Respuesta :
Answer:
a) Approximately 0.2066 m
b) 0.00002568443 s
c) Approximately 188.87 s
Explanation:
a) The total energy of the electron = 190GeV
The rest energy of the electron = 0.51 MeV
[tex]\dfrac{19 \ GeV}{0.51 \ MeV} = 37254.902 = \dfrac{1}{\sqrt{1 + \dfrac{u^2}{c^2} } }[/tex]
[tex]\left ( 1 - \dfrac{u^2}{c^2} \right ) = \dfrac{1}{37254.902^2} } }[/tex]
[tex]\left ( 1 - \dfrac{1}{37254.902^2} } } \right ) = \dfrac{u^2}{c^2}[/tex]
u = √(299792458² × (1 - 1/37254.902²)) = 299792457.892
β = v/c = 299792457.892/299792458 = 0.99999999964
[tex]\beta = \sqrt{1 - \left ( \dfrac{\Delta x}{\Delta x'} \right )^2}[/tex]
Δx = √(Δx'²×(1 - β²)) = √(7700²×(1 - 0.99999999964²)) ≈ 0.2066 m
The length of the tube in the electron's frame ≈ 0.2066 m
b) The time in the frame of the electron is given as follows;
Δt' = 7700/299792457.892 = 0.00002568443 s
c) The time in the frame of the tube is given as follows;
[tex]\beta = \sqrt{1 - \left ( \dfrac{\Delta t'}{\Delta t} \right )^2}[/tex]
1 - β² = (Δt'/Δt)²
Δt = √(Δt'/(1 - β²)) = √(0.00002568443 /(1 - 0.99999999964²)) ≈ 188.87 s.
