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UGC NET PAPER 1 · UNIT 05 · MODULE 5: AVERAGES, WORK AND MOTION · TOPIC 14

Time, Speed and Distance: Trains, Boats and Races

Distance = speed x time, then trains, boats and streams and races.

  • 7 sub-topics
  • 6 min read
  • Topic 14 of 17

What this topic covers

7 sub-topics, in the order to study them. Tap one to jump to it.

  1. §OverviewThe syllabus names "time and distance" directly, so this topic is very likely to appear.
  2. 14.1Core formulasConcept Formula Speed, distance, time S = D/T, D = S x T, T = D/S km/h to m/s Multiply by 5/18 m/s to km/h Multiply by 18/5 Equal distance: speed and time Inversely…
  3. 14.2TrainsCovers own length, square root of t2 : square root of t1.
    • own length
  4. 14.3Boats and streamsCovers Downstream speed.
    • Downstream speed
    • Upstream speed
  5. 14.4Races and circular tracks"A beats B by x metres in a race of d metres" means that when A has run d metres, B has run (d - x) metres. The speed ratio is d : (d - x). "A beats B by t seconds" means B…
  6. 14.5Worked examplesExample 1.
    • Example 1
    • 54 km/h
    • 20 seconds
    • Example 2
    • 12 s
  7. 14.6TricksCovers Convert units first, Use relative speed, Same distance, different speed, Train and platform, Check units of the answer.
    • Convert units first
    • Use relative speed
    • Same distance, different speed
    • Train and platform
    • Check units of the answer
  8. 14.7TrapsTaking the plain mean of two speeds as the average speed when the distances are equal. Using the sum of speeds when the objects move in the same direction. Forgetting to add…

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§Overview#

The syllabus names "time and distance" directly, so this topic is very likely to appear. Everything starts from one relation: Distance = Speed x Time.

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14.1Core formulas#

In brief

Concept Formula Speed, distance, time S = D/T, D = S x T, T = D/S km/h to m/s Multiply by 5/18 m/s to km/h Multiply by 18/5 Equal distance: speed and time Inversely…

Concept Formula
Speed, distance, time S = D/T, D = S x T, T = D/S
km/h to m/s Multiply by 5/18
m/s to km/h Multiply by 18/5
Equal distance: speed and time Inversely proportional (speeds a : b means times b : a)
Average speed (total distance/total time) General formula, always valid
Average speed for two equal distances at a and b 2ab/(a + b)
Average speed for three equal distances at a, b, c 3abc/(ab + bc + ca)
Relative speed, opposite directions S1 + S2
Relative speed, same direction S1 - S2 (larger minus smaller)
Time to meet or catch up Distance between them/relative speed
Late or early arrival at two speeds S1 and S2 (S1 > S2) Distance = S1 x S2/(S1 - S2) x time difference

Useful conversions: 18 km/h = 5 m/s, 36 km/h = 10 m/s, 54 km/h = 15 m/s, 72 km/h = 20 m/s, 90 km/h = 25 m/s, 108 km/h = 30 m/s.

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14.2Trains#

In brief

Covers own length, square root of t2 : square root of t1.

  • own length

  • A train crossing a pole, a signal or a standing man covers its own length: time = length/speed.

  • A train crossing a platform, bridge or tunnel covers its own length plus the length of the platform: time = (L_train + L_platform)/speed.

  • Two trains crossing each other: distance = sum of the two lengths.

    • Opposite directions: time = (L1 + L2)/(S1 + S2).

    • Same direction (faster overtaking slower): time = (L1 + L2)/(S1 - S2).

  • A train passing a man walking or running: use the relative speed (train speed minus the man's speed if he walks the same way, plus if he walks the opposite way), with the train's length as the distance.

  • Two trains start at the same time from A and B towards each other. If after meeting they take t1 and t2 hours to reach B and A, the ratio of their speeds is square root of t2 : square root of t1.

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14.3Boats and streams#

In brief

Covers Downstream speed.

  • Downstream speed
  • Upstream speed

  • Boat speed in still water = u. Stream speed = v.

  • Downstream speed = u + v. Upstream speed = u - v.

  • u = (downstream + upstream)/2. v = (downstream - upstream)/2.

  • Over the same distance, the times for the upstream and downstream trips are inversely proportional to the speeds.

  • Average speed over a round trip is not u; it is 2 x (u + v)(u - v)/(2u) = (u squared - v squared)/u.

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14.4Races and circular tracks#

In brief

"A beats B by x metres in a race of d metres" means that when A has run d metres, B has run (d - x) metres. The speed ratio is d : (d - x). "A beats B by t seconds" means B…

  • "A beats B by x metres in a race of d metres" means that when A has run d metres, B has run (d - x) metres. The speed ratio is d : (d - x).

  • "A beats B by t seconds" means B takes t more seconds than A for the same distance.

  • On a circular track of length L, two runners starting together meet for the first time after L/(S1 + S2) when running in opposite directions and after L/(S1 - S2) when running in the same direction.

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14.5Worked examples#

In brief

Example 1.

  • Example 1
  • 54 km/h
  • 20 seconds
  • Example 2
  • 12 s
  • 60 s
  • Example 3
  • 48 km/h

Example 1. A train 150 m long passes a pole in 10 seconds. Find its speed in km/h, and the time it takes to cross a 150 m platform.

  • Speed = 150/10 = 15 m/s = 15 x 18/5 = 54 km/h.

  • Crossing the platform: (150 + 150)/15 = 20 seconds.

Example 2. Two trains of lengths 120 m and 180 m run at 54 km/h and 36 km/h. Find the time they take to cross each other going in opposite directions, and in the same direction.

  • Opposite: relative speed 90 km/h = 25 m/s. Time = 300/25 = 12 s.

  • Same direction: relative speed 18 km/h = 5 m/s. Time = 300/5 = 60 s.

Example 3. A man covers equal distances at 60 km/h and at 40 km/h. Find the average speed.

  • 2 x 60 x 40/(60 + 40) = 48 km/h.

Example 4. Two cars 200 km apart start together towards each other at 60 km/h and 40 km/h. When do they meet?

  • Relative speed = 100 km/h. Time = 200/100 = 2 hours.

Example 5. A boat travels at 12 km/h in still water and the stream flows at 3 km/h. How long does a round trip of 45 km each way take?

  • Downstream speed 15, so 45/15 = 3 hours. Upstream speed 9, so 45/9 = 5 hours.

  • Total time = 8 hours.

Example 6. A boat goes 20 km downstream in 2 hours and returns in 4 hours. Find the boat's speed in still water and the stream speed.

  • Downstream = 10 km/h and upstream = 5 km/h. Boat speed = (10 + 5)/2 = 7.5 km/h. Stream speed = (10 - 5)/2 = 2.5 km/h.

Example 7. A boy reaches school 10 minutes late at 4 km/h and 5 minutes early at 5 km/h. Find the distance.

  • Time difference = 15 minutes = 1/4 hour. Distance = (4 x 5)/(5 - 4) x 1/4 = 5 km.

Example 8. In a 100 m race A beats B by 10 m, and B beats C by 10 m. By how much does A beat C?

  • A : B = 100 : 90 and B : C = 100 : 90. So A : C = 10000 : 8100.

  • When A runs 100 m, C runs 81 m. A beats C by 19 m.

Example 9. Two trains start at the same time from A and B towards each other. After meeting, one takes 9 hours to reach B and the other takes 4 hours to reach A. Find the ratio of their speeds.

  • Ratio = square root of 4 : square root of 9 = 2 : 3. (The train that takes 9 hours after meeting is the slower one, so its speed matches the smaller root. Always pair each train with the square root of the other train's time.)

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14.6Tricks#

In brief

Covers Convert units first, Use relative speed, Same distance, different speed, Train and platform, Check units of the answer.

  • Convert units first
  • Use relative speed
  • Same distance, different speed
  • Train and platform
  • Check units of the answer

  • Convert units first: make all speeds m/s or all km/h before doing anything else.

  • Use relative speed for any two moving objects; it turns two moving objects into one.

  • Same distance, different speed: use the speed ratio and the inverse time ratio.

  • Train and platform: always add lengths before dividing by speed.

  • Check units of the answer: if the option list mixes minutes and hours, convert before choosing.

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14.7Traps#

In brief

Taking the plain mean of two speeds as the average speed when the distances are equal. Using the sum of speeds when the objects move in the same direction. Forgetting to add…

  • Taking the plain mean of two speeds as the average speed when the distances are equal.

  • Using the sum of speeds when the objects move in the same direction.

  • Forgetting to add the platform length to the train's length.

  • Treating the speed of the boat in still water as the average speed of a round trip.

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Test yourself on Time, Speed and Distance: Trains, Boats and Races

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