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Speed has no sense of direction unlike the velocity. Relative speed is the speed of one object as observed from another moving object. Questions on train are the classic examples of relative speed and in all these questions it is assumed that trains move parallel to each other â€“ whether in the same direction or the opposite direction. Thus, we shall see how the relative speed is calculated and using it we come to know the time taken by the trains to cross each other and some other like aspects.

**Important Formulas - Problems on Trains**

**1.** x km/hr = (xÃ—5)/18 m/s

**2.** y m/s = (yÃ—18)/5 km/hr

**3.** Speed = distance/time, that is, s = d/t

**4.** velocity = displacement/time, that is, v = d/t

**5. **Time taken by a train x meters long to pass a pole or standing man or a post = Time taken by the train to travel x meters.

**6. **Time taken by a train x meters long to pass an object of length y meters = Time taken by the train to travel (x + y) metres.

**7.** Suppose two trains or two objects are moving in the same direction at v1 m/s and v2 m/s where v1 > v2, then their relative speed = (v1 â€“ v2) m/s

**8.** Suppose two trains or two objects are moving in opposite directions at v1 m/s and v2 m/s , then their relative speed = (v1+ v2) m/s

**9. **Assume two trains of length x metres and y metres are moving in opposite directions at v1 m/s and v2 m/s, Then The time taken by the trains to cross each other = (x+y) / (v1+v2) seconds

**10.** Assume two trains of length x metres and y metres are moving in the same direction at at v1 m/s and v2 m/s where v1 > v2, Then The time taken by the faster train to cross the slower train = (x+y) / (v1-v2) seconds

**11.** Assume that two trains (objects) start from two points P and Q towards each other at the same time and after crossing they take p and q seconds to reach Q and P respectively. Then, A's speed: B's speed = âˆšq: âˆšp

__Solved Examples__

| |

A. 190 metres | B. 160 metres |

C. 200 metres
| D. 120 metres |

**Explanation :**

Speed of the train, v = 40 km/hr = 40000/3600 m/s = 400/36 m/s

Time taken to cross, t = 18 s

Distance Covered, d = vt = (400/36)Ã— 18 = 200 m

Distance covered is equal to the length of the train = 200 m

| |

A. 120 sec | B. 99 s |

C. 89 s | D. 80 s |

**Answer :** Option C

**Explanation :**

v = 240/24 (where v is the speed of the train) = 10 m/s

t = (240+650)/10 = 89 seconds

| |

A. 10.8 s | B. 12 s |

C. 9.8 s | D. 8 s |

**Answer : **Option A

**Explanation :**

Distance = 140+160 = 300 m

Relative speed = 60+40 = 100 km/hr = (100Ã—10)/36 m/s

Time = distance/speed = 300 / (100Ã—10)/36 = 300Ã—36 / 1000 = 3Ã—36/10 = 10.8 s

| |

A. 79.2 km/hr | B. 69 km/hr |

C. 74 km/hr | D. 61 km/hr |

**Answer : **Option A

**Explanation :**

Let x is the length of the train and v is the speed

Time taken to move the post = 8 s

=> x/v = 8

=> x = 8v --- (1)

Time taken to cross the platform 264 m long = 20 s

(x+264)/v = 20

=> x + 264 = 20v ---(2)

Substituting equation 1 in equation 2, we get

8v +264 = 20v

=> v = 264/12 = 22 m/s

= 22Ã—36/10 km/hr = 79.2 km/hr

| |

A. 2 : 3 | B. 2 :1 |

C. 4 : 3 | D. 3 : 2 |

**Answer :** Option C

**Explanation :**

Ratio of their speeds = Speed of first train : Speed of second train

= âˆš16: âˆš 9

= 4:3

| |

A. 320 m | B. 190 m |

C. 210 m | D. 230 m |

**Answer : **Option D

**Explanation :**

Relative speed = 120+80 = 200 kmph = 200Ã—10/36 m/s = 500/9 m/s

time = 9 s

Total distance covered = 270 + x where x is the length of other train

(270+x)/9 = 500/9

=> 270+x = 500

=> x = 500-270 = 230 meter

| |

A. 10.30 a.m | B. 10 a.m. |

C. 9.10 a.m. | D. 11 a.m. |

**Answer : **Option B

**Explanation :**

Assume both trains meet after x hours after 7 am

Distance covered by train starting from P in x hours = 20x km

Distance covered by train starting from Q in (x-1) hours = 25(x-1)

Total distance = 110

=> 20x + 25(x-1) = 110

=> 45x = 135

=> x= 3 Means, they meet after 3 hours after 7 am, ie, they meet at 10 am

| |

A. 42 | B. 36 |

C. 28 | D. 20 |

**Answer : **Option B

**Explanation :**

Distance covered = 120+120 = 240 m

Time = 12 s

Let the speed of each train = v. Then relative speed = v+v = 2v

2v = distance/time = 240/12 = 20 m/s

Speed of each train = v = 20/2 = 10 m/s

= 10Ã—36/10 km/hr = 36 km/hr

| |

A. 270 m | B. 245 m |

C. 235 m | D. 220 m |

**Answer : **Option B

**Explanation :**

Assume the length of the bridge = x meter

Total distance covered = 130+x meter

total time taken = 30 s

speed = Total distance covered /total time taken = (130+x)/30 m/s

=> 45 Ã— (10/36) = (130+x)/30

=> 45 Ã— 10 Ã— 30 /36 = 130+x

=> 45 Ã— 10 Ã— 10 / 12 = 130+x

=> 15 Ã— 10 Ã— 10 / 4 = 130+x

=> 15 Ã— 25 = 130+x = 375

=> x = 375-130 =245

| |

A. 182 km/hr | B. 180 km/hr |

C. 152 km/hr | D. 169 km/hr |

**Answer :** Option A

**Explanation :**

Length of the train, l = 150m

Speed of the man, Vm= 2 km/hr

Relative speed, Vr = total distance/time = (150/3) m/s = (150/3) Ã— (18/5) = 180 km/hr

Relative Speed = Speed of train, Vt - Speed of man (As both are moving in the same direction)

=> 180 = Vt - 2 => Vt = 180 + 2 = 182 km/hr

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