Motion in Two and Three Dimensions Problems

This section provides 100 problems to test your understanding of motion in two and three dimensions, including position, displacement, velocity, acceleration, projectile motion, uniform circular motion, and relative motion. 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 multi-dimensional motion, a key topic for JEE/NEET success.

Numerical Problems

1. A particle moves from position ${\overrightarrow{r}}_1=2\hat{i}+3\hat{j}m$ to ${\overrightarrow{r}}_2=6\hat{i}+8\hat{j}m$ in $4s$. Calculate the magnitude of the average velocity.

   • (a) $1.25m/s$
   • (b) $2.5m/s$
   • (c) $3.75m/s$
   • (d) $5.0m/s$

2. A particle’s position is given by $\overrightarrow{r}(t)=(4t)\hat{i}+(3t^2)\hat{j}m$. Find the magnitude of the velocity at $t=2s$.

   • (a) $10m/s$
   • (b) $12m/s$
   • (c) $14m/s$
   • (d) $16m/s$

3. A particle has velocity $\overrightarrow{v}(t)=(6t)\hat{i}+(2t^2)\hat{j}m/s$. Calculate the magnitude of the acceleration at $t=1s$.

   • (a) $4m/s^2$
   • (b) $6m/s^2$
   • (c) $8m/s^2$
   • (d) $10m/s^2$

4. A projectile is launched at $v_0=40m/s$ at $\theta ={30}^{\circ }$ ($g=9.8m/s^2$). Calculate the time of flight.

   • (a) $2.0s$
   • (b) $3.0s$
   • (c) $4.0s$
   • (d) $5.0s$

5. A ball is thrown at $v_0=50m/s$ at $\theta ={45}^{\circ }$ ($g=9.8m/s^2$). Find the maximum height.

   • (a) $50m$
   • (b) $63.78m$
   • (c) $75m$
   • (d) $100m$

6. A projectile is launched at $v_0=30m/s$ at $\theta ={60}^{\circ }$ ($g=9.8m/s^2$). Calculate the range.

   • (a) $70m$
   • (b) $79.5m$
   • (c) $90m$
   • (d) $100m$

7. A car moves in a circle of radius $20m$ at a speed of $5m/s$. Find the centripetal acceleration.

   • (a) $1.25m/s^2$
   • (b) $2.5m/s^2$
   • (c) $3.75m/s^2$
   • (d) $5.0m/s^2$

8. A satellite orbits at $r=7\times {10}^{6}m$ with period $T=5400s$. Calculate the speed.

   • (a) $7500m/s$
   • (b) $8000m/s$
   • (c) $8500m/s$
   • (d) $9000m/s$

9. A boat moves at ${\overrightarrow{v}}_b=6\hat{i}m/s$ in a river with current ${\overrightarrow{v}}_r=4\hat{j}m/s$. Find the magnitude of the boat’s velocity relative to the ground.

   • (a) $6m/s$
   • (b) $7.21m/s$
   • (c) $8m/s$
   • (d) $10m/s$

10. A particle’s position in 3D is $\overrightarrow{r}(t)=(2t)\hat{i}+(3t)\hat{j}+(t^2)\hat{k}m$. Find the speed at $t=2s$.

    • (a) $5m/s$
    • (b) $6m/s$
    • (c) $7m/s$
    • (d) $8m/s$

11. A projectile is launched at $v_0=60m/s$ at $\theta ={30}^{\circ }$ ($g=9.8m/s^2$). Find the horizontal velocity at $t=2s$.

    • (a) $30m/s$
    • (b) $45m/s$
    • (c) $51.96m/s$
    • (d) $60m/s$

12. A ball is thrown at $v_0=20m/s$ at $\theta ={45}^{\circ }$ ($g=9.8m/s^2$). Find the vertical velocity at $t=1s$.

    • (a) $4.14m/s$
    • (b) $5.14m/s$
    • (c) $6.14m/s$
    • (d) $7.14m/s$

13. A particle moves in a circle of radius $10m$ with $\omega =2rad/s$. Calculate the centripetal acceleration.

    • (a) $20m/s^2$
    • (b) $30m/s^2$
    • (c) $40m/s^2$
    • (d) $50m/s^2$

14. Two cars have velocities ${\overrightarrow{v}}_1=30\hat{i}m/s$ and ${\overrightarrow{v}}_2=20\hat{i}+10\hat{j}m/s$. Find the magnitude of the velocity of the second car relative to the first.

    • (a) $10m/s$
    • (b) $14.14m/s$
    • (c) $20m/s$
    • (d) $30m/s$

15. A projectile is launched at $v_0=80m/s$ at $\theta ={60}^{\circ }$ ($g=9.8m/s^2$). Calculate the time to reach maximum height.

    • (a) $5.0s$
    • (b) $6.0s$
    • (c) $7.0s$
    • (d) $8.0s$

16. A ball is thrown at $v_0=25m/s$ at $\theta ={30}^{\circ }$ ($g=9.8m/s^2$). Find the range.

    • (a) $50m$
    • (b) $55.2m$
    • (c) $60m$
    • (d) $65m$

17. A particle moves in a circle of radius $5m$ at $v=3m/s$. Find the angular velocity.

    • (a) $0.4rad/s$
    • (b) $0.6rad/s$
    • (c) $0.8rad/s$
    • (d) $1.0rad/s$

18. An airplane flies at ${\overrightarrow{v}}_a=200\hat{i}m/s$ in a wind ${\overrightarrow{v}}_w=50\hat{j}m/s$. Find the magnitude of the airplane’s velocity relative to the ground.

    • (a) $200m/s$
    • (b) $206.16m/s$
    • (c) $225m/s$
    • (d) $250m/s$

19. A particle’s position is $\overrightarrow{r}(t)=(5t)\hat{i}+(2t^2)\hat{j}+(t)\hat{k}m$. Find the magnitude of the acceleration at $t=1s$.

    • (a) $2m/s^2$
    • (b) $4m/s^2$
    • (c) $6m/s^2$
    • (d) $8m/s^2$

20. A projectile is launched at $v_0=70m/s$ at $\theta ={45}^{\circ }$ ($g=9.8m/s^2$). Find the maximum height.

    • (a) $100m$
    • (b) $125.26m$
    • (c) $150m$
    • (d) $175m$

21. A particle moves in a circle of radius $15m$ with a period of $T=3s$. Calculate the speed.

    • (a) $25.13m/s$
    • (b) $31.42m/s$
    • (c) $37.71m/s$
    • (d) $43.98m/s$

22. Two objects have velocities ${\overrightarrow{v}}_1=10\hat{i}+5\hat{j}m/s$ and ${\overrightarrow{v}}_2=5\hat{i}-5\hat{j}m/s$. Find the magnitude of the velocity of the first relative to the second.

    • (a) $5m/s$
    • (b) $7.07m/s$
    • (c) $10m/s$
    • (d) $12.5m/s$

23. A projectile is launched at $v_0=100m/s$ at $\theta ={30}^{\circ }$ ($g=9.8m/s^2$). Calculate the range.

    • (a) $800m$
    • (b) $883.7m$
    • (c) $900m$
    • (d) $1000m$

24. A particle moves in a circle of radius $8m$ with $\omega =4rad/s$. Find the centripetal acceleration.

    • (a) $64m/s^2$
    • (b) $96m/s^2$
    • (c) $128m/s^2$
    • (d) $160m/s^2$

25. A car moves at ${\overrightarrow{v}}_c=15\hat{i}m/s$ while rain falls at ${\overrightarrow{v}}_r=-8\hat{j}m/s$. Find the magnitude of the rain’s velocity relative to the car.

    • (a) $15m/s$
    • (b) $17m/s$
    • (c) $19m/s$
    • (d) $21m/s$

26. A projectile is launched at $v_0=45m/s$ at $\theta ={60}^{\circ }$ ($g=9.8m/s^2$). Find the time of flight.

    • (a) $4.0s$
    • (b) $5.0s$
    • (c) $6.0s$
    • (d) $7.0s$

27. A ball is thrown at $v_0=35m/s$ at $\theta ={45}^{\circ }$ ($g=9.8m/s^2$). Find the range.

    • (a) $100m$
    • (b) $125m$
    • (c) $132.5m$
    • (d) $150m$

28. A particle moves in a circle of radius $25m$ at $v=10m/s$. Find the period.

    • (a) $10.47s$
    • (b) $12.57s$
    • (c) $15.71s$
    • (d) $18.85s$

29. An airplane flies at ${\overrightarrow{v}}_a=250\hat{i}m/s$ in a wind ${\overrightarrow{v}}_w=-40\hat{j}m/s$. Find the magnitude of the airplane’s velocity relative to the ground.

    • (a) $250m/s$
    • (b) $253.55m/s$
    • (c) $260m/s$
    • (d) $270m/s$

30. A particle’s position is $\overrightarrow{r}(t)=(t^2)Hati+(2t)\hat{j}m$. Find the magnitude of the velocity at $t=3s$.

    • (a) $4m/s$
    • (b) $6m/s$
    • (c) $8m/s$
    • (d) $10m/s$

31. A projectile is launched at $v_0=90m/s$ at $\theta ={30}^{\circ }$ ($g=9.8m/s^2$). Find the vertical velocity at $t=2s$.

    • (a) $25.4m/s$
    • (b) $30.4m/s$
    • (c) $35.4m/s$
    • (d) $40.4m/s$

32. A particle moves in a circle of radius $12m$ with a period of $T=4s$. Calculate the centripetal acceleration.

    • (a) $18.85m/s^2$
    • (b) $22.62m/s^2$
    • (c) $26.39m/s^2$
    • (d) $30.16m/s^2$

33. Two particles have velocities ${\overrightarrow{v}}_1=15\hat{i}+10\hat{j}m/s$ and ${\overrightarrow{v}}_2=10\hat{i}+15\hat{j}m/s$. Find the magnitude of the velocity of the first relative to the second.

    • (a) $5m/s$
    • (b) $7.07m/s$
    • (c) $10m/s$
    • (d) $12.5m/s$

34. A projectile is launched at $v_0=55m/s$ at $\theta ={60}^{\circ }$ ($g=9.8m/s^2$). Find the maximum height.

    • (a) $75m$
    • (b) $100m$
    • (c) $116.33m$
    • (d) $125m$

35. A car moves in a circle of radius $30m$ with $\omega =3rad/s$. Find the speed.

    • (a) $60m/s$
    • (b) $75m/s$
    • (c) $90m/s$
    • (d) $105m/s$

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Conceptual Problems

36. What is the main difference between motion in 1D and 2D?

• (a) 1D involves scalars, 2D involves vectors
• (b) 1D involves vectors, 2D involves scalars
• (c) 1D has one component, 2D has two components
• (d) 1D has two components, 2D has one component

37. What does a zero vertical velocity indicate in projectile motion?

• (a) Launch point
• (b) Maximum height
• (c) Landing point
• (d) Midpoint of flight

38. What is the direction of centripetal acceleration in uniform circular motion?

• (a) Tangent to the circle
• (b) Away from the center
• (c) Toward the center
• (d) Along the velocity

39. What does relative velocity ${\overrightarrow{v}}_{A/B}$ represent?

• (a) Velocity of A relative to the ground
• (b) Velocity of B relative to the ground
• (c) Velocity of A as observed by B
• (d) Velocity of B as observed by A

40. What is the unit of angular velocity in SI units?

• (a) $m/s$
• (b) $rad/s$
• (c) $m/s^2$
• (d) $s$

41. At what angle is the range of a projectile maximum on level ground?

• (a) ${30}^{\circ }$
• (b) ${45}^{\circ }$
• (c) ${60}^{\circ }$
• (d) ${90}^{\circ }$

42. What happens to the horizontal velocity of a projectile during flight (no air resistance)?

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

43. What is the physical significance of centripetal acceleration?

• (a) Changes speed of the object
• (b) Changes direction of the velocity
• (c) Changes magnitude of the velocity
• (d) Changes position of the object

44. How is relative velocity calculated?

• (a) ${\overrightarrow{v}}_A+{\overrightarrow{v}}_B$
• (b) ${\overrightarrow{v}}_A-{\overrightarrow{v}}_B$
• (c) ${\overrightarrow{v}}_A\times {\overrightarrow{v}}_B$
• (d) ${\overrightarrow{v}}_A\cdot {\overrightarrow{v}}_B$

45. What is the dimension of centripetal acceleration?

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

46. What does a zero horizontal displacement indicate in projectile motion?

• (a) Launched vertically
• (b) Launched horizontally
• (c) Maximum height
• (d) Midpoint of flight

47. What happens to the velocity vector in uniform circular motion?

• (a) Magnitude changes, direction constant
• (b) Magnitude constant, direction changes
• (c) Both magnitude and direction change
• (d) Neither magnitude nor direction changes

48. Which quantity is a vector in projectile motion?

• (a) Speed
• (b) Time of flight
• (c) Velocity
• (d) Range

49. What is the significance of the period in circular motion?

• (a) Time for one complete revolution
• (b) Time for half a revolution
• (c) Speed of the object
• (d) Radius of the circle

50. What does a negative relative velocity component indicate?

• (a) Objects moving in the same direction
• (b) Objects moving in opposite directions
• (c) Objects at rest
• (d) Objects perpendicular to each other

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Derivation Problems

51. Derive the expression for instantaneous velocity in 3D motion using the position vector.

52. Derive the expression for instantaneous acceleration in 2D motion using the velocity vector.

53. Derive the time of flight for a projectile launched at an angle on level ground.

54. Derive the maximum height of a projectile launched at an angle.

55. Derive the range of a projectile on level ground.

56. Derive the centripetal acceleration for uniform circular motion.

57. Derive the period of circular motion in terms of speed and radius.

58. Derive the relative velocity ${\overrightarrow{v}}_{A/B}$ using position vectors.

59. Derive the angular velocity in terms of linear speed and radius.

60. Derive the banking angle for a car on a curved road without friction.

61. Derive the speed of a satellite in circular orbit using the period.

62. Derive the trajectory equation of a projectile ($y$ as a function of $x$).

63. Derive the magnitude of the velocity vector in 3D motion.

64. Derive the relative position over time for two moving objects.

65. Derive the centripetal acceleration using angular velocity.

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NEET-style Conceptual Problems

66. What is the unit of displacement in 2D motion?

• (a) $m/s$
• (b) $m$
• (c) $m/s^2$
• (d) $s$

67. What does a zero vertical velocity in projectile motion indicate?

• (a) Launch point
• (b) Maximum height
• (c) Landing point
• (d) Midpoint of flight

68. Which quantity remains constant in the horizontal direction for a projectile (no air resistance)?

• (a) Acceleration
• (b) Velocity
• (c) Position
• (d) Time

69. What is the direction of the velocity vector in uniform circular motion?

• (a) Toward the center
• (b) Away from the center
• (c) Tangent to the circle
• (d) Along the radius

70. What does relative velocity ${\overrightarrow{v}}_{A/B}$ represent?

• (a) Velocity of A relative to the ground
• (b) Velocity of B relative to the ground
• (c) Velocity of A as observed by B
• (d) Velocity of B as observed by A

71. What is the dimension of angular velocity?

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

72. What is the role of centripetal acceleration in circular motion?

• (a) Changes the speed
• (b) Changes the direction of velocity
• (c) Changes the position
• (d) Changes the angular velocity

73. What happens to the vertical velocity of a projectile at maximum height?

• (a) It is zero
• (b) It equals the initial velocity
• (c) It equals $g$
• (d) It is negative

74. Why is the horizontal acceleration zero in projectile motion (no air resistance)?

• (a) No force acts horizontally
• (b) Gravity acts horizontally
• (c) Velocity is zero
• (d) Position is constant

75. What is the unit of centripetal acceleration?

• (a) $m/s$
• (b) $m/s^2$
• (c) $rad/s$
• (d) $m$

76. What does a constant speed in circular motion imply?

• (a) Zero acceleration
• (b) Constant centripetal acceleration
• (c) Changing centripetal acceleration
• (d) Zero velocity

77. Which kinematic equation is used to find the range of a projectile?

• (a) $v=u+at$
• (b) $x=ut+\frac{1}{2}a{t}^2$
• (c) $v^2=u^2+2ax$
• (d) $x=\frac{1}{2}(u+v)t$

78. What is the direction of acceleration in uniform circular motion?

• (a) Tangent to the circle
• (b) Toward the center
• (c) Away from the center
• (d) Along the velocity

79. What does a negative relative velocity component indicate?

• (a) Objects moving in the same direction
• (b) Objects moving in opposite directions
• (c) Objects at rest
• (d) Objects perpendicular to each other

80. What is the dimension of velocity in 2D motion?

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

81. What is the role of the time of flight in projectile motion?

• (a) Time to reach maximum height
• (b) Total time in the air
• (c) Time to travel the range
• (d) Time to change direction

82. What happens to the velocity vector at the maximum height of a projectile?

• (a) Only horizontal component remains
• (b) Only vertical component remains
• (c) Both components are zero
• (d) Both components change

83. Why is centripetal acceleration necessary in circular motion?

• (a) To change speed
• (b) To change direction
• (c) To change position
• (d) To change angular velocity

84. What does relative motion describe?

• (a) Motion of an object relative to the ground
• (b) Motion of one object as observed by another
• (c) Motion of an object in a straight line
• (d) Motion of an object in a circle

85. What is the unit of the period in circular motion?

• (a) $m$
• (b) $s$
• (c) $m/s$
• (d) $rad/s$

86. What does a zero horizontal velocity indicate in projectile motion?

• (a) Launched vertically
• (b) Launched horizontally
• (c) Maximum height
• (d) Midpoint of flight

87. What is the physical significance of $\frac{v^2}{r}$ in circular motion?

• (a) Angular velocity
• (b) Centripetal acceleration
• (c) Linear velocity
• (d) Period

88. How is the velocity vector oriented in projectile motion at launch?

• (a) Parallel to the ground
• (b) Perpendicular to the ground
• (c) At an angle to the ground
• (d) Zero

89. What is the dimension of relative velocity?

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

90. Why does the vertical velocity change in projectile motion?

• (a) Due to horizontal acceleration
• (b) Due to gravity
• (c) Due to air resistance
• (d) Due to initial speed

91. What is the role of angular velocity in circular motion?

• (a) Measures linear speed
• (b) Measures rate of rotation
• (c) Measures centripetal acceleration
• (d) Measures period

92. What does a zero centripetal acceleration indicate?

• (a) Circular motion
• (b) Straight-line motion
• (c) Zero velocity
• (d) Zero speed

93. What is the physical significance of ${\overrightarrow{v}}_A-{\overrightarrow{v}}_B$?

• (a) Absolute velocity
• (b) Relative velocity
• (c) Angular velocity
• (d) Centripetal acceleration

94. What is the dimension of the period in circular motion?

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

95. Why is the range of a projectile dependent on $\sin 2\theta$?

• (a) It accounts for horizontal velocity
• (b) It accounts for vertical velocity
• (c) It accounts for both components
• (d) It accounts for gravity

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NEET-style Numerical Problems

96. A projectile is launched at $v_0=65m/s$ at $\theta ={45}^{\circ }$ ($g=9.8m/s^2$). What is the time of flight?

• (a) $6.0s$
• (b) $7.0s$
• (c) $8.0s$
• (d) $9.0s$

97. A car moves in a circle of radius $40m$ at $v=8m/s$. What is the centripetal acceleration?

• (a) $1.6m/s^2$
• (b) $2.0m/s^2$
• (c) $2.4m/s^2$
• (d) $2.8m/s^2$

98. A boat moves at ${\overrightarrow{v}}_b=10\hat{i}m/s$ in a river with current ${\overrightarrow{v}}_r=6\hat{j}m/s$. What is the magnitude of the boat’s velocity relative to the ground?

• (a) $11.66m/s$
• (b) $12m/s$
• (c) $13m/s$
• (d) $14m/s$

99. A projectile is launched at $v_0=85m/s$ at $\theta ={30}^{\circ }$ ($g=9.8m/s^2$). What is the maximum height?

• (a) $75m$
• (b) $92.6m$
• (c) $100m$
• (d) $125m$

100. Two particles have velocities ${\overrightarrow{v}}_1=25\hat{i}m/s$ and ${\overrightarrow{v}}_2=15\hat{i}+20\hat{j}m/s$. What is the magnitude of the velocity of the second relative to the first?
     - (a) $10m/s$
     - (b) $15m/s$
     - (c) $20m/s$
     - (d) $22.36m/s$

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