Relativity Problems

This section provides 100 problems to test your understanding of special relativity, including calculations of the Lorentz factor, time dilation, length contraction, relativistic velocity, momentum, kinetic energy, and mass-energy conversion, as well as applications like nuclear reactions and spacecraft navigation. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering modern physics, a key topic for JEE/NEET success.

Numerical Problems

1. A spaceship moves at $v = 0.6c$ relative to Earth. Calculate the Lorentz factor $\gamma$.

   • (a) 1.24
   • (b) 1.25
   • (c) 1.26
   • (d) 1.27

2. A clock on a spaceship moving at $v = 0.8c$ measures 5 s. Calculate the time interval in the Earth frame.

   • (a) 8.32 s
   • (b) 8.33 s
   • (c) 8.34 s
   • (d) 8.35 s

3. A rod has a proper length $L_0 = 10 , \text{m}$ and moves at $v = 0.5c$ relative to an observer. Calculate its length in the observer’s frame.

   • (a) 8.65 m
   • (b) 8.66 m
   • (c) 8.67 m
   • (d) 8.68 m

4. A rocket moves at $v = 0.6c$ relative to Earth. A projectile is fired from the rocket at $u' = 0.4c$ in the same direction. Calculate the projectile’s speed in the Earth frame.

   • (a) 0.806c
   • (b) 0.807c
   • (c) 0.808c
   • (d) 0.809c

5. A particle with rest mass $m_0 = 1 , \text{kg}$ moves at $v = 0.8c$. Calculate its relativistic momentum.

   • (a) $3.99 \times 10^8 , \text{kg·m/s}$
   • (b) $4.00 \times 10^8 , \text{kg·m/s}$
   • (c) $4.01 \times 10^8 , \text{kg·m/s}$
   • (d) $4.02 \times 10^8 , \text{kg·m/s}$

6. An electron ($m_0 = 9.11 \times 10^{-31} , \text{kg}$) moves at $v = 0.99c$. Calculate its kinetic energy in MeV.

   • (a) 1.54 MeV
   • (b) 1.55 MeV
   • (c) 1.56 MeV
   • (d) 1.57 MeV

7. A nuclear reaction has a mass defect of $\Delta m = 0.02 , \text{kg}$. Calculate the energy released in joules.

   • (a) $1.79 \times 10^{15} , \text{J}$
   • (b) $1.80 \times 10^{15} , \text{J}$
   • (c) $1.81 \times 10^{15} , \text{J}$
   • (d) $1.82 \times 10^{15} , \text{J}$

8. A spaceship moves at $v = 0.9c$. A clock on the ship measures 2 s. Calculate the time in the Earth frame.

   • (a) 4.57 s
   • (b) 4.58 s
   • (c) 4.59 s
   • (d) 4.60 s

9. A rod ($L_0 = 20 , \text{m}$) moves at $v = 0.6c$. Calculate its length in the observer’s frame.

   • (a) 15.99 m
   • (b) 16.00 m
   • (c) 16.01 m
   • (d) 16.02 m

10. A rocket at $v = 0.7c$ fires a projectile at $u' = 0.5c$ in the same direction. Calculate the projectile’s speed in the Earth frame.

    • (a) 0.909c
    • (b) 0.910c
    • (c) 0.911c
    • (d) 0.912c

11. A proton ($m_0 = 1.67 \times 10^{-27} , \text{kg}$) moves at $v = 0.95c$. Calculate its momentum.

    • (a) $1.59 \times 10^{-18} , \text{kg·m/s}$
    • (b) $1.60 \times 10^{-18} , \text{kg·m/s}$
    • (c) $1.61 \times 10^{-18} , \text{kg·m/s}$
    • (d) $1.62 \times 10^{-18} , \text{kg·m/s}$

12. A particle ($m_0 = 2 , \text{kg}$, $v = 0.5c$). Calculate its kinetic energy in joules.

    • (a) $2.07 \times 10^{16} , \text{J}$
    • (b) $2.08 \times 10^{16} , \text{J}$
    • (c) $2.09 \times 10^{16} , \text{J}$
    • (d) $2.10 \times 10^{16} , \text{J}$

13. A photon has energy $E = 3 , \text{MeV}$. Calculate its momentum.

    • (a) $1.59 \times 10^{-21} , \text{kg·m/s}$
    • (b) $1.60 \times 10^{-21} , \text{kg·m/s}$
    • (c) $1.61 \times 10^{-21} , \text{kg·m/s}$
    • (d) $1.62 \times 10^{-21} , \text{kg·m/s}$

14. A spaceship ($v = 0.99c$) measures a time interval of 1 s. Calculate the time in the Earth frame.

    • (a) 7.08 s
    • (b) 7.09 s
    • (c) 7.10 s
    • (d) 7.11 s

15. A rod ($L_0 = 5 , \text{m}$) moves at $v = 0.8c$. Calculate its length in the observer’s frame.

    • (a) 2.99 m
    • (b) 3.00 m
    • (c) 3.01 m
    • (d) 3.02 m

16. A rocket at $v = 0.4c$ fires a projectile at $u' = 0.3c$ in the same direction. Calculate the projectile’s speed in the Earth frame.

    • (a) 0.649c
    • (b) 0.650c
    • (c) 0.651c
    • (d) 0.652c

17. A particle ($m_0 = 1 , \text{kg}$, $v = 0.9c$). Calculate its total energy in joules.

    • (a) $2.06 \times 10^{17} , \text{J}$
    • (b) $2.07 \times 10^{17} , \text{J}$
    • (c) $2.08 \times 10^{17} , \text{J}$
    • (d) $2.09 \times 10^{17} , \text{J}$

18. A muon ($v = 0.98c$, $\tau_0 = 2.2 , \mu\text{s}$). Calculate its lifetime in the lab frame.

    • (a) 11.0 $\mu$s
    • (b) 11.1 $\mu$s
    • (c) 11.2 $\mu$s
    • (d) 11.3 $\mu$s

19. A rod ($L_0 = 15 , \text{m}$) moves at $v = 0.95c$. Calculate its length in the observer’s frame.

    • (a) 4.67 m
    • (b) 4.68 m
    • (c) 4.69 m
    • (d) 4.70 m

20. A spaceship at $v = 0.85c$ fires a projectile at $u' = 0.6c$ in the same direction. Calculate the projectile’s speed in the Earth frame.

    • (a) 0.961c
    • (b) 0.962c
    • (c) 0.963c
    • (d) 0.964c

21. An electron ($m_0 = 9.11 \times 10^{-31} , \text{kg}$) moves at $v = 0.9c$. Calculate its momentum.

    • (a) $1.87 \times 10^{-22} , \text{kg·m/s}$
    • (b) $1.88 \times 10^{-22} , \text{kg·m/s}$
    • (c) $1.89 \times 10^{-22} , \text{kg·m/s}$
    • (d) $1.90 \times 10^{-22} , \text{kg·m/s}$

22. A particle ($m_0 = 3 , \text{kg}$, $v = 0.6c$). Calculate its kinetic energy in joules.

    • (a) $3.37 \times 10^{16} , \text{J}$
    • (b) $3.38 \times 10^{16} , \text{J}$
    • (c) $3.39 \times 10^{16} , \text{J}$
    • (d) $3.40 \times 10^{16} , \text{J}$

23. A photon has energy $E = 1 , \text{MeV}$. Calculate its momentum.

    • (a) $5.33 \times 10^{-22} , \text{kg·m/s}$
    • (b) $5.34 \times 10^{-22} , \text{kg·m/s}$
    • (c) $5.35 \times 10^{-22} , \text{kg·m/s}$
    • (d) $5.36 \times 10^{-22} , \text{kg·m/s}$

24. A spaceship ($v = 0.7c$) measures 3 s. Calculate the time in the Earth frame.

    • (a) 4.19 s
    • (b) 4.20 s
    • (c) 4.21 s
    • (d) 4.22 s

25. A rod ($L_0 = 8 , \text{m}$) moves at $v = 0.4c$. Calculate its length in the observer’s frame.

    • (a) 7.41 m
    • (b) 7.42 m
    • (c) 7.43 m
    • (d) 7.44 m

26. A rocket at $v = 0.5c$ fires a projectile at $u' = 0.2c$ in the same direction. Calculate the projectile’s speed in the Earth frame.

    • (a) 0.636c
    • (b) 0.637c
    • (c) 0.638c
    • (d) 0.639c

27. A particle ($m_0 = 1 , \text{kg}$, $v = 0.95c$). Calculate its total energy in joules.

    • (a) $2.87 \times 10^{17} , \text{J}$
    • (b) $2.88 \times 10^{17} , \text{J}$
    • (c) $2.89 \times 10^{17} , \text{J}$
    • (d) $2.90 \times 10^{17} , \text{J}$

28. A muon ($v = 0.99c$, $\tau_0 = 2.2 , \mu\text{s}$). Calculate its lifetime in the lab frame.

    • (a) 15.5 $\mu$s
    • (b) 15.6 $\mu$s
    • (c) 15.7 $\mu$s
    • (d) 15.8 $\mu$s

29. A rod ($L_0 = 12 , \text{m}$) moves at $v = 0.9c$. Calculate its length in the observer’s frame.

    • (a) 5.22 m
    • (b) 5.23 m
    • (c) 5.24 m
    • (d) 5.25 m

30. A rocket at $v = 0.8c$ fires a projectile at $u' = 0.7c$ in the same direction. Calculate the projectile’s speed in the Earth frame.

    • (a) 0.974c
    • (b) 0.975c
    • (c) 0.976c
    • (d) 0.977c

31. A spacecraft converts 0.05 kg of mass to energy for propulsion. Calculate the energy released in joules.

    • (a) $4.49 \times 10^{15} , \text{J}$
    • (b) $4.50 \times 10^{15} , \text{J}$
    • (c) $4.51 \times 10^{15} , \text{J}$
    • (d) $4.52 \times 10^{15} , \text{J}$

32. A particle ($m_0 = 1 , \text{kg}$, $v = 0.99c$). Calculate its kinetic energy in joules.

    • (a) $5.49 \times 10^{17} , \text{J}$
    • (b) $5.50 \times 10^{17} , \text{J}$
    • (c) $5.51 \times 10^{17} , \text{J}$
    • (d) $5.52 \times 10^{17} , \text{J}$

33. A spaceship ($v = 0.5c$) measures 4 s. Calculate the time in the Earth frame.

    • (a) 4.61 s
    • (b) 4.62 s
    • (c) 4.63 s
    • (d) 4.64 s

34. A rod ($L_0 = 25 , \text{m}$) moves at $v = 0.85c$. Calculate its length in the observer’s frame.

    • (a) 13.05 m
    • (b) 13.06 m
    • (c) 13.07 m
    • (d) 13.08 m

35. A proton ($m_0 = 1.67 \times 10^{-27} , \text{kg}$, $v = 0.9c$). Calculate its total energy in joules.

    • (a) $3.44 \times 10^{-10} , \text{J}$
    • (b) $3.45 \times 10^{-10} , \text{J}$
    • (c) $3.46 \times 10^{-10} , \text{J}$
    • (d) $3.47 \times 10^{-10} , \text{J}$

Conceptual Problems

36. What does the first postulate of special relativity state?

• (a) Speed of light varies in different frames
• (b) Laws of physics are the same in all inertial frames
• (c) Time is absolute across all frames
• (d) Space is absolute across all frames

37. What does the second postulate of special relativity state?

• (a) Speed of light depends on the source’s motion
• (b) Speed of light is constant in all inertial frames
• (c) Speed of light is zero in moving frames
• (d) Speed of light varies with observer’s motion

38. What is the unit of the Lorentz factor $\gamma$?

• (a) Meter
• (b) Second
• (c) Dimensionless
• (d) Joule

39. What happens to a clock moving relative to an observer?

• (a) Ticks faster
• (b) Ticks slower
• (c) Stops ticking
• (d) No change

40. What does length contraction affect?

• (a) Length perpendicular to motion
• (b) Length parallel to motion
• (c) Both directions
• (d) No direction

41. What is the unit of relativistic momentum $p$?

• (a) kg·m/s
• (b) Joule
• (c) Watt
• (d) Newton

42. What does a larger $\gamma$ indicate?

• (a) Lower speed
• (b) Higher speed
• (c) No speed
• (d) Constant speed

43. What happens to an object’s length as its speed approaches $c$?

• (a) Increases
• (b) Decreases to zero
• (c) Remains the same
• (d) Becomes infinite

44. What does $E = mc^2$ represent?

• (a) Kinetic energy
• (b) Mass-energy equivalence
• (c) Potential energy
• (d) Momentum

45. What is the dimension of relativistic energy $E$?

• (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
  - (c) $[\text{L} \text{T}^{-2}]$
  - (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

46. What does proper time $\Delta t_0$ represent?

    • (a) Time in the observer’s frame
    • (b) Time in the moving frame
    • (c) Time in the rest frame of the event
    • (d) Time in all frames

47. What is the significance of $\frac{u' + v}{1 + \frac{u' v}{c^2}}$?

    • (a) Classical velocity addition
    • (b) Relativistic velocity addition
    • (c) Time dilation
    • (d) Length contraction

48. What happens to kinetic energy as $v \to c$?

    • (a) Becomes zero
    • (b) Approaches infinity
    • (c) Remains constant
    • (d) Decreases

49. What does $E_0 = m_0 c^2$ represent?

    • (a) Kinetic energy
    • (b) Rest energy
    • (c) Total energy
    • (d) Momentum

50. How does relativity assist in spacecraft navigation?

    • (a) Increases speed
    • (b) Accounts for time dilation in timing systems
    • (c) Reduces mass
    • (d) Increases length

Derivation Problems

51. Derive the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$.

52. Derive the time dilation formula $\Delta t = \gamma \Delta t_0$.

53. Derive the length contraction formula $L = \frac{L_0}{\gamma}$.

54. Derive the relativistic velocity addition formula $u = \frac{u' + v}{1 + \frac{u' v}{c^2}}$.

55. Derive the relativistic momentum $p = \gamma m_0 v$.

56. Derive the relativistic kinetic energy $KE = (\gamma - 1) m_0 c^2$.

57. Derive the mass-energy equivalence $E = mc^2$.

58. Derive the energy-momentum relation $E^2 = (pc)^2 + (m_0 c^2)^2$.

59. Derive the time dilation for a moving muon’s lifetime.

60. Derive the relativistic momentum of a particle moving at $v = 0.9c$.

61. Derive the kinetic energy of a particle at low speeds ($v \ll c$).

62. Derive the total energy of a photon using $E = pc$.

63. Derive the length contraction for a moving rod at $v = 0.8c$.

64. Derive the velocity addition for a projectile fired from a moving rocket.

65. Derive the rest energy of an electron in eV.

NEET-style Conceptual Problems

66. What is the unit of energy $E$ in SI units?

    • (a) Joule
    • (b) Radian
    • (c) Hertz
    • (d) Watt

67. What does a clock at rest relative to an observer measure?

    • (a) Dilated time
    • (b) Proper time
    • (c) No time
    • (d) Infinite time

68. What is the relationship between $\gamma$ and speed $v$?

    • (a) $\gamma \propto v$
    • (b) $\gamma$ increases with $v$
    • (c) $\gamma$ decreases with $v$
    • (d) $\gamma$ is independent of $v$

69. What happens to simultaneity in different inertial frames?

    • (a) Always simultaneous
    • (b) Not simultaneous
    • (c) No effect
    • (d) Infinite simultaneity

70. What is the dimension of momentum $p$?

    • (a) $[\text{M} \text{L} \text{T}^{-1}]$
    • (b) $[\text{M} \text{L}^2 \text{T}^{-2}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

71. What does proper length $L_0$ represent?

    • (a) Length in the observer’s frame
    • (b) Length in the moving frame
    • (c) Length in the rest frame of the object
    • (d) Length in all frames

72. What is the role of relativity in GPS systems?

    • (a) Increases speed
    • (b) Corrects for time dilation effects
    • (c) Reduces mass
    • (d) Increases length

73. What happens to relativistic mass as $v \to c$?

    • (a) Becomes zero
    • (b) Approaches infinity
    • (c) Remains the same
    • (d) Decreases

74. Why does a muon’s lifetime increase in the lab frame?

    • (a) Due to length contraction
    • (b) Due to time dilation
    • (c) Due to mass increase
    • (d) Due to velocity addition

75. What is the unit of mass defect $\Delta m$?

    • (a) Kilogram
    • (b) Joule
    • (c) Watt
    • (d) Radian

76. What does a velocity $u = c$ indicate in velocity addition?

    • (a) Exceeds $c$
    • (b) Equals $c$ for light
    • (c) Becomes zero
    • (d) Becomes infinite

77. Which frame measures the proper time of an event?

    • (a) Moving frame
    • (b) Rest frame of the event
    • (c) Observer’s frame
    • (d) All frames

78. What is the effect of relativity on simultaneity?

    • (a) Absolute simultaneity
    • (b) Relative simultaneity
    • (c) No simultaneity
    • (d) Infinite simultaneity

79. What does a pseudo-force do in a non-inertial frame for relativity?

    • (a) Affects perceived time dilation
    • (b) Affects mass
    • (c) Creates energy
    • (d) Reduces momentum

80. What is the dimension of $\gamma m_0$?

    • (a) $[\text{M}]$
    • (b) $[\text{M} \text{L} \text{T}^{-1}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

81. What is the role of relativity in spacecraft propulsion?

    • (a) Increases speed
    • (b) Converts mass to energy for propulsion
    • (c) Reduces mass
    • (d) Increases length

82. What happens to an object’s length perpendicular to its motion?

    • (a) Contracts
    • (b) Expands
    • (c) Remains the same
    • (d) Becomes zero

83. Why does relativistic momentum increase with speed?

    • (a) Due to $p = \gamma m_0 v$
    • (b) Due to length contraction
    • (c) Due to time dilation
    • (d) Due to mass defect

84. What is the significance of $(\gamma - 1) m_0 c^2$?

    • (a) Rest energy
    • (b) Kinetic energy
    • (c) Total energy
    • (d) Momentum

85. What is the unit of speed $c$?

    • (a) m/s
    • (b) Joule
    • (c) Hertz
    • (d) Watt

86. What does a high relativistic energy indicate?

    • (a) Low speed
    • (b) High speed
    • (c) No speed
    • (d) Constant speed

87. What is the physical significance of $(pc)^2 + (m_0 c^2)^2$?

    • (a) Kinetic energy
    • (b) Total energy squared
    • (c) Momentum
    • (d) Mass defect

88. Why does the speed of light remain constant in all frames?

    • (a) Due to the first postulate
    • (b) Due to the second postulate
    • (c) Due to time dilation
    • (d) Due to length contraction

89. What is the dimension of $\frac{v}{c}$?

    • (a) Dimensionless
    • (b) $[\text{L} \text{T}^{-1}]$
    • (c) $[\text{M} \text{L} \text{T}^{-1}]$
    • (d) $[\text{L} \text{T}^{-2}]$

90. How does relativity affect nuclear reactions?

    • (a) Increases mass
    • (b) Converts mass defect to energy
    • (c) Reduces energy
    • (d) Increases momentum

91. What is the role of the Lorentz factor $\gamma$?

    • (a) Measures speed
    • (b) Relates quantities between inertial frames
    • (c) Reduces energy
    • (d) Increases length

92. What does a dilated time interval indicate?

    • (a) Faster clock
    • (b) Slower clock in the moving frame
    • (c) No clock
    • (d) Constant clock

93. What is the physical significance of $m_0 c^2$?

    • (a) Kinetic energy
    • (b) Rest energy
    • (c) Total energy
    • (d) Momentum

94. What is the dimension of $E = mc^2$?

    • (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
    • (b) $[\text{M} \text{L} \text{T}^{-1}]$
    • (c) $[\text{L} \text{T}^{-2}]$
    • (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

95. Why does relativistic velocity addition ensure $u \leq c$?

    • (a) Due to $\frac{u' v}{c^2}$ term
    • (b) Due to length contraction
    • (c) Due to time dilation
    • (d) Due to mass increase

NEET-style Numerical Problems

96. A spaceship moves at $v = 0.7c$. Calculate the Lorentz factor $\gamma$.

    • (a) 1.39
    • (b) 1.40
    • (c) 1.41
    • (d) 1.42

97. A clock on a spaceship ($v = 0.9c$) measures 1 s. Calculate the time in the Earth frame.

    • (a) 2.28 s
    • (b) 2.29 s
    • (c) 2.30 s
    • (d) 2.31 s

98. A rod ($L_0 = 10 , \text{m}$) moves at $v = 0.6c$. Calculate its length in the observer’s frame.

    • (a) 7.99 m
    • (b) 8.00 m
    • (c) 8.01 m
    • (d) 8.02 m

99. A rocket at $v = 0.8c$ fires a projectile at $u' = 0.5c$ in the same direction. Calculate the projectile’s speed in the Earth frame.

    • (a) 0.945c
    • (b) 0.946c
    • (c) 0.947c
    • (d) 0.948c

100. A nuclear reaction has a mass defect $\Delta m = 0.01 , \text{kg}$. Calculate the energy released in joules.
     - (a) $8.99 \times 10^{14} , \text{J}$
     - (b) $9.00 \times 10^{14} , \text{J}$
     - (c) $9.01 \times 10^{14} , \text{J}$
     - (d) $9.02 \times 10^{14} , \text{J}$

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