Center of Mass and Linear Momentum Problems

This section provides 100 problems to test your understanding of center of mass and linear momentum, including the definition and calculation of the center of mass, linear momentum and its conservation, impulse and its relation to momentum change, and collisions (elastic and inelastic) with applications. 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 system dynamics, a key topic for JEE/NEET success.

Numerical Problems

1. Two particles, $m_1=3kg$ at $(2,1)$ and $m_2=5kg$ at $(4,5)$, form a system. Calculate the x-coordinate of the center of mass.

   • (a) $3.0$
   • (b) $3.25$
   • (c) $3.5$
   • (d) $3.75$

2. A uniform rod of mass $6kg$ and length $3m$ lies along the x-axis from $x=0$ to $x=3m$. Calculate the position of the center of mass along the x-axis.

   • (a) $1.0m$
   • (b) $1.5m$
   • (c) $2.0m$
   • (d) $2.5m$

3. A $4kg$ cart moving at $5m/s$ collides with a stationary $2kg$ cart on a frictionless surface. They stick together. Calculate their final velocity.

   • (a) $2.0m/s$
   • (b) $2.5m/s$
   • (c) $3.0m/s$
   • (d) $3.33m/s$

4. A $0.3kg$ ball moving at $6m/s$ to the right collides with a $0.7kg$ ball moving at $2m/s$ to the left. Calculate the total momentum before collision.

   • (a) $0.4kg\cdot m/s$
   • (b) $0.8kg\cdot m/s$
   • (c) $1.2kg\cdot m/s$
   • (d) $1.6kg\cdot m/s$

5. A $1kg$ ball initially at rest is struck by a force $F=50N$ for $0.1s$. Calculate the final velocity of the ball.

   • (a) $3m/s$
   • (b) $4m/s$
   • (c) $5m/s$
   • (d) $6m/s$

6. A $2kg$ ball moving at $8m/s$ collides elastically with a $2kg$ ball at rest in 1D. Calculate the final velocity of the first ball.

   • (a) $0m/s$
   • (b) $2m/s$
   • (c) $4m/s$
   • (d) $6m/s$

7. A $5kg$ object explodes into two pieces: $2kg$ at $4m/s$ to the right and $3kg$ at $v$ to the left. Initial velocity was zero. Calculate $v$.

   • (a) $2.0m/s$
   • (b) $2.67m/s$
   • (c) $3.0m/s$
   • (d) $3.5m/s$

8. A $0.5kg$ ball dropped from rest hits the ground after $2s$ and rebounds with velocity $15m/s$ upward ($g=9.8m/s^2$). Calculate the impulse.

   • (a) $15.0kg\cdot m/s$
   • (b) $16.5kg\cdot m/s$
   • (c) $17.5kg\cdot m/s$
   • (d) $18.5kg\cdot m/s$

9. A $3kg$ block moving at $4m/s$ on a frictionless surface has a $1kg$ block dropped onto it from rest. They move together. Calculate the final velocity.

   • (a) $2.5m/s$
   • (b) $3.0m/s$
   • (c) $3.5m/s$
   • (d) $4.0m/s$

10. A $1kg$ ball moving at $5m/s$ collides with a $2kg$ ball moving at $1m/s$ in the same direction. After collision, the $1kg$ ball moves at $2m/s$. Calculate the final velocity of the $2kg$ ball.

    • (a) $1.0m/s$
    • (b) $1.5m/s$
    • (c) $2.0m/s$
    • (d) $2.5m/s$

11. Three particles, $m_1=2kg$ at $(0,0)$, $m_2=3kg$ at $(4,0)$, $m_3=5kg$ at $(0,2)$, form a system. Calculate the y-coordinate of the center of mass.

    • (a) $0.8$
    • (b) $1.0$
    • (c) $1.2$
    • (d) $1.4$

12. A $0.2kg$ ball moving at $10m/s$ is struck by an impulse $J=4kg\cdot m/s$ in the same direction. Calculate the final velocity.

    • (a) $20m/s$
    • (b) $25m/s$
    • (c) $30m/s$
    • (d) $35m/s$

13. A $4kg$ cart moving at $6m/s$ collides with a stationary $8kg$ cart on a frictionless surface. They stick together. Calculate their final velocity.

    • (a) $1.5m/s$
    • (b) $2.0m/s$
    • (c) $2.5m/s$
    • (d) $3.0m/s$

14. A $1kg$ ball moving at $8m/s$ collides elastically with a $3kg$ ball at rest in 1D. Calculate the final velocity of the $1kg$ ball.

    • (a) $-2m/s$
    • (b) $0m/s$
    • (c) $2m/s$
    • (d) $4m/s$

15. A $6kg$ object explodes into two pieces: $2kg$ at $3m/s$ to the right and $4kg$ at $v$ to the left. Initial velocity was zero. Calculate $v$.

    • (a) $1.0m/s$
    • (b) $1.5m/s$
    • (c) $2.0m/s$
    • (d) $2.5m/s$

16. Two particles, $m_1=4kg$ at $(1,3)$ and $m_2=4kg$ at $(5,3)$, form a system. Calculate the x-coordinate of the center of mass.

    • (a) $2.5$
    • (b) $3.0$
    • (c) $3.5$
    • (d) $4.0$

17. A $0.4kg$ ball moving at $5m/s$ to the right collides with a $0.6kg$ ball moving at $3m/s$ to the left. Calculate the total momentum before collision.

    • (a) $0.2kg\cdot m/s$
    • (b) $0.4kg\cdot m/s$
    • (c) $0.6kg\cdot m/s$
    • (d) $0.8kg\cdot m/s$

18. A $2kg$ object moving at $4m/s$ is slowed to $2m/s$ by a constant force in $0.5s$. Calculate the force.

    • (a) $-4.0N$
    • (b) $-6.0N$
    • (c) $-8.0N$
    • (d) $-10.0N$

19. A $1kg$ ball moving at $6m/s$ collides elastically with a $1kg$ ball moving at $2m/s$ in the same direction. Calculate the final velocity of the first ball.

    • (a) $2m/s$
    • (b) $3m/s$
    • (c) $4m/s$
    • (d) $5m/s$

20. A $5kg$ cart moving at $3m/s$ collides with a stationary $3kg$ cart on a frictionless surface. They stick together. Calculate their final velocity.

    • (a) $1.5m/s$
    • (b) $1.875m/s$
    • (c) $2.0m/s$
    • (d) $2.25m/s$

21. A $0.5kg$ ball dropped from rest hits the ground after $1s$ and rebounds with velocity $7m/s$ upward ($g=9.8m/s^2$). Calculate the impulse.

    • (a) $7.0kg\cdot m/s$
    • (b) $7.5kg\cdot m/s$
    • (c) $8.0kg\cdot m/s$
    • (d) $8.5kg\cdot m/s$

22. A $3kg$ block moving at $5m/s$ on a frictionless surface has a $2kg$ block dropped onto it from rest. They move together. Calculate the final velocity.

    • (a) $2.5m/s$
    • (b) $3.0m/s$
    • (c) $3.5m/s$
    • (d) $4.0m/s$

23. A $2kg$ object explodes into two pieces: $1kg$ at $6m/s$ to the right and $1kg$ at $v$ to the left. Initial velocity was zero. Calculate $v$.

    • (a) $4m/s$
    • (b) $5m/s$
    • (c) $6m/s$
    • (d) $7m/s$

24. Two particles, $m_1=1kg$ at $(0,2)$ and $m_2=3kg$ at $(4,6)$, form a system. Calculate the y-coordinate of the center of mass.

    • (a) $4.5$
    • (b) $5.0$
    • (c) $5.5$
    • (d) $6.0$

25. A $0.6kg$ ball moving at $4m/s$ to the right collides with a $0.4kg$ ball moving at $2m/s$ to the left. Calculate the total momentum before collision.

    • (a) $1.2kg\cdot m/s$
    • (b) $1.4kg\cdot m/s$
    • (c) $1.6kg\cdot m/s$
    • (d) $1.8kg\cdot m/s$

26. A $1kg$ ball moving at $7m/s$ is struck by an impulse $J=2kg\cdot m/s$ in the same direction. Calculate the final velocity.

    • (a) $8m/s$
    • (b) $9m/s$
    • (c) $10m/s$
    • (d) $11m/s$

27. A $3kg$ cart moving at $4m/s$ collides with a stationary $6kg$ cart on a frictionless surface. They stick together. Calculate their final velocity.

    • (a) $1.0m/s$
    • (b) $1.33m/s$
    • (c) $1.5m/s$
    • (d) $1.67m/s$

28. A $2kg$ ball moving at $5m/s$ collides elastically with a $4kg$ ball at rest in 1D. Calculate the final velocity of the $2kg$ ball.

    • (a) $-1.67m/s$
    • (b) $0m/s$
    • (c) $1.67m/s$
    • (d) $2.5m/s$

29. A $4kg$ object explodes into two pieces: $1kg$ at $5m/s$ to the right and $3kg$ at $v$ to the left. Initial velocity was zero. Calculate $v$.

    • (a) $1.0m/s$
    • (b) $1.5m/s$
    • (c) $1.67m/s$
    • (d) $2.0m/s$

30. Three particles, $m_1=1kg$ at $(1,1)$, $m_2=2kg$ at $(3,1)$, $m_3=3kg$ at $(1,4)$, form a system. Calculate the x-coordinate of the center of mass.

    • (a) $1.5$
    • (b) $1.67$
    • (c) $1.83$
    • (d) $2.0$

31. A $0.8kg$ ball moving at $3m/s$ to the right collides with a $0.2kg$ ball moving at $1m/s$ to the left. Calculate the total momentum before collision.

    • (a) $2.0kg\cdot m/s$
    • (b) $2.2kg\cdot m/s$
    • (c) $2.4kg\cdot m/s$
    • (d) $2.6kg\cdot m/s$

32. A $3kg$ object moving at $6m/s$ is slowed to $2m/s$ by a constant force in $0.4s$. Calculate the force.

    • (a) $-15.0N$
    • (b) $-12.0N$
    • (c) $-10.0N$
    • (d) $-8.0N$

33. A $1kg$ ball moving at $4m/s$ collides elastically with a $2kg$ ball moving at $2m/s$ in the same direction. Calculate the final velocity of the $1kg$ ball.

    • (a) $1.33m/s$
    • (b) $2.0m/s$
    • (c) $2.67m/s$
    • (d) $3.0m/s$

34. A $5kg$ cart moving at $2m/s$ collides with a stationary $1kg$ cart on a frictionless surface. They stick together. Calculate their final velocity.

    • (a) $1.5m/s$
    • (b) $1.67m/s$
    • (c) $1.83m/s$
    • (d) $2.0m/s$

35. A $0.4kg$ ball dropped from rest hits the ground after $1.5s$ and rebounds with velocity $12m/s$ upward ($g=9.8m/s^2$). Calculate the impulse.

    • (a) $10.0kg\cdot m/s$
    • (b) $10.5kg\cdot m/s$
    • (c) $11.0kg\cdot m/s$
    • (d) $11.5kg\cdot m/s$

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Conceptual Problems

36. What does the center of mass represent?

• (a) The geometric center of an object
• (b) The mass-weighted average position
• (c) The point of maximum mass
• (d) The point of minimum velocity

37. When is linear momentum conserved?

• (a) When net external force is zero
• (b) When kinetic energy is conserved
• (c) When potential energy is conserved
• (d) When velocity is constant

38. What does the impulse-momentum theorem state?

• (a) Impulse equals kinetic energy change
• (b) Impulse equals momentum change
• (c) Impulse equals force change
• (d) Impulse equals velocity change

39. What characterizes an elastic collision?

• (a) Momentum is conserved, kinetic energy is not
• (b) Momentum and kinetic energy are conserved
• (c) Objects stick together
• (d) Kinetic energy is conserved, momentum is not

40. What is the unit of linear momentum?

• (a) $kg\cdot m/s$
• (b) $N\cdot s$
• (c) $J$
• (d) $kg\cdot m/s^2$

41. What happens to the center of mass of a system with no external forces?

• (a) It accelerates
• (b) It moves with constant velocity
• (c) It remains stationary
• (d) It oscillates

42. What does a perfectly inelastic collision imply?

• (a) Kinetic energy is conserved
• (b) Objects stick together
• (c) Momentum is not conserved
• (d) Velocity is zero after collision

43. What is the physical significance of impulse?

• (a) Change in kinetic energy
• (b) Change in momentum
• (c) Change in velocity
• (d) Change in force

44. How does mass affect linear momentum?

• (a) Momentum is inversely proportional to mass
• (b) Momentum is directly proportional to mass
• (c) Momentum does not depend on mass
• (d) Momentum depends on mass squared

45. What is the dimension of impulse?

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

46. What does a zero total momentum imply?

• (a) System is at rest
• (b) System’s center of mass is at rest or moving uniformly
• (c) System has no mass
• (d) System has no velocity

47. What is the significance of $m\overrightarrow{v}$?

• (a) Kinetic energy
• (b) Potential energy
• (c) Linear momentum
• (d) Angular momentum

48. What happens in an explosion with zero initial velocity?

• (a) Momentum is not conserved
• (b) Total momentum remains zero
• (c) Kinetic energy is conserved
• (d) Velocity becomes zero

49. What does the center of mass of a uniform object depend on?

• (a) Mass distribution
• (b) Geometric shape
• (c) Velocity of the object
• (d) External forces

50. How does impulse affect an object at rest?

• (a) Increases its kinetic energy only
• (b) Changes its momentum, giving it velocity
• (c) Changes its potential energy
• (d) Does not affect its momentum

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Derivation Problems

51. Derive the center of mass position for a system of two particles.

52. Derive the center of mass of a uniform rod.

53. Derive the conservation of linear momentum for two particles.

54. Derive the impulse-momentum theorem.

55. Derive the final velocity in a perfectly inelastic collision.

56. Derive the final velocities in a 1D elastic collision.

57. Derive the center of mass velocity in terms of total momentum.

58. Derive the impulse for a ball rebounding off a surface.

59. Derive the final velocity after an explosion of a stationary object.

60. Derive the center of mass position for a three-particle system.

61. Derive the force from impulse in a constant-force scenario.

62. Derive the momentum change in a 2D collision.

63. Derive the final velocities for an elastic collision with equal masses.

64. Derive the impulse delivered in a perfectly inelastic collision.

65. Derive the center of mass motion under external forces.

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NEET-style Conceptual Problems

66. What is the unit of impulse in SI units?

• (a) $kg\cdot m/s$
• (b) $N\cdot m$
• (c) $J$
• (d) $kg\cdot m/s^2$

67. What does a zero center of mass velocity indicate?

• (a) System is accelerating
• (b) System’s total momentum is zero
• (c) System is moving uniformly
• (d) System has no mass

68. Which quantity is conserved in all collisions?

• (a) Kinetic energy
• (b) Potential energy
• (c) Linear momentum
• (d) Angular momentum

69. What happens to kinetic energy in a perfectly inelastic collision?

• (a) Fully conserved
• (b) Partially lost
• (c) Completely lost
• (d) Increased

70. What is the dimension of linear momentum?

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

71. What does the center of mass of a system represent?

• (a) Point of maximum velocity
• (b) Mass-weighted average position
• (c) Geometric center
• (d) Point of zero momentum

72. What is the role of impulse in a collision?

• (a) Changes kinetic energy
• (b) Changes momentum
• (c) Changes potential energy
• (d) Changes velocity direction only

73. What happens to the velocity of the center of mass after an explosion?

• (a) Increases
• (b) Decreases
• (c) Remains the same if no external forces act
• (d) Becomes zero

74. Why is momentum conserved in collisions?

• (a) Kinetic energy is conserved
• (b) Net external force is zero
• (c) Potential energy is conserved
• (d) Velocity is constant

75. What is the unit of force in terms of impulse?

• (a) $kg\cdot m/s$
• (b) $N$
• (c) $J$
• (d) $kg\cdot m/s^2$

76. What does a constant momentum imply?

• (a) No net external force
• (b) Constant kinetic energy
• (c) Constant potential energy
• (d) Constant velocity of each particle

77. Which type of collision conserves both momentum and kinetic energy?

• (a) Perfectly inelastic
• (b) Inelastic
• (c) Elastic
• (d) Explosive

78. What is the direction of impulse in a collision?

• (a) Always opposite to velocity
• (b) Same as the change in momentum
• (c) Perpendicular to velocity
• (d) Along the initial velocity

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

• (a) Conserves momentum
• (b) Affects momentum calculations
• (c) Provides centripetal force
• (d) Reduces friction

80. What is the dimension of the center of mass position?

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

81. What is the role of momentum in an explosion?

• (a) Increases kinetic energy
• (b) Remains conserved
• (c) Decreases potential energy
• (d) Increases velocity

82. What happens to the center of mass during a collision?

• (a) Moves with constant velocity if no external forces act
• (b) Accelerates
• (c) Stops moving
• (d) Oscillates

83. Why does impulse change momentum?

• (a) It applies a force over time
• (b) It changes kinetic energy
• (c) It changes potential energy
• (d) It changes mass

84. What does a positive impulse imply?

• (a) Momentum decreases
• (b) Momentum increases in the direction of the impulse
• (c) Velocity becomes zero
• (d) Force is zero

85. What is the unit of the mass in momentum calculations?

• (a) $kg$
• (b) $N$
• (c) $J$
• (d) $m/s$

86. What does a zero impulse indicate?

• (a) No change in momentum
• (b) No change in kinetic energy
• (c) No change in velocity
• (d) No change in mass

87. What is the physical significance of $\overrightarrow{F}\mathrm{\Delta }t$?

• (a) Work done
• (b) Kinetic energy
• (c) Impulse
• (d) Potential energy

88. Why does the center of mass simplify system dynamics?

• (a) It acts as the point of maximum velocity
• (b) It represents the system as a single particle
• (c) It eliminates external forces
• (d) It increases momentum

89. What is the dimension of force in terms of impulse?

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

90. How does mass affect the center of mass position?

• (a) Positions are weighted by mass
• (b) Positions are independent of mass
• (c) Positions depend on mass squared
• (d) Positions depend on velocity

91. What is the role of conservation in collisions?

• (a) Determines kinetic energy
• (b) Determines final velocities using momentum
• (c) Determines potential energy
• (d) Determines impulse

92. What does a zero final velocity in an elastic collision indicate?

• (a) Objects stick together
• (b) One object transfers all its velocity
• (c) Kinetic energy is not conserved
• (d) Momentum is not conserved

93. What is the physical significance of $\sum m_i{\overrightarrow{v}}_i$?

• (a) Total kinetic energy
• (b) Total potential energy
• (c) Total linear momentum
• (d) Total angular momentum

94. What is the dimension of velocity in momentum?

• (a) $[\text{L}{\text{T}}^{-1}]$
• (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 momentum depend on direction?

• (a) It is a scalar quantity
• (b) It is a vector quantity
• (c) It depends on mass only
• (d) It depends on time

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NEET-style Numerical Problems

96. A $2kg$ cart moving at $4m/s$ collides with a stationary $1kg$ cart on a frictionless surface. They stick together. What is their final velocity?

• (a) $2.0m/s$
• (b) $2.67m/s$
• (c) $3.0m/s$
• (d) $3.5m/s$

97. A $0.5kg$ ball moving at $6m/s$ to the right collides with a $0.5kg$ ball moving at $2m/s$ to the left. What is the total momentum before collision?

• (a) $1.0kg\cdot m/s$
• (b) $1.5kg\cdot m/s$
• (c) $2.0kg\cdot m/s$
• (d) $2.5kg\cdot m/s$

98. A $1kg$ ball initially at rest is struck by a force $F=40N$ for $0.05s$. What is the final velocity?

• (a) $1.5m/s$
• (b) $2.0m/s$
• (c) $2.5m/s$
• (d) $3.0m/s$

99. A $2kg$ ball moving at $3m/s$ collides elastically with a $2kg$ ball at rest in 1D. What is the final velocity of the second ball?

• (a) $1.5m/s$
• (b) $2.0m/s$
• (c) $2.5m/s$
• (d) $3.0m/s$

100. A $3kg$ object explodes into two pieces: $1kg$ at $4m/s$ to the right and $2kg$ at $v$ to the left. Initial velocity was zero. What is $v$?
     - (a) $1.5m/s$
     - (b) $2.0m/s$
     - (c) $2.5m/s$
     - (d) $3.0m/s$

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