TRB PG Assistant - Lorentz transformation - time dilation, length contraction, velocity addition law, momentum and energy in relativistic mechanics, centre-of-mass system for two relativistic particles Question Bank with Options

1. The concept of 'relativistic mass' (m = γm₀) is useful but sometimes discouraged in modern physics because:

A) It is incorrect.

B) It obscures the fact that only energy and momentum change with velocity, not rest mass.

C) It leads to contradictions with quantum mechanics.

D) It is only valid for speeds below 0.5c.

2. In the centre-of-mass system for two relativistic particles, if particle 1 has energy E₁ and momentum p₁, and particle 2 has energy E₂ and momentum p₂, then:

A) E₁ = E₂, p₁ = p₂

B) E₁ + E₂ = constant, p₁ + p₂ = 0

C) E₁ = E₂, p₁ = -p₂

D) E₁ = E₂ = m₀c²

3. What is the relativistic momentum of a particle with rest mass m₀ and energy E?

A) p = E/c

B) p = E²/c

C) p = E*c

D) p = m₀v

4. The relativistic addition of velocities is necessary because the Galilean addition law:

A) Is only valid for speeds less than c.

B) Leads to speeds greater than c when adding velocities close to c.

C) Does not conserve momentum.

D) Is only valid in non-inertial frames.

5. Which of the following physical quantities is invariant under Lorentz transformations?

A) Time interval

B) Length

C) Mass

D) Spacetime interval

6. The speed of light 'c' in vacuum is invariant, meaning it is the same for all inertial observers. This is a fundamental postulate of:

A) Newtonian mechanics

B) Quantum mechanics

C) Special relativity

D) General relativity

7. When considering the centre-of-mass system for two relativistic particles, the total energy in this frame is often denoted by W. If the particles are created from rest and move apart, W represents the:

A) Sum of their rest masses.

B) Sum of their relativistic masses.

C) Total kinetic energy.

D) Sum of their rest energies.

8. In relativistic mechanics, the energy of a particle is always greater than or equal to its rest energy. The difference is its kinetic energy, which is always:

A) Negative

B) Zero

C) Positive

D) Imaginary

9. If the velocity of a particle is v = 0.99c, the Lorentz factor γ will be approximately:

A) 1

B) 7

C) 10

D) 100

10. The Lorentz contraction affects the length of an object only along the dimension:

A) Perpendicular to its motion.

B) Parallel to its motion.

C) In all dimensions equally.

D) It affects volume, not length.

11. In the context of relativistic mechanics, the term 'proper time' (τ) refers to the time measured by:

A) An observer in a stationary frame.

B) An observer moving at relativistic speeds.

C) A clock that is stationary relative to the observer measuring it.

D) A clock that is moving relative to the observer measuring it.

12. What is the relativistic momentum of a photon, which has zero rest mass?

A) 0

B) m₀c

C) E/c

D) E*c

13. If an observer A sees a rod of length L₀ moving parallel to its length with velocity v, and observer B is at rest with the rod, observer A will measure the rod's length as:

A) L₀ / γ

B) L₀ * γ

C) L₀

D) 0

14. The Lorentz transformation is a linear transformation that preserves the spacetime interval Δs² = (cΔt)² - (Δx)² - (Δy)² - (Δz)². This means that special relativity is based on the geometry of:

A) Euclidean space

B) Minkowski spacetime

C) Riemannian geometry

D) Curved spacetime

15. In the centre-of-mass system of two particles, the total kinetic energy is minimized when:

A) The particles are moving at maximum speed.

B) The particles are at rest.

C) The particles have equal and opposite momenta.

D) The particles have zero momentum.

16. The velocity addition formula ensures that the speed of light c is the ultimate speed limit. If you add any velocity less than c to c, the result is:

A) Greater than c

B) Equal to c

C) Less than c

D) Zero

17. Consider a particle at rest with mass m₀. Its total energy is E₀ = m₀c². If it is accelerated to a velocity v, its new total energy E will be:

A) Less than E₀

B) Equal to E₀

C) Greater than E₀

D) Zero

18. If a particle's speed is v, its relativistic mass is often described as m = γm₀. This 'relativistic mass' concept emphasizes that:

A) Mass increases with velocity.

B) Mass decreases with velocity.

C) Mass is constant regardless of velocity.

D) Mass becomes infinite at the speed of light.

19. The Lorentz factor γ is always greater than or equal to 1. What is its value when v = 0?

A) 0

B) 1

C) ∞

D) Undefined

20. In relativistic mechanics, the concept of 'simultaneity' is relative. Two events that are simultaneous in one inertial frame may:

A) Always be simultaneous in all other inertial frames.

B) Never be simultaneous in any other inertial frame.

C) Not be simultaneous in another inertial frame moving relative to the first.

D) Only be simultaneous if the events occur at the same location.

21. If a particle has zero rest mass (m₀ = 0), its total energy is given by E = pc. This applies to:

A) Electrons

B) Protons

C) Neutrinos

D) Photons

22. The length contraction formula L = L₀/γ implies that L₀ is the length measured:

A) By an observer moving relative to the object.

B) By an observer in the object's rest frame.

C) When the object is at rest.

D) When the object is moving at the speed of light.

23. The time dilation formula Δt = γΔt₀ implies that Δt is the time interval measured by:

A) The observer at rest with the clock.

B) The observer moving with the clock.

C) An observer in a frame moving relative to the clock.

D) An observer at the speed of light.

24. If an event occurs at (x, t) in one inertial frame, what are its coordinates (x', t') in a frame moving with velocity v along the x-axis relative to the first frame, according to Lorentz transformation?

A) x' = x - vt, t' = t

B) x' = γ(x - vt), t' = γ(t - vx/c²)

C) x' = x/γ, t' = t/γ

D) x' = x + vt, t' = t

25. The Lorentz transformation equations reduce to the Galilean transformation equations under which condition?

A) When v approaches c.

B) When v is much greater than c.

C) When v is much less than c.

D) When v = c.

26. If two identical relativistic particles of rest mass m₀ collide head-on and stop in the CMS, what is the total energy in the CMS?

A) 2m₀c²

B) 2γm₀c²

C) 2(γ-1)m₀c²

D) 0

27. Why is the concept of a centre-of-mass system particularly important in relativistic collisions?

A) It simplifies calculations of energy and momentum conservation.

B) It is the only frame where energy is conserved.

C) It is the only frame where momentum is conserved.

D) It is the frame where time dilation is most pronounced.

28. In the centre-of-mass system for two relativistic particles, if the particles have momenta p₁ and p₂, then:

A) p₁ = p₂

B) p₁ = -p₂

C) p₁ = 0 and p₂ = 0

D) p₁ + p₂ = c

29. Consider two relativistic particles colliding. The centre-of-mass system (CMS) is defined as the inertial frame in which the total momentum of the system is:

A) Maximum.

B) Minimum.

C) Zero.

D) Equal to the momentum of the heavier particle.

30. The relativistic energy-momentum relation states that E² = (pc)² + (m₀c²)². What is E in this equation?

A) Kinetic energy

B) Rest energy

C) Total relativistic energy

D) Work done

31. Which of the following is NOT a consequence of special relativity?

A) Time dilation

B) Length contraction

C) Mass-energy equivalence

D) Gravitational lensing

32. What is the rest energy of a particle with rest mass m₀?

A) E = 0

B) E = m₀c²

C) E = γm₀c²

D) E = m₀c²/√(1 - v²/c²)

33. The famous equation E=mc² relates mass and energy. In relativistic mechanics, the total energy E of a particle with rest mass m₀ and velocity v is given by:

A) E = m₀c²

B) E = γm₀c²

C) E = (γ - 1)m₀c²

D) E = m₀v²/2

34. The relativistic kinetic energy of a particle is the difference between its total relativistic energy and its rest energy. What is the relativistic kinetic energy (KE)?

A) KE = m₀c²

B) KE = γm₀c²

C) KE = (γ - 1)m₀c²

D) KE = m₀v²

35. In relativistic mechanics, as the velocity v of a particle approaches the speed of light c, its momentum:

A) Approaches zero.

B) Remains constant.

C) Approaches infinity.

D) Becomes imaginary.

36. The relativistic momentum of a particle with rest mass m₀ and velocity v is given by:

A) p = m₀v

B) p = m₀v / √(1 - v²/c²)

C) p = m₀v * √(1 - v²/c²)

D) p = m₀c² / v

37. Calculate the denominator in the relativistic velocity addition formula for speeds u and v: (u+v)/(denominator).

A) 1 + uv/c²

B) 1 - uv/c²

C) √(1 - v²/c²)

D) c²

38. If a spaceship moving at 0.9c relative to Earth launches a probe forward at 0.8c relative to the spaceship, what is the speed of the probe relative to Earth, according to the relativistic velocity addition law?

A) 0.9c + 0.8c = 1.7c

B) 0.9c + 0.8c = 1.7c / (1 + (0.9c)(0.8c)/c²)

C) 0.9c + 0.8c = 1.7c / (1 - (0.9c)(0.8c)/c²)

D) 0.9c - 0.8c = 0.1c

39. The classical Galilean velocity addition states that if observer A sees object P moving at speed u, and observer B moves past A at speed v, then B sees P moving at speed u+v. The relativistic velocity addition law differs when:

A) The speeds are much less than the speed of light.

B) The speeds are comparable to the speed of light.

C) The motion is in three dimensions.

D) The observers are accelerating.

40. According to the relativistic velocity addition law, if two objects are moving towards each other with speeds u and v relative to a third frame, and u and v are close to c, their relative speed is:

A) u + v

B) (u + v) / (1 + uv/c²)

C) (u + v) / (1 - uv/c²)

D) u - v

41. Which of the following statements about length contraction is correct?

A) An object appears shorter in its rest frame than to a moving observer.

B) An object appears shorter to an observer when it moves parallel to its length.

C) An object appears longer to an observer when it moves parallel to its length.

D) Length contraction is only observed for objects moving perpendicular to their length.

42. From the perspective of the muon itself (in its own rest frame), its lifetime is observed to be:

A) Dilated.

B) Contracted.

C) Its proper lifetime.

D) Infinite.

43. A muon created in the upper atmosphere travels towards Earth at a speed close to the speed of light. Due to time dilation, an observer on Earth observes the muon's lifetime to be:

A) Shorter than its proper lifetime.

B) Longer than its proper lifetime.

C) Equal to its proper lifetime.

D) Zero.

44. Length contraction in special relativity occurs along the direction:

A) Perpendicular to the direction of motion.

B) Parallel to the direction of motion.

C) Both perpendicular and parallel to the direction of motion.

D) It is an illusion and does not occur.

45. Consider a rod of length L₀ at rest in a reference frame. What is the length L measured by an observer moving with velocity v relative to the rod?

A) L = L₀

B) L = γL₀

C) L = L₀ / γ

D) L = L₀ * v/c

46. If a spaceship travels at a significant fraction of the speed of light, an observer on Earth will measure the spaceship's clock to be:

A) Running faster than their own clock.

B) Running slower than their own clock.

C) Running at the same rate as their own clock.

D) Stopped.

47. The phenomenon where a moving clock runs slower than an identical clock at rest relative to an observer is known as:

A) Length contraction

B) Time dilation

C) Velocity addition

D) Mass-energy equivalence

48. What is the term 'γ' (gamma) in the Lorentz transformation equations, where v is the relative velocity between frames and c is the speed of light?

A) The gamma function

B) The Lorentz factor, given by 1/√(1 - v²/c²)

C) The gyromagnetic ratio

D) The gamma decay constant

49. In the Lorentz transformation, if an observer is at rest and measures a time interval Δt, what is the time interval Δt' measured by an observer moving with velocity v relative to the first observer?

A) Δt' = Δt

B) Δt' = γΔt

C) Δt' = Δt / γ

D) Δt' = Δt * v/c

50. What is the primary purpose of the Lorentz transformation equations in special relativity?

A) To describe the motion of objects in a gravitational field.

B) To relate the space and time coordinates of an event as measured by two observers in inertial frames moving at a constant velocity relative to each other.

C) To explain the wave-particle duality of light.

D) To derive the laws of thermodynamics.

Lorentz transformation - time dilation, length contraction, velocity addition law, momentum and energy in relativistic mechanics, centre-of-mass system for two relativistic particles - Question Bank