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

Lab Methods - Standard Deviation

The magnitude of the force F on an object can be determined by measuring both the mass m of an object and the magnitude of its acceleration a, where F = ma. Assume that these measurements are uncorrelated and normally distributed. if the standard deviations of the measurements of the mass and acceleration are σ_m and σ_a respectively, then σ_F/F is

A.

B.

C.

D.

E.

(GR9277 #48)

Solution:

Multiplying two quantities with uncertainty:

u = x ⋅ y



For F = ma:



Answer: C

Nuclear & Particle Physics - Selection Rules

A transition in which one photon is radiated by the electron in a hydrogen atom when the electron's wave function changes from ψ₁ to ψ₂ is forbidden if ψ₁ and ψ₂

A. have opposite parity
B. are orthogonal to each other
C. are zero at the center of the atomic nucleus
D. are both spherically symmetrical
E. are associated with different angular momenta

(GR8677 #48)

Solution:

Selection rules:

1.	Principal quantum number	:	∆n = anything
2.	Orbital angular momentum	:	∆l = ±1
3.	Magnetic quantum number	:	∆ml = 0, ±1
4.	Spin	:	∆s = 0
5.	Total angular momentum	:	∆j = 0, ±1, but j = 0 ↛j = 0

A. FALSE
It’s not related to the selection rules.

B. FALSE
In any transition, eigenstates should always be mutually orthogonal.

C. FALSE.
Eigenstates zero at the center → l 0 could change, for example from 3 to 2. This is allowed.

D. TRUE.
If both initial and final states have spherically symmetrical wave functions, then they have the same angular momentum. l = 0 → l = 0 is forbidden.

E. FALSE.
The selection rules require l to change.

Answer: D

Thermal Physics - Equipartition Law

A gaseous mixture of O₂ (molecular mass 32 u) and N₂ (molecular mass 28 u) is maintained at constant temperature. What is the ratio v_rms(N₂)/v_rms(O₂) of the root-mean-square speeds of the molecules?

A. 7/8
B. √(7/8)
C. √(8/7)
D. (8/7)²
E. ln (8/7)

(GR0177 #48)

Solution:

Kinetic energy, E = ¹/₂ mv²

Equipartition of energy, E = ³/₂ kT 

¹/₂ mv² = ³/₂ kT

v² = 3kT/m

v_rms = √(3kT/m)

v_rms ∝ √(1/m)

v_rmsN2 / v_rmsO2 

= √(m_O2 / m_N2) 

= √(32u/28u)

= √(8/7)

Answer: C