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# Special Relativity watch

1. A pion has a lifetime 26.0 ns when at rest. In order for it to travel 10m before decaying, how fast must it move?

Do you guys get about 0.79c
2. let t1 be time of pion as seen be special relativity
let t2 be time of pion as seen by pion

let l1 be length travleed as seen by special relativity
let l2 be length traveled as seen by pion

t1=t2/(x) l1=x.l2 note i have used x to represent the srqt (1-(v^2/c^2))

where v is velocity of pion

but v= l1/t1 = x^2 . l2/t2 x^2 = (1-(v^2/c^2))

bring this down and divide to get

v/(x^2)= l2/t2 = 10/(26x10^-9)

v/(x^2) = v/(1-(v^2)/c^2)) = v - ((c^2)/v) = 1/v(v^2-c^2)

so

1/v(v^2-c^2)=10/(26x10^-9)

i dont have a calculator at hand so let 10/(26x10^-9) = r for simplicity

therefore

1/v(v^2-c^2)=r
v^2-c^2=rv
v^2-rv-c^2=0

this quadratic equation should give 2 real roots - 1 of which will make sense the other won't i.e. it's greater than c or really small OR it might give a repeated root in which case u r dncing

really hope this helps

i also really hope it's correct
3. 0.7885c is what I got I think.
4. Yes thank you Sunil, I have handed it in by about 17:00

I just wanted to check

5. (Original post by latentcorpse)
let t1 be time of pion as seen be special relativity
let t2 be time of pion as seen by pion

let l1 be length travleed as seen by special relativity
let l2 be length traveled as seen by pion

t1=t2/(x) l1=x.l2 note i have used x to represent the srqt (1-(v^2/c^2))

where v is velocity of pion

but v= l1/t1 = x^2 . l2/t2 x^2 = (1-(v^2/c^2))

bring this down and divide to get

v/(x^2)= l2/t2 = 10/(26x10^-9)

v/(x^2) = v/(1-(v^2)/c^2)) = v - ((c^2)/v) = 1/v(v^2-c^2)

so

1/v(v^2-c^2)=10/(26x10^-9)

i dont have a calculator at hand so let 10/(26x10^-9) = r for simplicity

therefore

1/v(v^2-c^2)=r
v^2-c^2=rv
v^2-rv-c^2=0

this quadratic equation should give 2 real roots - 1 of which will make sense the other won't i.e. it's greater than c or really small OR it might give a repeated root in which case u r dncing

really hope this helps

i also really hope it's correct
I did it like this

t=L/vy=(L/v)Sqrt(1-v^2/c^2)

(t^2)(v^2)/(L^2)=1-v^2/c^2

(t^2)(c^2)(v^2)/(L^2)=(c^2)-(v^2)

Let (t^2)(c^2)/(L^2)=a

a(v^2)=(c^2)-(v^2)

(v^2)(a+1)=(c^2)

v=cSqrt(1/(a+1))

Which yields v=0.79c

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