Special Relativity - Relativistic Speed

If a charged pion that decay in 10⁻⁸ second in its own rest frame is to travel 30 meters in the laboratory before decaying, the pion’s speed must be most nearly

A. 0.43 × 10⁸ m/s
B. 2.84 × 10⁸ m/s
C. 2.90 × 10⁸ m/s
D. 2.98 × 10⁸ m/s
E. 3.00 × 10⁸ m/s

(GR0177 #33)

Solution:

v = v₀/γ 
with γ = 1/(1 − ^v2/_c²)^½ 

Given:
t₀ = 10⁻⁸ second 
L₀ = 30 meters
v₀ = L₀/t₀ = 30 × 10⁸ = 10(3 × 10⁸) = 10c

v = v₀(1 − ^v2/_c²)^½
v² = v₀²(1 − ^v2/_c²) = (v₀² − v²v₀²/c²) = 100c² − 100v²
101v² = 100c²
v = (100/101)c = 0.99c = 0.99(3 × 10⁸) ≈ 2.98 × 10⁸ m/s

Answer: D