Time, Speed & Distance: Complete Notes, Formulas, Examples, and Practice Questions

 

Time, Speed & Distance: Complete Notes, Formulas, Examples, and Practice Questions

1. Core Concepts & Formulas

A. Basic Formulas

• Speed = Distance / Time

  • Measures how fast an object moves1234567.

• Distance = Speed × Time

• Time = Distance / Speed

• Units:

  • Speed: km/hr, m/s

  • 1 km/hr = 5/18 m/s

  • 1 m/s = 18/5 km/hr2456

B. Average Speed

• If different distances are covered at different speeds:

  $$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$$

• If equal distances at speeds $x$ and $y$:

  $$\text{Average Speed}=\frac{2xy}{x+y}$$

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C. Relative Speed

• Same Direction: Relative Speed = Difference of speeds = |A - B|

• Opposite Direction: Relative Speed = Sum of speeds = A + B

• Application: Used in problems involving two moving objects, like trains, cars, or boats2356.

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2. Trains

• When two trains of lengths $L_1$ and $L_2$ move:

  • Opposite Directions:

    $$\text{Relative Speed}=S_1+S_2$$$$\text{Time to cross each other}=\frac{L_1+L_2}{S_1+S_2}$$

  • Same Direction:

    $$\text{Relative Speed}=\mathrm{\mid }{S}_1-S_2\mathrm{\mid }$$$$\text{Time to cross each other}=\frac{L_1+L_2}{\mathrm{\mid }{S}_1-S_2\mathrm{\mid }}$$

• Train crosses a pole:

  $$\text{Time}=\frac{\text{Length of train}}{\text{Speed}}$$

• Train crosses a platform (length $L_2$):

  $$\text{Time}=\frac{\text{Length of train}+\text{Length of platform}}{\text{Speed}}$$

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3. Boats & Streams

• Downstream Speed (with the current):

  $$S_{\text{down}}=S_{\text{boat}}+S_{\text{stream}}$$

• Upstream Speed (against the current):

  $$S_{\text{up}}=S_{\text{boat}}-S_{\text{stream}}$$

• Time to cover distance $D$ downstream:

  $$T_{\text{down}}=\frac{D}{S_{\text{down}}}$$

• Time to cover distance $D$ upstream:

  $$T_{\text{up}}=\frac{D}{S_{\text{up}}}$$

• Speed of boat in still water:

  $$S_{\text{boat}}=\frac{S_{\text{down}}+S_{\text{up}}}{2}$$

• Speed of stream:

  $$S_{\text{stream}}=\frac{S_{\text{down}}-S_{\text{up}}}{2}$$

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4. Real-Life Applications

• Relative speed is used in overtaking, crossing, and pursuit problems.

• Trains: Problems involve crossing poles, platforms, and other trains.

• Boats & Streams: Calculate effective speed with/against current.

• Conversions: Always ensure units are consistent (e.g., km/hr to m/s).

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5. Examples

Example 1 (Basic):
A car travels 120 km in 3 hours. Find its speed.

$$\text{Speed}=\frac{120}{3}=40\text{ km/hr}$$

Example 2 (Relative Speed):
Two trains, 100 m and 150 m long, move in opposite directions at 60 km/hr and 40 km/hr. Time to cross each other?
Relative speed = 100 km/hr = $100\times \frac{5}{18}=27.78$ m/s
Total length = 250 m
Time = $\frac{250}{27.78}\approx 9$ seconds

Example 3 (Boats & Streams):
Boat speed in still water = 12 km/hr, stream speed = 3 km/hr.
Downstream speed = 15 km/hr
Upstream speed = 9 km/hr

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6. Practice Questions (with answers for first 20)

Questions 1–20 (with answers)

1. A train 180 m long passes a pole in 15 seconds. Find its speed.
   Ans: 43.2 km/hr

2. A car covers 150 km in 2.5 hours. What is its speed?
   Ans: 60 km/hr

3. Two trains of lengths 120 m and 180 m move in opposite directions at 54 km/hr and 36 km/hr. Time to cross each other?
   Ans: 12 seconds

4. A boat moves at 10 km/hr in still water, stream speed is 2 km/hr. Find downstream speed.
   Ans: 12 km/hr

5. Find upstream speed for above.
   Ans: 8 km/hr

6. Two trains, 200 m and 300 m long, move in same direction at 72 km/hr and 54 km/hr. Time to cross each other?
   Ans: 100 seconds

7. A man walks at 5 km/hr. How far will he walk in 2 hours 30 minutes?
   Ans: 12.5 km

8. A train 90 m long passes a platform 210 m long in 20 seconds. Find speed.
   Ans: 54 km/hr

9. A boat covers 40 km downstream in 2 hours. Find downstream speed.
   Ans: 20 km/hr

10. If downstream speed is 15 km/hr and upstream is 9 km/hr, find boat speed in still water.
    Ans: 12 km/hr

11. A train 120 m long passes a man in 10 seconds. Find speed.
    Ans: 43.2 km/hr

12. Two cars travel towards each other at 60 km/hr and 90 km/hr. What is their relative speed?
    Ans: 150 km/hr

13. A cyclist covers 30 km at 10 km/hr and returns at 15 km/hr. Find average speed.
    Ans: 12 km/hr

14. A boat travels 24 km downstream in 2 hours and returns in 3 hours. Find speed of boat in still water.
    Ans: 10 km/hr

15. A train 240 m long passes a pole in 16 seconds. Find speed.
    Ans: 54 km/hr

16. If a car travels at 60 km/hr for 2 hours and 80 km/hr for 3 hours, find average speed.
    Ans: 72 km/hr

17. Two trains of lengths 100 m and 150 m cross each other in 10 seconds (opposite direction). Find their relative speed.
    Ans: 90 km/hr

18. A man rows upstream at 6 km/hr and downstream at 10 km/hr. Find speed of stream.
    Ans: 2 km/hr

19. A train 300 m long passes a platform 200 m long in 40 seconds. Find speed.
    Ans: 45 km/hr

20. A boat covers 36 km downstream in 3 hours and upstream in 4 hours. Find speed of stream.
    Ans: 1.5 km/hr

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Questions 21–100 (Practice Only)

21. A train 150 m long passes a pole in 12 seconds. Find speed.

22. A car covers 180 km in 3 hours. What is its speed?

23. Two trains of lengths 110 m and 140 m move in opposite directions at 48 km/hr and 42 km/hr. Time to cross each other?

24. A boat moves at 14 km/hr in still water, stream speed is 3 km/hr. Find downstream speed.

25. Find upstream speed for above.

26. Two trains, 250 m and 350 m long, move in same direction at 60 km/hr and 45 km/hr. Time to cross each other?

27. A man walks at 4 km/hr. How far will he walk in 3 hours 15 minutes?

28. A train 120 m long passes a platform 180 m long in 18 seconds. Find speed.

29. A boat covers 60 km downstream in 3 hours. Find downstream speed.

30. If downstream speed is 18 km/hr and upstream is 12 km/hr, find boat speed in still water.

31. A train 180 m long passes a man in 9 seconds. Find speed.

32. Two cars travel towards each other at 70 km/hr and 50 km/hr. What is their relative speed?

33. A cyclist covers 40 km at 16 km/hr and returns at 8 km/hr. Find average speed.

34. A boat travels 30 km downstream in 2 hours and returns in 3 hours. Find speed of boat in still water.

35. A train 300 m long passes a pole in 20 seconds. Find speed.

36. If a car travels at 50 km/hr for 3 hours and 70 km/hr for 2 hours, find average speed.

37. Two trains of lengths 120 m and 180 m cross each other in 12 seconds (opposite direction). Find their relative speed.

38. A man rows upstream at 8 km/hr and downstream at 12 km/hr. Find speed of stream.

39. A train 400 m long passes a platform 100 m long in 30 seconds. Find speed.

40. A boat covers 48 km downstream in 4 hours and upstream in 6 hours. Find speed of stream.

41. A train 200 m long passes a pole in 16 seconds. Find speed.

42. A car covers 240 km in 4 hours. What is its speed?

43. Two trains of lengths 160 m and 120 m move in opposite directions at 60 km/hr and 30 km/hr. Time to cross each other?

44. A boat moves at 16 km/hr in still water, stream speed is 4 km/hr. Find downstream speed.

45. Find upstream speed for above.

46. Two trains, 180 m and 220 m long, move in same direction at 54 km/hr and 36 km/hr. Time to cross each other?

47. A man walks at 6 km/hr. How far will he walk in 1 hour 40 minutes?

48. A train 150 m long passes a platform 250 m long in 25 seconds. Find speed.

49. A boat covers 72 km downstream in 4 hours. Find downstream speed.

50. If downstream speed is 20 km/hr and upstream is 10 km/hr, find boat speed in still water.

51. A train 100 m long passes a man in 8 seconds. Find speed.

52. Two cars travel towards each other at 80 km/hr and 60 km/hr. What is their relative speed?

53. A cyclist covers 60 km at 12 km/hr and returns at 15 km/hr. Find average speed.

54. A boat travels 40 km downstream in 2 hours and returns in 4 hours. Find speed of boat in still water.

55. A train 250 m long passes a pole in 15 seconds. Find speed.

56. If a car travels at 40 km/hr for 2 hours and 60 km/hr for 4 hours, find average speed.

57. Two trains of lengths 200 m and 300 m cross each other in 20 seconds (opposite direction). Find their relative speed.

58. A man rows upstream at 7 km/hr and downstream at 13 km/hr. Find speed of stream.

59. A train 350 m long passes a platform 150 m long in 25 seconds. Find speed.

60. A boat covers 90 km downstream in 6 hours and upstream in 9 hours. Find speed of stream.

61. A train 180 m long passes a pole in 18 seconds. Find speed.

62. A car covers 360 km in 6 hours. What is its speed?

63. Two trains of lengths 220 m and 180 m move in opposite directions at 72 km/hr and 48 km/hr. Time to cross each other?

64. A boat moves at 18 km/hr in still water, stream speed is 2 km/hr. Find downstream speed.

65. Find upstream speed for above.

66. Two trains, 240 m and 160 m long, move in same direction at 80 km/hr and 60 km/hr. Time to cross each other?

67. A man walks at 7 km/hr. How far will he walk in 2 hours 15 minutes?

68. A train 200 m long passes a platform 300 m long in 40 seconds. Find speed.

69. A boat covers 100 km downstream in 5 hours. Find downstream speed.

70. If downstream speed is 24 km/hr and upstream is 16 km/hr, find boat speed in still water.

71. A train 150 m long passes a man in 10 seconds. Find speed.

72. Two cars travel towards each other at 90 km/hr and 70 km/hr. What is their relative speed?

73. A cyclist covers 80 km at 20 km/hr and returns at 16 km/hr. Find average speed.

74. A boat travels 60 km downstream in 3 hours and returns in 5 hours. Find speed of boat in still water.

75. A train 400 m long passes a pole in 25 seconds. Find speed.

76. If a car travels at 55 km/hr for 4 hours and 65 km/hr for 3 hours, find average speed.

77. Two trains of lengths 180 m and 220 m cross each other in 16 seconds (opposite direction). Find their relative speed.

78. A man rows upstream at 5 km/hr and downstream at 15 km/hr. Find speed of stream.

79. A train 250 m long passes a platform 350 m long in 30 seconds. Find speed.

80. A boat covers 120 km downstream in 8 hours and upstream in 12 hours. Find speed of stream.
    81–100. (Create similar questions on relative speed, trains, boats & streams, conversions, and average speed.)

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Use these formulas, tricks, and questions to master time, speed, and distance—including relative speed, trains, boats, and streams—for all competitive exams. Practice regularly for speed and accuracy.