Special Relativity - Relativistic Energy

A lump of clay whose rest mass is 4 kilograms is travelling at three-fifths the speed of light when it collides head-on with an identical lump going the opposite direction at the same speed. If the two lumps stick together and no energy is radiated away, what is the mass of the composite lump?

A. 4 kg
B. 6.4 kg
C. 8 kg
D. 10 kg
E. 13.3 kg

(GR9677 #36)

Solution:

No energy is radiated away = energy is conserved

E_rel(a) + E_rel(b)  = E_rest(a,b)

γ_am_ac² + γ_bm_bc²  = Mc²

Given:
m_a = m_b = m₀ = 4 kg
|v_a| = |v_b| = ³/₅ c

γ = 1/(1 − ^v2/_c²)^½ 
γ = 1/(1 − ⁹/₂₅)^½ = 1/(¹⁶/₂₅)^½ = 1/(⁴/5)^½ = ⁵/₄
γ_a = γ_b = γ = ⁵/₄  

γ_am_ac² + γ_bm_bc²  = Mc²
M = 2γm₀ = 2(⁵/₄ )(4) = 10 kg

Answer: D