JEE Main - Relative velocity and motion in a plane Online Test (MCQ Quiz)

1. Two bodies A and B move with velocities v_A and v_B respectively. The relative velocity of A with respect to B is given by:

A) v_A + v_B B) v_B - v_A C) v_A - v_B D) v_A * v_B

2. A boat is moving with velocity (3i + 4j) km/hr in a river. The river is flowing with velocity (-3i + 4j) km/hr. The velocity of the boat relative to the river bank is:

A) 6i km/hr B) 8j km/hr C) 0i + 8j km/hr D) 6i + 8j km/hr

3. A particle moves in the xy-plane such that its position vector is given by r(t) = (2t^2 - t)i + (3t - 5)j. What is its velocity vector at t = 2 seconds?

A) 7i + 3j B) 6i + 3j C) 8i + 3j D) 5i + 3j

4. If the velocity of rain is v_r = (-2i - 3j) km/hr and the velocity of a man walking is v_m = (i - j) km/hr, the velocity of rain relative to the man is:

A) -i - 2j B) -3i - 2j C) -i - 4j D) -3i - 4j

5. A particle is projected with velocity v_0 at an angle theta with the horizontal. Its velocity vector in the air is:

A) v_0 cos(theta) i + v_0 sin(theta) j B) v_0 cos(theta) i + (v_0 sin(theta) - gt) j C) v_0 i + v_0 j D) v_0 cos(theta) i

6. Two particles A and B are moving in the xy-plane. Particle A has velocity v_A = (3i + 2j) m/s and particle B has velocity v_B = (-i + 4j) m/s. The relative velocity of A with respect to B is:

A) 4i - 2j B) 2i + 6j C) 4i + 6j D) 2i - 2j

7. A plane is flying with velocity (100i + 20j) km/hr. A wind blows with velocity (20i + 50j) km/hr. The velocity of the plane relative to the ground is:

A) 120i + 70j B) 80i - 30j C) 100i + 20j D) 120i + 50j

8. The position of a particle is given by x(t) = 5t^3 - 2t^2 + 4, y(t) = 3t^2 - 6t + 1. The velocity vector at t=1 is:

A) 9i - 3j B) 11i - 0j C) 9i + 0j D) 11i - 3j

9. A car moves north at 30 m/s. Another car moves east at 40 m/s. The velocity of the second car relative to the first car is:

A) 40i - 30j B) 30i + 40j C) -30i + 40j D) 30i - 40j

10. The acceleration of a particle moving in a plane is given by a = (2i + 3j) m/s^2. If its initial velocity is zero, its velocity at t = 5 seconds will be:

A) 10i + 15j B) 2i + 3j C) 5i + 5j D) 7i + 8j

11. A boat heads due north at 20 m/s in a river flowing east at 10 m/s. The velocity of the boat relative to the bank is:

A) 20i + 10j B) 10i + 20j C) -10i + 20j D) 20i - 10j

12. Two particles P and Q are moving with velocities v_P = (2i - j) m/s and v_Q = (i + 3j) m/s. The velocity of P relative to Q is:

A) i - 4j B) 3i + 2j C) i + 4j D) 3i - 4j

13. A projectile is fired with initial velocity u at an angle alpha to the horizontal. The horizontal component of its velocity remains constant, while the vertical component changes due to gravity. This describes motion in:

A) One dimension B) Two dimensions C) Three dimensions D) Zero dimensions

14. A bird flies with velocity (5i - 12j) km/hr. It is flying in a wind blowing with velocity (3i + 4j) km/hr. The velocity of the bird relative to the ground is:

A) 8i - 8j B) 2i - 16j C) 8i - 16j D) 2i - 8j

15. If v_A = (v_x1, v_y1) and v_B = (v_x2, v_y2), then v_A - v_B is:

A) (v_x1 - v_x2, v_y1 - v_y2) B) (v_x1 + v_x2, v_y1 + v_y2) C) (v_x1 * v_x2, v_y1 * v_y2) D) (v_x1 / v_x2, v_y1 / v_y2)

16. A particle's position is given by r(t) = t^3 i + (2t^2 - 5t) j. What is its acceleration vector at t = 3 seconds?

A) 18i + 4j B) 6t i + 4j C) 18i + 0j D) 6i + 4j

17. A ship is moving with velocity (4i - 3j) km/hr. A submarine is moving with velocity (-2i + 5j) km/hr. The velocity of the ship relative to the submarine is:

A) 6i - 8j B) 2i + 2j C) 6i + 2j D) 4i - 3j

18. A cyclist is moving north at 15 m/s. A car is moving west at 20 m/s. The velocity of the car relative to the cyclist is:

A) -20i - 15j B) 20i + 15j C) -20i + 15j D) 20i - 15j

19. The velocity of a particle is given by v(t) = (2t i + 3t^2 j). What is its displacement from t=1 to t=3?

A) 8i + 24j B) 4i + 26j C) 8i + 26j D) 4i + 24j

20. A person walks on a moving walkway. The person walks at 2 m/s relative to the walkway, and the walkway moves at 1 m/s relative to the ground. If the person walks in the same direction as the walkway, their speed relative to the ground is:

A) 1 m/s B) 2 m/s C) 3 m/s D) 4 m/s

21. Two cars A and B are moving on a straight road. Car A moves north at 60 km/hr. Car B moves south at 80 km/hr. The velocity of car A relative to car B is:

A) 140 km/hr North B) 20 km/hr North C) 140 km/hr South D) 20 km/hr South

22. If a particle's position is described by r(t) = (t^2 - 1)i + (2t + 3)j, its velocity vector at t=0 is:

A) 0i + 2j B) 2t i + 2j C) 0i + 0j D) -1i + 3j

23. A boat can move at 10 m/s in still water. It wants to cross a river flowing at 5 m/s. If the boat heads perpendicular to the river flow, its velocity relative to the bank will be:

A) 10 m/s B) 5 m/s C) sqrt(125) m/s D) 15 m/s

24. A particle has velocity v = (3i + 4j) m/s. What is its speed?

A) 3 m/s B) 4 m/s C) 5 m/s D) 7 m/s

25. The position vector of a particle is r(t) = (t^2 i + t j). Its velocity vector is v(t) = 2t i + j. The acceleration vector is:

A) 2i B) 2j C) 2i + j D) t i + 2j

26. A plane flies north at 200 km/hr. The wind blows east at 50 km/hr. The velocity of the plane relative to the air is:

A) 200i + 50j B) -200i + 50j C) 200i - 50j D) -200i - 50j

27. Two trains A and B are moving on parallel tracks. Train A moves east at 30 m/s. Train B moves west at 40 m/s. The velocity of train A relative to train B is:

A) 70 m/s East B) 10 m/s East C) 70 m/s West D) 10 m/s West

28. A boat is steered due west at 15 km/hr in a river flowing north at 20 km/hr. The velocity of the boat relative to the river bank is:

A) -15i + 20j B) 15i + 20j C) -15i - 20j D) 15i - 20j

29. If the velocity of a particle is given by v = (3i + 4j) m/s, its direction of motion with respect to the positive x-axis is given by an angle theta such that:

A) tan(theta) = 3/4 B) tan(theta) = 4/3 C) sin(theta) = 3/5 D) cos(theta) = 4/5

30. A particle's position is given by x(t) = 2t^2 and y(t) = 3t. Its velocity vector is:

A) 4t i + 3j B) 2t i + 3j C) 4t i + 3t j D) 2t i + 3t j

31. A swimmer can swim at 5 m/s in still water. He swims across a river of width 100m flowing at 10 m/s. If he swims perpendicular to the flow, his resultant velocity relative to the bank is:

A) 5 m/s B) 10 m/s C) sqrt(125) m/s D) 15 m/s

32. The velocity of a car is v_c = (10i + 20j) m/s. The velocity of a motorcycle is v_m = (5i + 15j) m/s. The velocity of the car relative to the motorcycle is:

A) 5i + 5j B) 15i + 35j C) 5i + 35j D) 15i + 5j

33. A particle moves in the xy-plane with velocity v = (3t^2 - 2t)i + (4t - 1)j. Its acceleration vector at t=2 is:

A) 8i + 3j B) 10i + 3j C) 8i + 4j D) 10i + 4j

34. A projectile is launched with velocity v_0 at angle theta. The vertical component of its velocity at time t is:

A) v_0 cos(theta) B) v_0 sin(theta) C) v_0 sin(theta) - gt D) v_0 cos(theta) - gt

35. A boat travels north at 30 km/hr. A wind blows southwest at 40 km/hr. The velocity of the boat relative to the ground is approximately:

A) 10 km/hr North B) 10 km/hr South C) 20 km/hr North D) 20 km/hr South

36. Two particles A and B have velocities v_A = (2i + 3j) m/s and v_B = (i + 2j) m/s. The relative velocity v_B - v_A is:

A) i + j B) 3i + 5j C) -i - j D) i - j

37. The position of a particle is given by r(t) = (5t i + 10t^2 j). The magnitude of its velocity at t=1 is:

A) 5 B) 10 C) 15 D) 20

38. A person walks east at 4 km/hr. The river flows north at 3 km/hr. The velocity of the person relative to the river bank is:

A) 4i + 3j B) -4i + 3j C) 4i - 3j D) -4i - 3j

39. If the position of a particle is given by x = t^2 and y = 2t, its velocity vector is:

A) 2t i + 2j B) t^2 i + 2t j C) 2i + 2j D) 2t i + 2t j

40. A plane's airspeed is 300 km/hr. It is flying due north. The wind is blowing from the west at 50 km/hr. The velocity of the plane relative to the ground is:

A) 300i + 50j B) -50i + 300j C) 50i + 300j D) 300i - 50j

41. Two particles A and B are moving in the xy-plane with velocities v_A = (3i - 2j) m/s and v_B = (i + 4j) m/s. The relative velocity of B with respect to A is:

A) -2i + 6j B) 2i - 6j C) -2i - 6j D) 4i + 2j

42. A particle's position is given by r(t) = (t^3 - 3t)i + (2t^2 + 4t)j. Its acceleration vector at t=2 is:

A) 12i + 4j B) 6t i + 4t j C) 12i + 8j D) 6i + 4j

43. A boat heads upstream at 10 m/s. The river flows downstream at 5 m/s. The velocity of the boat relative to the bank is:

A) 5 m/s upstream B) 5 m/s downstream C) 15 m/s upstream D) 15 m/s downstream

44. A projectile is thrown with velocity v = (3i + 4j) m/s. The horizontal component of velocity is:

A) 3 m/s B) 4 m/s C) 5 m/s D) 7 m/s

45. If the velocity of rain is v_r = (-3i - 4j) km/hr and a cyclist is moving with v_c = (i - j) km/hr, the velocity of rain relative to the cyclist is:

A) -2i - 3j B) -4i - 3j C) -2i - 5j D) -4i - 5j

46. A particle moves in the xy-plane. Its velocity components are v_x = 2t and v_y = 3. If its initial position is (0,0), its position at t=2 is:

A) (4, 6) B) (2, 6) C) (4, 3) D) (2, 3)

47. A ship is moving at 20 knots due east. A current flows north at 10 knots. The resultant velocity of the ship relative to the seabed is:

A) sqrt(500) knots B) sqrt(300) knots C) sqrt(100) knots D) sqrt(200) knots

48. The velocity vector of a particle is given by v = (2t i + 4j). Its acceleration vector is:

A) 2i + 4j B) 2i C) 4j D) t i + 4j

49. A swimmer can swim at 6 m/s in still water. He wants to cross a river of width 120 m flowing at 8 m/s. To cross the river in the shortest possible time, he should swim:

A) Upstream at an angle B) Downstream at an angle C) Perpendicular to the river flow D) Against the river flow

50. Two particles P and Q are moving with velocities v_P = (4i + 3j) m/s and v_Q = (2i + j) m/s. The relative velocity of P with respect to Q is:

A) 2i + 2j B) 6i + 4j C) 2i + 4j D) 6i + 2j

Relative velocity and motion in a plane - Online Test

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