Time and Speed - Train and Boat

Question: A train 240 meters long passes a pole in 16 seconds. How long will it take to pass a platform that is 360 meters long?

Solution:
Train's speed = Distance/Time = 240/16 = 15 m/s

Total distance to pass the platform,
= Length of train + Length of platform
= 240 m + 360 m
= 600 m

∴ Required time = Distance/Speed
= 600/15
= 40 seconds

∴ The train will take 40 seconds to pass the platform.

Question: A takes 2 hours 30 minutes more than B to walk 40 km. If A doubles his speed, then he can make it in 1 hour less than B. What is the average time taken by A and B to walk a 40 km distance?

Solution: 
(40/A) - (40/B) = 5/2 . . . . . . . (i)
(40/B) - (40/2A) = 1 . . . . . . . (ii)

Equation (i) + Equation (ii)
(40/A) - (40/2A) = (5/2) - 1
⇒ (40/A) - (40/2A) = 7/2
⇒ 40/2A = 7/2
⇒ A = 40/7

Putting the value of A in equation (i)
7 - (40/B) = 5/2
⇒ 7 - (5/2) = 40/B
⇒ 9/2 = 40/B
⇒ B = 80/9

Now, 
Total time = [40/(40/7)] + [40/(80/9)]
= 7 + (9/2)
= 23/2

Required average time = (23/2)/2
= 23/4
= 5 hours 45 minutes. 

Question: Rahim is twice as fast as Karim and Karim is three times as fast as Jamal. The distance covered by Jamal in 6 hours will be covered by Rahim in how many hours?

Solution:
Let speed of Jamal = x
Then Speed of Karim = 3x
Then Speed of Rahim = 6x

∴ Ratio of the speed of Rahim and Jamal = 6 : 1

So, The greater the speed, the less time is taken for the journey.

Rahim’s speed is 6 times that of Jamal.
 So Rahim will take (1/6) of the total time taken by Jamal to cover the same distance.

So, Time taken by Rahim
= 6 × (1/6) hours
= 1 hour
= 60 minutes

Question: A man sitting in a train is counting the pillars of electricity. The distance between two pillars is 50 meters, and the speed of the train is 45 km/hr. In 4 hours, how many pillars will he count?

Solution:
Distance covered by the train in 4 hours = (Speed × Time)
= (45 × 4) km
= 180 km
= 180000 m

∴ Number of pillars counted by the man = (180000/50) + 1
= (3600 + 1)
= 3601

Average speed = total distance / total time
Total distance covered = 6 miles; total time = 45 minutes = 0.75 hours
Average speed = 6/ 0.75 = 8 miles/hour

Question: A boat running downstream covers a distance of 40 km in 4 hours, while covering the same distance upstream, it takes 5 hours. What is the speed of the boat in still water?

Solution:
স্রোতের অনুকূলে নৌকার গতিবেগ (Downstream Rate):
= অতিক্রান্ত দূরত্ব/সময়
= 40 কিমি/4 ঘন্টা
= 10 কিমি/ঘন্টা

স্রোতের প্রতিকূলে নৌকার গতিবেগ (Upstream Rate):
= অতিক্রান্ত দূরত্ব/সময়
= 40 কিমি/5 ঘন্টা
= 8 কিমি/ঘন্টা

স্থির পানিতে নৌকার গতিবেগ = (স্রোতের অনুকূলে গতিবেগ + স্রোতের প্রতিকূলে গতিবেগ)/2
= (10 + 8)/2 কিমি/ঘন্টা
= 18/2 কিমি/ঘন্টা
= 9 কিমি/ঘন্টা

সুতরাং, স্থির পানিতে নৌকাটির গতিবেগ হলো 9 কিমি/ঘন্টা।

Speed = 45 km/hr = 45 × 5/18
= 25/2 m/s
Distance travelled = Length of the train + Length of the platform
= 360 + 140
= 500 metre.
Time taken to cross the platform = 500/(25/2)
= 40 seconds

Let the total distance covered = x
= Length of journey
According to the question,
x/2 × 1/40 + x/2 × 1/60 = 16
x/80 + x/120 = 160
(3x + 2x)/240 = 16
5x = 16 × 240
x = (16 × 240)/5
x = 16 × 48
x = 768 km

Question: An ambulance moving at 33 km/h is 45 m behind a school bus. After 15 s, it is 30 m ahead of the bus. Find the speed of the school bus.

Solution: 
Relative Speed = Total distance/Total time
= (30 + 45)/15
= 75/15 
= 5 m/s
= 5 × (18/5)
= 18 km/h

Now,
Relative Speed = Speed of ambulance - Speed of school bus
∴ Speed of school bus = Speed of ambulance - Relative speed
= 33 - 18
= 15 km/h

Question: A train is 100 meter long and is running at the speed of 30 km per hour. Find the time it will take to pass a man standing at a crossing.

Solution: 
Given that, 
Length of train, L = 100 meters
Speed of train, v = 30 km/h
= 30 × (1000/3600) m/s
= 30 × (5/18) m/s
= 150/18 m/s 
= 25/3 m/s

We know, 
Time taken = Distance/Speed
= 100/(25/3) m/s
= 100 × (3/25)
= 300/25
= 12 seconds

So the train takes 12 seconds to pass the man standing at the crossing.

Question: A person walks at 14 km/h instead of 10 km/h and in the same time covers 20 km more distance. Find the actual distance traveled.

Solution: 
Given that,
Usual speed = 10 km/h
Increased speed = 14 km/h
Extra distance covered = 20 km

Let the actual distance traveled be = d km

Now,
Time = d/10 
And, At 14 km/h, distance covered in the same time = d + 20 
So,
⇒ d/10 = (d + 20)/14
⇒ 14d = 10d + 200
⇒ 14d - 10d = 200
⇒ 4d = 200
⇒ d = 200/4
∴ d = 50 km

The actual distance traveled is 50 km.

Speed of the first train
= 120/10 m/sec.
= 12 m/sec.

Speed of the second train
= 120/15 m/sec.
= 8 m/sec.

Relative speed = 12 + 8 = 20 m/sec.

∴ Required time
= (120 + 120)/20 sec.
= 240/20 sec.
= 12 sec.

Question: A train running at the speed of 60 km/hr crosses a pole in 9 seconds. Find the length of the train.

Solution: 
Given that, 
Speed of train, v = 60 km/h
Time to cross a pole, t = 9 seconds

We know, 
Length of the train = speed × time
= (50/3) × 9
= (50 × 9)/3
= 450/3
= 150 meters

So the length of the train is 150 meters.

Distance covered by the first plane till 2 P.M.
i.e., in 2 hrs = (500 × 2) miles
= 1000 miles.
Distance covered by the second plane till 2 P.M.
i.e., in 1(1/2) hrs = (400 × 3/2) miles
= 600 miles.
∴ Required distance
= AB =√{(1000)² + (600)²}
=√(1000000 + 360000)
=√1360000
= 200√34 miles
= 200 × 5.83
= 1166 miles.

Let the speeds of the two trains be x m/sec and y m/sec respectively.
Then, length of the first train = 27x metres
and length of the second train = 17y metres
∴ 27x+17y/(x+y) = 23
⇒ 27x+17y = 23x+23y
⇒ 4x=6y
⇒ x/y= 3/2

Previous speed = x
A/Q,
32/x - 32/(x + 4) = 4
Or, 32x + 128 - 32x/x(x + 4) = 4
Or, (x + 4)x = 32
Or, x² + 4x - 32 = 0
Or, x² + 8x - 4x - 32 = 0
Or, (x + 8) (x - 4) = 0
So, previous speed is 4 kmph.
And, Present speed will be 4 + 4 = 8 kmph.

Question: A man completes a journey in 10 hours. He travels the first half of the journey at the rate of 30 km/hr and the second half at the rate of 50 km/hr. Find the total journey in km.

Solution:
ধরা যাক, মোট যাত্রার দূরত্ব হলো d কিমি।
তাহলে, যাত্রার প্রথম অর্ধেকের দূরত্ব হবে d/2 কিমি
এবং দ্বিতীয় অর্ধেকের দূরত্বও হবে d/2 কিমি।

প্রথম অর্ধেক যাত্রায়,
সময় = দূরত্ব/গতিবেগ
= (d/2)/30 ঘন্টা
= d/60 ঘন্টা

দ্বিতীয় অর্ধেক যাত্রায়,
সময় = দূরত্ব/গতিবেগ
= (d/2)/50 ঘন্টা
= d/100 ঘন্টা

প্রশ্নমতে,
(d/60) + (d/100) = 10
⇒ (5d + 3d)/300 = 10
⇒ 8d/300 = 10
⇒ 8d = 10 × 300
⇒ 8d = 3000
⇒ d = 3000/8
⇒ d = 375 কিমি

∴ মোট যাত্রার দূরত্ব 375 কিলোমিটার।

Speed of the train relative to man
= 125/10 m/sec
= 25/2 m/sec
= (25/2)×(18/5) km/hr
= 45km/hr
Let the speed of the train be x km/hr. Then, relative speed=(x−5)km/hr
∴x−5= 45
⇒x = 50km/hr

Question: A 280 metre long train crosses a platform thrice its length in 50 seconds. What is the speed of the train in km/hr?

Solution:
Given that,
Length of the train = 280 m
Length of the platform = 3 × 280 = 840 m

∴ Total distance to be covered
= Train length + Platform length
= 280 + 840
= 1120m
And, Time taken = 50 seconds

∴  Speed = Distance​/Time
= 1120/50
= 22.4 m/s
= 22.4 × 3.6  ;[1 m/s = 3.6 km/h]
= 80.64 km/h

∴ The speed of the train is 80.64 km/h.

Let the length of the tunnel = x meter
Then, distance = (800 + x) meter.

Given,
Time = 1 minute = 60 seconds.
Speed = 78 km/hr
= 78 × (5/18)
= (65/3) m/s

According to the question,
800 + x = 60 × (65/3)
⇒ 800 + x = 1300
⇒ x = 500 meter.

So, the length of the tunnel = 500 meter

Question: A car travels a distance of 60 km in 1 hour and 40 minutes. How much faster, in kilometers per hour, on an average, must it travel to make the same trip in 20 minutes less time?

সমাধান:
দেওয়া আছে,
মোট অতিক্রান্ত দূরত্ব = 60 কিমি।
প্রাথমিক সময় = 1 ঘন্টা 40 মিনিট = 1 + 40/60 ঘন্টা = 1 + 2/3 = 5/3 ঘন্টা।

প্রাথমিক গতিবেগ = দূরত্ব/সময়
= 60 / (5/3) কিমি/ঘন্টা
= (60 × 3/5) কিমি/ঘন্টা
= 36 কিমি/ঘন্টা।

20 মিনিট কম সময় = 20/60 = 1/3 ঘন্টা কম।
∴ নতুন সময় = (প্রাথমিক সময় - কমানো সময়)
= 5/3 - 1/3 ঘন্টা
= 4/3 ঘন্টা।

∴ নতুন গতিবেগ = 60/(4/3) কিমি/ঘন্টা
= (60 × 3/4) কিমি/ঘন্টা
= 45 কিমি/ঘন্টা।

গতি বৃদ্ধি করতে হবে = (নতুন গতিবেগ - প্রাথমিক গতিবেগ)
= (45 - 36) কিমি/ঘন্টা
= 9 কিমি/ঘন্টা।

∴ গাড়িটিকে গড়ে 9 কিমি/ঘন্টা বেশি গতিতে যেতে হবে।

Speed of the first train = 48 km/hr.
Let the length of the first train = 2x meter.
Speed of the second train = 42 km/hr.
Let the length of the second train = x meter.
Distance = (2x + x) = 3x meter.
Time = 12 seconds
Relative speed = 48 + 42 = 90 km/hr.
= 90 × (5/18) = 25 m/s.
3x = 25 × 12
⇒ x = 100 meter.
Length of the first train = 200
Time is taken to cross the platform = 45 seconds.
Speed of first train = 48 km/hr
= 48 × (5/18)
= 40/3 m/s.
Let the length of the platform = y meter.
Distance = 200 + y meter.
⇒ 200 + y = 45 × (40/3)
⇒ 200 + y = 600
⇒ y = 400 meter.

Question: Over a span of 9 hours, a man traveled 61 km, part of it on foot at 4 km/hr and part by bike at 9 km/hr. What distance did he walk?

Solution:
Let the time in which he travelled on foot = x hr
Then the time in which he travelled on bicycle =(9 - x) hr
distance = speed × time
⇒ 4x + 9(9 - x) = 61
⇒ 4x + 81 - 9x = 61
⇒ 5x = 20
⇒ x = 4

∴ The distance travelled on foot = 4 × 4 = 16 km

Let,
The speed is x km/hr.
Then,
Distance covered in 1 hr. 40 min
i.e., 1(2/3) hrs. = 5x/3 km.
Remaining Distance = {24 - (5x/3)}
∴ 5x/3 = 5/7{24 - (5x/3)
⇒ 5x/3 = 5/7{(72 - 5x)/3}
⇒ 7x = 72 - 5x
⇒ 12x = 72
⇒ x = 72/12
⇒ x = 6.
Hence, Speed = 6 km/hr
= {6 × (5/18)} m/s
= 5/3 m/s
= 1(2/3) m/s.

Speed downstream = (22 + 4) = 26 kmph
Time = 24 minutes
= 24/60 hour = 2/5 hour
Distance travelled = Time × speed
= (2/5) × 26 = 10.4 km

Assume, distance traveled by P in x hrs = 25 x km -----(1)
distance traveled by Q in (x-1) hrs = 20 (x-1) km -----(2)

Adding (1) & (2),
25 x + 20 (x -1) = 275
x = 6.5 hrs
(x -1) = (6.5 -1) = 5.5 hrs

Time at which they cross each other = 9 a.m. + 5.5hrs = 2.30 p.m.
The two motorcycle riders cross each other at 2.30 p.m.

We know,
Distance(D) = Speed(S) × Time(T)
∴ S = D/T ; T = D/S

Let distance traveled by cat before dog catches it be D
We know, time for which Dog and Cat ran is the same
∴ T = T
∴ D/5 = (D + 80)/7
⇒ 7D = 5D + 400
⇒ 7D - 5D = 400
⇒ 2D = 400
⇒ D = 200 m.

Question: A train crosses two bridges that are 600 meters and 200 meters long in 60 seconds and 40 seconds respectively. What is the length of the train?

Solution:
We know,
To cross a bridge, a train must cover the length of the bridge along with its own length.

Let the length of the train = x meters

Then,
For the first bridge, the distance covered by the train = (x + 600) meters
And,
For the second bridge, the distance covered by the train = (x + 200) meters

According to the question,
(x + 600)/60 = (x + 200)/40
⇒ 40(x + 600) = 60(x + 200)

⇒ 40x + 24000 = 60x + 12000
⇒ 60x - 40x = 24000 - 12000
⇒ 20x = 12000
⇒ x = 12000/20 
⇒ x = 600

∴ The length of the train is 600 meters.