(a) 492.2 ns
The lifetime of the particle in the frame of reference of the laboratory is given by
![T= \frac{T_0}{\sqrt{1-(\frac{v}{c})^2}}](https://tex.z-dn.net/?f=T%3D%20%5Cfrac%7BT_0%7D%7B%5Csqrt%7B1-%28%5Cfrac%7Bv%7D%7Bc%7D%29%5E2%7D%7D)
where
is the proper lifetime
v is the speed of the particle
c is the speed of the light
For this particle we have:
![T_0 = 161 ns](https://tex.z-dn.net/?f=T_0%20%3D%20161%20ns)
![v=0.945c](https://tex.z-dn.net/?f=v%3D0.945c)
Substituting into the equation, we get:
![T= \frac{161}{\sqrt{1-(0.945)^2}}=492.2 ns](https://tex.z-dn.net/?f=T%3D%20%5Cfrac%7B161%7D%7B%5Csqrt%7B1-%280.945%29%5E2%7D%7D%3D492.2%20ns)
(b) 139.5 m
In the reference frame of the laboratory, the distance travelled by the particle is given by
![L = v T](https://tex.z-dn.net/?f=L%20%3D%20v%20T)
where
v is the speed of the particle
T is the lifetime of the particle in the laboratory frame of reference (found in part a)
The speed of the particle is
![v=0.945c=0.945(3\cdot 10^8)=2.84\cdot 10^8 m/s](https://tex.z-dn.net/?f=v%3D0.945c%3D0.945%283%5Ccdot%2010%5E8%29%3D2.84%5Ccdot%2010%5E8%20m%2Fs)
The lifetime of the particle in the laboratory's frame of reference is
![T=492.2 ns = 492.2\cdot 10^{-9}s](https://tex.z-dn.net/?f=T%3D492.2%20ns%20%3D%20492.2%5Ccdot%2010%5E%7B-9%7Ds)
And substituting into the equation, we find:
![L=(2.84\cdot 10^8)(492.2\cdot 10^{-9})=139.5 m](https://tex.z-dn.net/?f=L%3D%282.84%5Ccdot%2010%5E8%29%28492.2%5Ccdot%2010%5E%7B-9%7D%29%3D139.5%20m)
(c) 426.5 m
Here we have to calculate the distance travelled by the particle in its frame of reference. This can be calculated by using the equation
![L=L_0 \sqrt{1-(\frac{v}{c})^2}](https://tex.z-dn.net/?f=L%3DL_0%20%5Csqrt%7B1-%28%5Cfrac%7Bv%7D%7Bc%7D%29%5E2%7D)
where
is the distance travelled measured by an observer moving with the particle
L is the distance travelled measured by an observer in the laboratory
We already know that
L = 139.5 m
So, solving the formula for
,
![L_0 = \frac{L}{\sqrt{1-(\frac{v}{c})^2}}=\frac{139.5}{\sqrt{1-(0.945)^2}}=426.5 m](https://tex.z-dn.net/?f=L_0%20%3D%20%5Cfrac%7BL%7D%7B%5Csqrt%7B1-%28%5Cfrac%7Bv%7D%7Bc%7D%29%5E2%7D%7D%3D%5Cfrac%7B139.5%7D%7B%5Csqrt%7B1-%280.945%29%5E2%7D%7D%3D426.5%20m)