Peoples Physics Book Version-2

http://www.ck12.org Chapter 23. Special and General Relativity Version 2

23.2 Relativity Example

Question: The muon particle(μ−)has a half-life of 2. 20 × 10 −^6 s. Most of these particles are produced in the
atmosphere, a good 5-20km above Earth, yet we see them all the time in our detectors here on Earth. In this problem
you will find out how it is possible that these particles make it all the way to Earth with such a short lifetime.

a) Calculate how far muons could travel before half decayed, without using relativity and assuming a speed of 0. 999 c
(i.e. 99.9% of the speed of light)

b) Now calculateγfor this muon.

c) Calculate its ’relativistic’ half-life.

d) Now calculate the distance before half decayed using relativistic half-life and express it in kilometers. (This has
been observed experimentally. This first experimental verification of time dilation was performed by Bruno Rossi at
Mt. Evans, Colorado in 1939.)

Answer:

a) To calculate the distance that the muon particle could travel we will use the equation for distance and then plug in
the known values to get the answer.

d=v×t= (. 999 × 3 × 108 m/s)×( 2. 20 × 10 −^6 s) = 659. 34 m

b) To solve forγ, we must first solve forβ.

β=

v

c

=

. 999 × 3 × 108 m/s
3 × 108 m/s

=. 999

Now we can solve forγ.

γ=

1

√

1 −β^2

=

1

√

1 −. 9992

= 22. 4

c) To calculate the muon particle’s relativistic half-life, we will use theγvalue we calculated in part b) and the
equation for determining relativistic half-life.

T′=γT= 22. 4 × 2. 20 × 10 −^6 s= 4. 92 × 10 −^5

d) To calculate the distance the muon particle can travel we will use the same distance equation but we will use the
new half-life instead of the non-relativistic half-life.

d=vt= (. 999 × 3 × 108 m/s)( 4. 92 × 10 −^5 s) = 14 ,700m

Now we will convert this into km.

14700m×

1km

1000m

= 14 .7km