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

Special Relativity - Relativistic Energy

A free electron (rest mass m_e = 0.5 MeV/c²) has a total energy of 1.5 MeV. Its momentum p in units of MeV/c is about

A. 0.86
B. 1.0
C. 1.4
D. 1.5
E. 2.0

(GR9277 #85)

Solution:

E² = p²c² + m₀²c⁴

1.5² = p²c² + (0.5/c²)²c⁴

 p²c² = 1.5² − 0.5² = 2

p = ^√2/_c ≈ ^1.4/_c

Answer: C

Special Relativity - Speed of Kaon

A positive kaon (K⁺) has a rest mass of 494 MeV/c² , whereas a proton has a rest mass of 938 MeV/c². If a kaon has a total energy that is equal to the proton rest energy, the speed of the kaon is most nearly

A. 0.25c
B. 0.40c
C. 0.55c
D. 0.70c
E. 0.85c

(GR8677 #20)

Solution:





Answer: E

Special Relativity - Relativistic Properties

If a newly discovered particle X moves with a speed equal to the speed of light in vacuum, then which of the following must be true?

A. The rest mass of X is zero
B. The spin of X equals the spin of a photon
C. The charge of X is carried on its surface
D. X does not spin
E. X cannot be detected

(GR8677 #68)

Solution:

Relativistic Energy: E² = p²c² + m₀²c⁴

If the particle moves with v = c, de Broglie relation: E = hf = pc

E² = E² + m₀²c⁴ 
m₀²c⁴ = E² − E² = 0
m₀ = 0

Answer: A

Special Relativity - Rest Mass

A particle leaving a cyclotron has a total relativistic energy of 10 GeV and a relativistic momentum of 8 GeV/c. What is the rest mass of this particle? 

A. 0.25 GeV/c²
B. 1.20 GeV/c²
C. 2.00 GeV/c²
D. 6.00 GeV/c²
E. 16.0 GeV/c²

(GR0177 #79)

Solution:

E² = p²c² + m₀²c⁴

Given: E = 10 GeV and p = 8 GeV/c

100 = (64/c²)c² + m₀²c⁴
m₀²c⁴ = 100 − 64 = 36
m₀ = 6 GeV/c²

Answer: D