Example 28.1 from College Physics, 28.2 Simultaneity And Time Dilation

Calculating Δt 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 $v = 0 . 950 c$ . The muon then travels at constant velocity and lives $1.52 μ s$ 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 Figure 28.7.)

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.

Figure 28.7 A muon in the Earth’s atmosphere lives longer as measured by an Earth-bound observer than measured by the muon’s internal clock.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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,
$Δ t = {γ Δ t}_{0},$
28.12

where
$γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} .$
28.13

4) Plug the knowns into the equation.
First find $γ$ .
$\begin{array}{l} γ & = & \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \\ = & \frac{1}{\sqrt{1 - \frac{(0.950 c)^{2}}{c^{2}}}} \\ = & \frac{1}{\sqrt{1 - (0.950)^{2}}} \\ = & 3.20. \end{array}$
28.14

Use the calculated value of $γ$ to determine $Δ t$ .
$\begin{array}{l} Δ t & = & {γ Δ t}_{0} \\ = & (3.20) (1.52 μ s) \\ = & 4.87 μ s \end{array}$
28.15
Discussion
One implication of this example is that since $γ = 3 . 20$ at $95 . 0%$ of the speed of light ( $v = 0 . 950 c$ ), 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 $0.950 c$ is said to be highly relativistic.

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Solution

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