Another relativistic particle decay question

[Elvis 123456789]

Homework Statement

Unstable particles cannot live very long. Their mean life time t is defined by N(t) = N0e−t/τ , i.e., after a time of t, the number of particles left is N0/e. (For muons, τ=2.2µs.) Due to time dilation and length contraction, unstable particles can still travel far if their speeds are high enough.

For some particles, the mean life time is so small that it is more convenient to define τ using the quatity cτ (c is the speed of light). For example, the particle Λ has a cτ measured to be 7.89cm.

a) If the Λ is traveling at 0.5c, how many of L are left after traveling 7.89cm?

b) How far would the Λ’s have travelled, if only 50% of them are left?

c) (Extra) Derive the general expression of N(L)/N0 for the Λ particles, as a function of L (distance travelled) and the speed v (arbitrary, not just always 0.5c) of the Λ particles.

Homework Equations

N(t) = N0e−t/τ

t_e = t_Λ *γ

L_Λ = L_e/γ

t_e & L_e is the time and length measured in the Earth frame of reference

and t_Λ and L_Λ is the time and length measured in the lambda particle frame of reference

I did all the parts but I feel pretty unsure about it. These relativity questions just feel really ambiguous to me. I was hoping you guys could take a look and let me know if it seems ok. Thanks in advance!

The Attempt at a Solution

Parts a.) , b.), and c.) are in the attached image
[/B]
I assumed that the cτ = 7.89 cm is in the particle's FR

and for part a that the 7.89cm traveled was in the earth/lab FR

 

[TSny]

Your answers look correct to me.

But note that it would be nice to express your equations in terms of the defined quantity $c \tau_\Lambda$. Thus you can write $N = N_0 \exp \left(- \frac{L}{v \gamma \tau_\Lambda} \right)$ as $N = N_0 \exp \left(- \frac{L}{(v/c) \gamma (c \tau_\Lambda )} \right)$. Then you can just use the given value for $c \tau_\Lambda$ in the calculation for part (a) without having to find $v$ in m/s or $\tau_\Lambda$ in seconds.

For part (c), I think they want an equation for the ratio $N/N_0$, which just requires a little change in what you wrote. They might prefer the equation to be written in terms of the the quantity $c \tau_\Lambda$. But, maybe not.

 

[David Lewis]

Due to time dilation and length contraction, unstable particles can still travel far if their speeds are high enough.

If there were no time dilation or length contraction, unstable particles would still travel far if their speeds are high enough.