Answer:
Q1) Time taking by particle to travel the 40 km wrt. earth =
sec.
Q2) The distance traveled by particle where the particle created to surface of earth wrt. particle's frame = 3.84 km.
Q3) The time taking by particle to travel from where it is created to the surface of the earth =
sec.
Explanation:
Given :
Speed of particle wrt. earth ![v=0.99537c](https://tex.z-dn.net/?f=v%3D0.99537c)
Distance between where particle is created and earth surface = 40 km
we know that,
⇒ ![v = \frac{x}{t}](https://tex.z-dn.net/?f=v%20%3D%20%5Cfrac%7Bx%7D%7Bt%7D)
Where
,
, we know speed of light ![c = 3 \times10^{8}](https://tex.z-dn.net/?f=c%20%3D%203%20%5Ctimes10%5E%7B8%7D)
∴ ![t = \frac{x}{v}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7Bx%7D%7Bv%7D)
![= \frac{40 \times10^{3} }{0.99537\times3\times10^{8} }](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B40%20%5Ctimes10%5E%7B3%7D%20%7D%7B0.99537%5Ctimes3%5Ctimes10%5E%7B8%7D%20%7D)
![t = 1.34\times10^{-6} sec](https://tex.z-dn.net/?f=t%20%3D%201.34%5Ctimes10%5E%7B-6%7D%20sec)
∴ Thus, time taking by particle to travel the 40 km wrt. earth ![t = 1.34\times10^{-6} sec](https://tex.z-dn.net/?f=t%20%3D%201.34%5Ctimes10%5E%7B-6%7D%20sec)
According to the lorentz transformation,
⇒ ![l = l_{o} \sqrt{1-\frac{v^2}{c^2} }](https://tex.z-dn.net/?f=l%20%3D%20l_%7Bo%7D%20%5Csqrt%7B1-%5Cfrac%7Bv%5E2%7D%7Bc%5E2%7D%20%7D)
Where
improper length,
proper length (distance measured wrt. rest frame) = 40 km
![l = 40 \sqrt{1-\frac{v^2}{c^2 }](https://tex.z-dn.net/?f=l%20%3D%2040%20%5Csqrt%7B1-%5Cfrac%7Bv%5E2%7D%7Bc%5E2%20%7D)
![l = 40 \times 0.096](https://tex.z-dn.net/?f=l%20%3D%2040%20%5Ctimes%200.096)
km
∴ Thus, the distance traveled by particle where the particle created to surface of earth wrt. particle's frame = 3.84 km.
According to the time dilation,
![\Delta t = \frac{\Delta t_{o} }{\sqrt{1-\frac{v^2}{c^2} } }](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%20%5Cfrac%7B%5CDelta%20t_%7Bo%7D%20%7D%7B%5Csqrt%7B1-%5Cfrac%7Bv%5E2%7D%7Bc%5E2%7D%20%7D%20%7D)
Where
improper time (wrt. earth frame time)
,
proper time (wrt. particle frame).
![1.34\times10^{-6} = \frac{ \Delta t_{o}}{0.096}](https://tex.z-dn.net/?f=1.34%5Ctimes10%5E%7B-6%7D%20%3D%20%5Cfrac%7B%20%5CDelta%20t_%7Bo%7D%7D%7B0.096%7D)
![\Delta t_{o} = 1.285 \times10^{-5} sec](https://tex.z-dn.net/?f=%5CDelta%20t_%7Bo%7D%20%3D%201.285%20%5Ctimes10%5E%7B-5%7D%20sec)
Thus, the time taking by particle to travel from where it is created to the surface of the earth
.