look out the window?

Real-world connections

It may seem that special relativity has little effect on your life, but it is probably more important than you realize. One of the most common effects is through the Global Positioning System (GPS). Emergency vehicles, package delivery services, electronic maps, and communications devices are just a few of the common uses of GPS, and the GPS system could not work without taking into account relativistic effects. GPS satellites rely on precise time measurements to communicate. The signals travel at relativistic speeds. Without corrections for time dilation, the satellites could not communicate, and the GPS system would fail within minutes.

The twin paradox

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to the Earth would age less than her Earth-bound twin. Imagine the astronaut moving at such a velocity that γ = 30 . 0 size 12{γ="30" "." 0} {} , as in [link] . A trip that takes 2.00 years in her frame would take 60.0 years in her Earth-bound twin’s frame. Suppose the astronaut traveled 1.00 year to another star system. She briefly explored the area, and then traveled 1.00 year back. If the astronaut was 40 years old when she left, she would be 42 upon her return. Everything on the Earth, however, would have aged 60.0 years. Her twin, if still alive, would be 100 years old.

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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