Special Relativity - Relativistic Speed

The half life of π⁺ meson at rest is 2.5 × 10⁻⁸ second. A beam of  π⁺ mesons is generated at a point 15 meters from a detector. Only ½ of the π⁺mesons live to reach the detector. The speed of the π⁺ mesons is

A.¹/₂ c
B. √²/₅ c
C. ²/_√5 c
D. c
E. 2c

(GR9677 #48)

Solution:

v = v₀/γ 

with γ = (1 − ^v2/_c²)^−1/2

Given:

t₀ =  2.5 × 10⁻⁸s
L₀ = 15 m (measured in lab)

v₀ = L₀/t₀ = 15/(2.5 × 10⁻⁸) = 3/5 × 10⁹ =  (10/5)(3 × 10⁸) = 2c 

v₀/c = 2

v = v₀(1 − ^v2/_c²)^1/2
v² = v₀²(1 − ^v2/_c²) = v₀² − v²v₀²/c² = v₀² − 4v²
v² +  4v² = v₀²
5v² = 4c²
v = ²/_√5 c

Answer: C