A certain particle has a lifetime of when measured at rest. How far does it go before decaying if its speed was when it was created?
The speed of light is
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Time dilation defines, Δ t ′ = 1 − ( c 2 v 2 ) Δ t where Δ t is the time of particle's reference frame and Δ t ′ is the time of our reference frame. Switching Δ t = 1 . 0 0 × 1 0 − 7 , we obtain
Δ t ′ = 1 − ( c 2 ( 0 . 9 9 c ) 2 ) 1 . 0 0 × 1 0 − 7 ⇒ Δ t ′ = 1 − 0 . 9 9 2 1 . 0 0 × 1 0 − 7 = 7 . 0 8 8 1 × 1 0 − 7
So according to our reference frame the particle has traveled a distance s = 0 . 9 9 c × Δ t ′ = ( 0 . 9 9 × 3 × 1 0 8 ) m s − 1 × ( 7 . 0 8 8 1 × 1 0 − 7 ) s = 2 1 0 . 5 1 6 5 7