(a) 492.2 ns
The lifetime of the particle in the frame of reference of the laboratory is given by

where
is the proper lifetime
v is the speed of the particle
c is the speed of the light
For this particle we have:


Substituting into the equation, we get:

(b) 139.5 m
In the reference frame of the laboratory, the distance travelled by the particle is given by

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

The lifetime of the particle in the laboratory's frame of reference is

And substituting into the equation, we find:

(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

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
,
