Length contraction numerical problem

[Pushoam]

Homework Statement

Homework Equations

1) k.e. = (1/2)mv², non- relatvistic

2)K.E. = m₀ c² (ϒ -1), relativistic

3) L=L₀/ϒ , Length contraction

The Attempt at a Solution

(A) Taking 10¹⁹ eV to be the kinetic energy of the proton,
Non- relativistic calculation
(1/2)mv² = 10¹⁹ eV = 1.6

v = √[ (3.2/1.67)* 10²⁷ ] = 4.3 * 10¹³ m/s

How is kinetic energy defined relativistically?

K.E. = m₀ c² (ϒ -1)

We can't take 10¹⁹ eV as total energy of the proton as if we take so then we won't be able to calculate speed of the proton.

So, for calculating speed of the proton, I have to take 10¹⁹ eV as the kinetic energy of proton.

So,
ϒ² = [ {10¹⁹ eV/ m₀ c²}+1]²

m₀ c² = 939 MeV

Solving it gives,
1 - v²/c² = [939 * 10^-13]²

v ≈ c

Hence, w.r.t. galaxy frame the proton will take 10⁵ years to cross the galaxy.
B)

W.r.t. the proton frame,
the length of the galaxy is shortened to 10⁵ [939 * 10^-13] light years.Hence, the time taken = 10⁵ [939 * 10^-13] yearsIs this correct?

 

[Buzz Bloom]

Hence, the time taken = 105 [939 * 10-13] years

Hi Pushoam:

Your work looks OK to me. However, I suggest converting the (B) answer to seconds.

Regards,
Buzz