11. If a train takes 34 seconds to pass a platform which is 480 m long and 10 seconds to cross pole then how long will it take to pass a man running at a speed of 18 km/hr in the same direction?

यदि एक ट्रेन 480 मीटर लंबे प्लेटफॉर्म को पार करने में 34 सेकंड का समय लेती है और 10 सेकंड पोल को पार करने में लेती है, तो वह ट्रेन उसी दिशा में 18 किमी/घंटा की गति से दौड़ने वाले व्यक्ति को पार करने में कितना समय लेगी?

Option “B” is correct.

Given:

Length of platform = 480 m

Time taken to Cross platform = 34

Time taken to Cross pole = 10 Seconds

The man running at speed of 18 km/hr in same direction

Formula used:

Time = Distance/Speed

Concept:

When a train crosses a vertical object, Distance traveled by train = Length of train

When a train crosses a horizontal object, Distance traveled by train = Length of train + Length of the object

Calculation:

Let speed be S and distance be X

Case 1:

Train crossed pole in 10 seconds

∴ 10 = (X/S)

⇒ 10 × S = X

Case 2:

Train crossed platform of 480 m in 34 seconds

∴ 34 = (480 + X)/S

⇒ 34 = (480 + 10S)/S —(∵ 10 × S = X (From Equation 1))

⇒ (34 × S) – (10 × S) = 480

⇒ 24 × S = 480

⇒ S = 20 m/s

Hence, X = 200 m

Now, A man running at speed of 18 km/hr is the same direction

⇒ 18 km/hr = 5 m/s

∵ 1 km/hr = (5/18) m/s

Also,

Relative Speed = 20 – 5 = 15 m/s

⇒ Time = Distance/ Speed

⇒ Time = 200/15

⇒ Time = 40/3 = 13.33 m/s

12. A 150 meter long train at the speed of 60 km/hr crosses a man who is running in the opposite direction to it at the speed of 12 km/hr in x seconds. What is the value of x?

60 किमी/घंटे की गति से 150 मीटर लंबी रेलगाड़ी एक आदमी को x सेकंड में पार करती है, जो इसके विपरीत दिशा में 12 किमी/घंटे की गति से दौड़ रहा है। x का मान क्या है?

Option “C” is correct.

Given:

Length of the train = 150 m

Speed of the train = 60 km/hr

Speed of the man = 12 km/hr

The train crosses the man in x seconds

Formula:

If the speed of the two trains is x km/hr and y km/hr respectively and if x > y.

Relative speed, if the directions are opposite = (x + y) km/hr

Relative speed, if the direction is the same = (x – y) km/hr

Speed = Distance/Time

1 km/hr = 5/18 m/s

Calculation:

Relative speed of the train and the man, if both are running in opposite directions = 60 + 12 = 72 km/hr.

According to the question

72 × (5/18) = 150/x

⇒ 20 = 150/x

⇒ x = 150/20

⇒ x = 7.5 seconds

∴ Train crosses the man in 7.5 seconds.

13. The distance between two cities X and Y is 270 km. First train starts from X at 7:00 a.m. and travels towards Y at 40 km/hr. Second train starts from Y at 8:30 a.m. and travels towards X at 30 km/hr. At what time (in a.m.) will both the trains meet?

दो शहर X और Y के बीच की दूरी 270 किमी है। पहली ट्रेन X से पूर्वाह्न 7 :00 बजे शुरू होती है और Y की ओर 40 किमी/घंटा की गति से चलती है। दूसरी ट्रेन Y से पूर्वाह्न 8 :30 बजे शुरू होती है और X की ओर 30 किमी/घंटा की गति से चलती है। तो किस समय पर (पूर्वाह्न में) दोनों ट्रेन एक-दूसरे से मिलेंगी?

Option “C” is correct.

⇒ Total distance travelled by first train in 1 and half hour (from 7:00 am to 8:30 am) = speed × time = 40 × {1 + (1/2)} = 40 × (3/2) = 60
     km
⇒ Total distance to be covered now = 270 – 60 = 210 km

⇒ Relative speed = 40 + 30 = 70 km/hr

⇒ Time taken = distance/speed = 210/70 = 3 hr

∴ They will meet at 11:30 am ie. (8:30 am + 3 hrs)

14. What will be the ratio of time taken by Amit in crossing a train of length 300 m while moving in the same direction to the train and while moving in the opposite direction if the speed of the train is 60 km/h and the speed of Amit is 5 m/s.

यदि ट्रेन की गति 60 किमी/घंटा है और अमित की गति 5 मीटर/सेकंड है, तो अमित द्वारा 300 मीटर लम्बी ट्रेन को समान दिशा में और विपरीत दिशा में यात्रा करते हुए पार करने में लिए गये समय का अनुपात क्या होगा?

Option “D” is correct.

Speed of Amit 5 m/s = 5 x (18/5) km/h = 18 km/h

While moving in the same direction, relative speed = 60 – 18 = 42 km/hr

Taken time while moving in the same direction = 0.3/42 = 1/140 hr

While moving in the opposite direction, relative speed = 60 + 18 = 78 km/hr

Taken time while moving in the opposite direction = 0.3/78 = 1/260 hr

Required ratio = 1/140 ∶ 1/260 = 13 ∶ 7

15. Two trains of equal length travelling in opposite directions at 72 km/h and 108 km/h cross each other in 10 second. In how much time (in seconds) does the first train cross a platform of length 350 m?

72 किमी/घंटा और 108 किमी/घंटा पर विपरीत दिशाओं में यात्रा करने वाली समान लंबाई की दो रेलगाड़ी 10 सेकंड में एक दूसरे को पार करती हैं। पहली रेलगाड़ी कितने समय (सेकंड में) 350 मीटर की लंबाई के प्लेटफार्म को पार करती है?

Option “D” is correct.

Relative speed = 72 + 108 = 180 km/hr

Length of both train = 180 × (5/18) × 10 = 500 m

Length of train = 500/2 = 250

Time taken to cross platform = (250 + 350)/72 × 5/18 = 30 sec.

16. Two trains of same length are running on parallel tracks in the same direction at 54 km/h and 42 km/h respectively. The faster train passes the other train in 60 seconds. What is the length (in metres) of each train?

समान लम्बाई वाली दो ट्रेनें क्रमशः 54 किमी/घंटा और 42 किमी/घंटा की गति से समान दिशा में समानांतर ट्रैक पर चलती हैं। तेज गति वाली ट्रेन दूसरी ट्रेन को 60 सेकेंड में पार करती है। तो प्रत्येक ट्रेन की लम्बाई (मीटर में) क्या है?

Option “A” is correct.

Relative speed of fastest train = 54 – 42 = 12 km/hr = 12 × 5/18 = 10/3 m/s.

Let the length of the train be x metre.

As both the trains are running on parallel tracks in same direction, the distance covered by the fastest train will be x + x = 2x

We have,

Distance/Time = Speed

2x/60 = 10/3

x = 100 metre

∴ The length of each train = 100 metre.

17.A boy standing on a railway platform observes that a train going in one direction takes 7 seconds to pass him. Another train of same length going in the opposite direction as compare to the previous train takes 9 seconds to pass him. Find the time taken (in seconds) by trains to cross each other.

एक रेलवे प्लेटफॉर्म पर खड़ा एक लड़का देखता है कि एक दिशा में जा रही एक ट्रेन को गुज़रने में 7 सेकंड लगते हैं। पिछली ट्रेन की तुलना में विपरीत दिशा में जाने वाली समान लंबाई की एक और ट्रेन को गुज़रने में 9 सेकंड का समय लगता है। एक-दूसरे को पार करने के लिए ट्रेनों द्वारा लिया गया समय (सेकंड में) ज्ञात कीजिये।

Option “C” is correct.

Formula used:

Distance = speed × time

Calculation:

Let the length of each train be X m

Let the speed of 1st train be P m/sec and let the speed of 2nd train be Q m/sec

Now, according to question

X = P × 7

⇒ P = X/7        —-(1)

And

X = Q × 9

⇒ Q = X/9        —-(2)

When they cross each other they take time T (let)

So,

2X = (P + Q) × T

⇒ 2X = (X/7 + X/9) × T

⇒ 2X = (16X/63) × T

Hence, T = 63/8 seconds

18. A train, at its usual speed, crosses a 160 m long platform in 9 seconds. When the speed of the train is decreased by 20%, it is crossed by a car, running at 42 m/s in the same direction, in 20 seconds. Find the usual speed of the train. (Note- Length of the car is negligible as compared to the train.)

एक ट्रेन, अपनी सामान्य गति से, 9 सेकंड में 160 मीटर लंबा प्लेटफार्म पार करती है। जब ट्रेन की गति 20% तक कम हो जाती है, तो इसे 20 सेकंड में समान दिशा में 42 मीटर/सेकंड की गति से चलने वाली कार द्वारा पार कर लिया जाता है। ट्रेन की सामान्य गति ज्ञात कीजिए।

Option “A” is correct.

Let the usual speed of the train be ‘x’ m/s.

And the length of the train = ‘y’ m

From the question:

x = (y + 160)/9

y = 9x – 160      —- (1)

Now, the new speed of train = x × 80/100 = (4x/5) m/s

So, 42 – 4x/5 = y/20      —- (2)

From equations (1) and (2):

42 – 4x/5 = (9x – 160)/20

210 – 4x = (9x – 160)/4

840 – 16x = 9x – 160

x = 40

The usual speed of train = 40 m/s

19. A train crosses a stationary object in 25 sec. What is the length of the train if the speed of the train is 25 m/s?

एक ट्रेन किसी स्थिर वस्तु को 25 सेकेंड में पार करती है। यदि ट्रेन की गति 25 मीटर/सेकेंड है, तो ट्रेन की लम्बाई क्या है?

Option “B” is correct.

Let the length of train be X m

Now, according to question

X = 25 × 25

∴ The length of train is 625 m

20. A train crosses two platforms of length 1000 m and 600 m in 80 seconds and 60 seconds respectively. What is the length of the train?

एक ट्रेन 1000 मीटर और 600 मीटर वाले दो प्लेटफॉर्मों को क्रमशः 80 सेकेंड और 60 सेकेंड में पार करती है। तो ट्रेन की लम्बाई क्या है?

Option “B” is correct.

Let the length of the train be x m.

As we know,

Speed = (x + 1000)/80     —(1)

Speed = (x + 600)/60      —(2)

From equation (1) and equation (2)

(x + 1000)/80 = (x + 600)/60

⇒ 3 (x + 1000) = 4 (x + 600)

⇒ 3x + 3000 = 4x + 2400

⇒ 4x – 3x = 3000 – 2400

⇒ x = 600

Length of the train is 600 m.

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