Time and Speed - Train and Boat

Time taken by man if he did not stop
= 5 km/10 kmph
= 1/2 hr
= 30 min 
∵Man takes rest for 5 minutes on each km
Total rest time= 5×4= 20 min
Total travelling time:
= 30 min+20 min
= 50 min

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

Since the second train is 30 km/hr faster, it is moving at 120 km/hr
Now, let train A travel T hours before meeting train B.
Time for which train B travels = (T - 1/2) hrs.
This is because it starts half an hour late
Distance travelled is same.

∴ D = D
∴ 90 km/hr × T hrs = 120 km/hr × {T - (1/2} hrs
∴ 90T = 120T - 60
∴ T = 2 hours

Distance travelled by train A = 90 km/hr × 2 hours = 180km.
Thus they meet 180 km from starting point.

প্রশ্ন: দুইটি ট্রেন দুটি স্টেশন থেকে একসাথে যাত্রা শুরু করে পরস্পরের দিকে রওনা দেয়। স্টেশন দুটি ২০০ কিমি দূরে অবস্থিত। তারা একে অপরকে একটি স্টেশন থেকে ১১০ কিমি দূরত্বে অতিক্রম করে। ট্রেন দুইটির গতিবেগের অনুপাত কত?

সমাধান:
ধরি,
প্রথম ট্রেনের গতিবেগ = ক কিমি/ঘণ্টা
দ্বিতীয় ট্রেনের গতিবেগ = খ কিমি/ঘণ্টা

প্রথম ট্রেন অতিক্রম করে = ১১০ কিমি
দ্বিতীয় ট্রেন অতিক্রম করে = ২০০ - ১১০ = ৯০ কিমি

যেহেতু উভয় ট্রেন একসাথে শুরু করে এবং একই সময়ে একে অপরকে অতিক্রম করে, অতএব তাদের সময় সমান।

সময় = দূরত্ব/গতি

তাহলে,
১১০/ক = ৯০/খ
⇒ ক/খ = ১১০/৯০
⇒ ক : খ = ১১০ : ৯০
⇒ ক : খ = ১১ : ৯

∴ ট্রেন দুইটির গতির অনুপাত ১১ : ৯

Average speed= Total distance/Total time
Average speed= (10+12)/(10/12)+(12/10)
Average speed= 10.8 km/hr

Question: A boat travels 120 km upstream in 6 hours and the same distance downstream in 4 hours. What is the speed (in km/h) of the stream?

সমাধান:
ধরি, 
​স্থির জলে নৌকাটির গতিবেগ = x কিমি/ঘন্টা
এবং স্রোতের গতিবেগ = y কিমি/ঘন্টা

স্রোতের প্রতিকূলে গতিবেগ = (x - y) কিমি/ঘন্টা
স্রোতের অনুকূলে গতিবেগ = (x + y) কিমি/ঘন্টা

প্রশ্নমতে,
120/(x - y) = 6
⇒ x - y = 120/6
∴ x - y = 20 …………(1)

আবার,
120/(x + y) = 4
⇒ x + y = 120/4
∴ x + y = 30 …………(2)

(2) নং থেকে (1) নং সমীকরণ বিয়োগ করে পাই,
(x + y) - (x - y) = 30 - 20
⇒ x + y - x + y = 10
⇒ 2y = 10
⇒ y = 10/2
∴ y = 5

সুতরাং, স্রোতের গতিবেগ 5 কিমি/ঘন্টা।

Length of train = x metre (let)
Speed of train
5x + 1250 = 3x + 1500.
⇒ 5x – 3x = 1500 – 1250.
⇒ 2x = 250.
∴ x= 125 metres.

Question: A biker was riding at 60 km/h, but if he had gone at 80 km/h, he could’ve gone 100 km more in the same time. How far did he actually travel?

Solution:
Let, the actual distance travelled be a km.

Then,
a/60 = (a + 100)/80
⇒ a/6 = (a + 100)/8
⇒ 6(a + 100) = 8a
⇒ 6a + 600 = 8a
⇒ 8a - 6a = 600
⇒ 2a = 600
⇒ a = 600/2
⇒ a = 300 km

So the actual distance travelled = 300 km

Let the length of the train be x meters and its speed be y m/sec.
Then, xy = 20
⇒ y = x/20
∴ (x+100)/30 = x/20
⇒ 30x = 20x + 2000
⇒ 10x = 2000
⇒ x = 200 meters

Let the length of the tunnel = x metre
Then, distance = (800 + x) metre.
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 metre.
Hence, The length of the tunnel is 500 metre.

Question: A thief is noticed by a policeman from a distance of 300 meters. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km/hr and 12 km/hr respectively. What is the distance between them after 6 minutes?

সমাধান:
যেহেতু পুলিশ ও চোর একই দিকে দৌড়াচ্ছে, তাই তাদের আপেক্ষিক গতিবেগ (relative speed) হবে তাদের গতিবেগের পার্থক্য।
∴ আপেক্ষিক গতিবেগ = (12 - 10) কিমি/ঘন্টা
= 2 কিমি/ঘন্টা।
= (2 × 1000)/60 মিটার/মিনিট
= 200/6 মিটার/মিনিট

6 মিনিটে পুলিশ চোরের থেকে যে দূরত্ব কমিয়ে আনবে, তা হলো:
অতিক্রান্ত দূরত্ব = আপেক্ষিক গতিবেগ × সময়
= (200/6)× 6 
= 200 মিটার

সুতরাং, 6 মিনিট পরে তাদের মধ্যে অবশিষ্ট দূরত্ব = প্রাথমিক দূরত্ব - অতিক্রান্ত দূরত্ব
= 300 মিটার - 200 মিটার
= 100 মিটার

We know that, speed = Distance/Time
Therefore, Required distance = Speed × Time
=37 × 8
= 296km

Distance = 600 metres
Total Speed = 64 + 80 = 144 kmph (added because they are travelling in opposite directions)
In m/sec, speed = 144×(5/18) = 40 m/sec
Distance = Speed×Time
600 = 40×Time
Therefore, Time = 15 seconds

Suppose he moves 4 km downstream in x hours.
Then, Speed downstream = (4/x) km/hr
Speed upstream = (3/x) km/hr

∴ 48/(4/x) + 48/(3/x) = 14
⇒ 48x/4 + 48x/3 = 14
⇒ 48{(x/4) + (x/3)} = 14
⇒ 7x/12 = 14/48
⇒ 7x/12 = 7/24
⇒ 7x = (7/24) × 12
⇒ 7x = 7/2
⇒ 14x = 7
⇒ x = 7/14
⇒ x = 1/2

So, speed downstream = 7 km/hr
Rate of the stream = 1/2(8 - 6) km/hr = 1 km/hr.

Question: A man rows a boat a certain distance downstream in 9 hours, while it takes 18 hours to row the same distance upstream. How many hours will it take him to row three-fifth of the same distance in still water?

Solution: 
Let distance = d
So downstream speed = d/9​
And upstream speed = d/18​

We know, 
Speed in still water = (Downstream + Upstream)/2
= (d/9 + d/18)/2 
= {(2d + d)/18}/2
= (d/6)/2
= d/12
So, still water speed = d/12
Distance to travel = 3d/5

∴ Time = (3d/5)/(d/12) = (3 × 12)/5 = 36/5 = 7.2 hours

So it will take the man 7.2 hours to row three-fifth of the distance in still water. 

Given: Speed of the person = 5 km/hr, length of train = 100 m, speed of train = 60 km/hr
Speed of train relative to walking person = (60–5) = 55 km/hr
Convert km/hr into m/s
55 km/hr = 55 x(5/18) = 15.27 m/s
Distance to be covered by the train = 200 + 100 = 300 m
Therefore, time taken by the train to cross the person
= Distance over speed =300/15.27 = 19.64 sec

Average speed
= (2xy / x+y) km/hr
= (2×50×30 / 50+30) km/hr
= 37.5 km/hr

We know,
Man's/Boat's Speed = X
Stream/Current/River Speed = Y

∴ Downstream speed = X + Y
Upstream speed = X - Y

Downstream speed = Distance covered/Time taken
= 48/8 = 6 km/hr
Upstream Speed = Distance covered/Time taken
= 48/12 = 4 km/hr

X+Y = 6 km/hr and X-Y = 4 km/hr
Adding them we get,
X + Y + X - Y = 10 km/hr

∴ X=5 km/hr = Speed of Boat
Y=6 - 5 = 1 km/hr = Speed of stream

Here there are 2 cases
Case 1: accident happens at 50 km
Case 2: accident happens at 60 km
Difference between two cases is only for the 10 kms between 50 and 60.
Time difference of 10 minutes is only due to these 10 kms.
In case 1, 10 kms between 50 and 60 is covered at (3/4)^th Speed.
In case 2, 10 kms between 50 and 60 is covered at usual Speed.
So the usual Time 't' taken to cover 10 kms, can be found out as below.
(4/3)t – t = 10 mins [s = vt]
⇒ t = 30 mins, d = 10 kms
so usual
Speed = Distance/Time
= 10/30min
= 10/(1/2) km/hr.
= 20 km/hr.

Question: A boat covers 143 km upstream in 13 hours and the same distance downstream in 11 hours. What is the speed (in km/h) of the boat in still (without stream) water?

Solution:
Let, the speed of the boat in still water = x km/h
and the speed of the stream = y km/h

According to the question,
143/(x - y) = 13
⇒ 13(x - y) = 143
⇒ x - y = 11 ............(1)

Again,
143/(x + y) = 11
⇒ 11(x + y) = 143
⇒ x + y = 13 ............(2)

Adding (1) and (2):
x - y + x + y = 11 + 13
2x = 24
⇒ x = 12 km/h

We know,
Time taken by a train of length l metres to pass a stationary object of length b metres is
the time taken by the train to cover (l + b) metres.
Given that,
Time taken by a train of length 300 meters to pass a wall of length 600 meters is
the time taken by the train to covers (300+600) meters.
That is, 81 seconds.
We know that,
distance/time = speed
Therefore, required speed = (300+600)/81 m/sec
= 900/81 m/sec
Converting the above to km/hr
= (900/81) x (18/5) km/hr [m/s to km/hr conversion: a m/s = a x 18/5 km/hr.]
= 40 km/hr
Hence the answer is 40 km/hr.

Question: A man walks a at a rate of 10 mph. After every ten miles, he rests for 6 minutes. How much time does he take to walk 50 miles?

Solution:
The man needs time = 50/10 = 5 hours
= (5 × 60)
= 300 minutes

He will also rest 4 times after 10, 20, 30 and 40 miles
Total resting time = 4 × 6 = 24 minutes.

Total time = 300 + 24 = 324 minutes

Question: A train 300 m long is running at a speed of 90 km/hr. If it passes through a tunnel in 40 seconds, then the length of the tunnel is-

Solution:
ট্রেনের গতিবেগ = 90 কিমি/ঘন্টা
= (90 × 5/18) মিটার/সেকেন্ড
= 25 মিটার/সেকেন্ড

একটি সুড়ঙ্গ অতিক্রম করার সময় ট্রেনটি তার নিজের দৈর্ঘ্যের সাথে সুড়ঙ্গের দৈর্ঘ্যও অতিক্রম করে।
সুতরাং, মোট অতিক্রান্ত দূরত্ব = ট্রেনের গতিবেগ × সময়
= 25 মিটার/সেকেন্ড × 40 সেকেন্ড
= 1000 মিটার

ধরা যাক, সুড়ঙ্গটির দৈর্ঘ্য = L মিটার
মোট অতিক্রান্ত দূরত্ব = ট্রেনের দৈর্ঘ্য + সুড়ঙ্গের দৈর্ঘ্য
⇒ 1000 = 300 + L
⇒ L = 1000 - 300
⇒ L = 700 মিটার

সুতরাং, সুড়ঙ্গটির দৈর্ঘ্য হলো 700 মিটার।

Net height ascended in 2 min = (6 - 3) m = 3 m.
Net height ascended in 36 min = (3/2 × 36) = 54m.
In the 37th min,
the monkey ascends 6m and reaches the top.
Hence,
Total time taken = 37 minutes.

Question: A thief is noticed by a policeman from a distance of 200 m. The thief starts running and the policeman chases him. The thief and the policeman run at the rate of 10 km and 11 km per hour respectively. What is the distance between them after 6 minutes?

Solution:
Relative speed of the thief and policeman  =  (11 - 10) km/hr = 1 km/hr 
Distance covered in 6 minutes  = (1/60) × 6 km   = 1/10 km = 100 m
Therefore, Distance between the thief and policeman = (200 - 100) m = 100 m.

Question: A boat takes 12 hours to cover a certain distance upstream and 8 hours to cover the same distance downstream. What is the ratio between the speed of the boat and the speed of the water current, respectively?

Solution:
Let,
Upstream speed = x kmph
Downstream speed = y kmph

Since distance is same,
x × 12 = y × 8
⇒ 12x = 8y
⇒ y = 3x/2

Now,
Speed of boat = (y + x)/2
Speed of current = (y - x)/2

∴ Required ratio
= (y + x) : (y - x)
= [(3x/2) + x] : [(3x/2) - x]
= (5x/2) : (x/2)
= 5x : x
= 5 : 1

∴ The ratio of the speed of the boat to the speed of the water current is 5 : 1.

Given: Speed of passenger train = 55 km/hr, length of goods train (P) = 250, length of passenger train (Q)= 200m
Hint:
Time = (P+Q)/(V1+V2) sec
Goods train and the passenger train move in opposite direction. Hence, the relative speed is the addition of two speeds.
Convert 55 km/hr into m/s
55 x (5/18) = 15.277 m/s
Therefore,
10 = (250+200)/(15.27+V2)
V2 = 29.73 m/s 

Let the speed of the stream be x km/hr. Then,
Speed downstream = (15 + x) km/hr,
Speed upstream = (15 - x) km/hr

ATQ, 30/(15 + x) + 30/(15 - x) = 4(1/2)
Or, 900/(225 - X²) = 9/2
Or, 9X² = 225
Or, X²  = 25
∴ X = 5 km/hr

Total distance = Average speed × total time
So, 2 = 2/3×T
⇒ T = 3 hours
So, the whole journey took 3 hours
Since trip back took half as much time as the trip there, it took 2 hours to reach there and 1 hour to come back.
So, speed of the delivery cart while going to Lark Rise = distance/time = 1/2 mph

Question: A train 150 meters long passes a signal post in 15 seconds. How long will it take to pass a bridge that is 450 meters long?

Solution:
Train's speed = Distance/Time
= 150/15 = 10 m/s

Total distance to pass the bridge,
= Length of train + Length of bridge
= 150 m + 450 m
= 600 m

∴ Required time = Distance/Speed
= 600/10
= 60 seconds
​= 1 minute

∴ The train will take 60 seconds or 1 minute to pass platform.

Question: A man travels from his home to the office at 4 km/hr and reaches his office 5 min late. If the speed had been 5 km/hr he would have reached 10 min early. Find the distance from his home to the office?

Solution:
ধরি, বাড়ি থেকে অফিসের দূরত্ব = d কিমি।
4 কিমি/ঘণ্টা বেগে সময় লাগে = d/4 ঘণ্টা
5 কিমি/ঘণ্টা বেগে সময় লাগে = d/5 ঘণ্টা

সময়ের পার্থক্য = 5 মিনিট (দেরি) + 10 মিনিট (আগে) = 15 মিনিট
= 15/60 = 1/4 ঘণ্টা

প্রশ্নমতে,
(d/4) - (d/5) = 1/4
⇒ (5d - 4d) / 20 = 1/4
⇒ d / 20 = 1/4
⇒ d = 20 / 4
∴ d = 5

সুতরাং, বাড়ি থেকে অফিসের দূরত্ব 5 কিমি।

Question: A man walk at a speed of 10 km/h. After every 3 kilometers, he takes a rest of 5 minutes. How much time will he take to cover a distance of 15 km?

সমাধান:
দূরত্ব = 15 কিমি
গতিবেগ = 10 কিমি/ঘন্টা

∴ সময় = দূরত্ব/গতিবেগ
= 15 কিমি/10 কিমি/ঘন্টা
= 1.5 ঘন্টা
= 90 মিনিট

সে প্রতি 3 কিলোমিটারের পরে 5 মিনিট বিশ্রাম নেয়।
15 কিমি দূরত্বে সে (15/3) - 1 = 4 বার বিশ্রাম নেবে (3 কিমি, 6 কিমি, 9 কিমি এবং 12 কিমি অতিক্রম করার পর)।
শেষ কিলোমিটারের পরে তার আর বিশ্রাম নেওয়ার প্রয়োজন নেই।

মোট বিশ্রামের সময় = 4 × 5 মিনিট = 20 মিনিট।

∴ মোট সময় = হাঁটার সময় + বিশ্রামের সময়
= 90 মিনিট + 20 মিনিট
= 110 মিনিট
= 1 ঘন্টা 50 মিনিট

সুতরাং, লোকটি 15 কিমি দূরত্ব অতিক্রম করতে মোট 110 মিনিট বা 1 ঘন্টা 50 মিনিট সময় নেবে।

Let the speed of the boat in still water=x km/hr
Speed of the current = 2 km/hr

Then, speed downstream = (x + 2) km/hr
speed upstream = (x - 2) km/hr
Total time taken to travel 10 km upstream and back = 55 minutes
= (55/60) hr
= 11/12 hr.

According to question,
10/(x - 2) + 10/(x + 2) = 11/12
⇒ 120(x + 2)+120(x - 2) = 11(x² - 4)
⇒ 240x = 11x² - 44
⇒ 11x² - 240x - 44 = 0
⇒ 11x(x -22) + 2(x - 22) = 0
⇒ (x -22) (11x + 2) = 0
Since x cannot be negative.
So, x = 22 km/hr.

Hence, the Speed of the motorboat is 22 km/hr.

Question: A thief committed a crime and escaped from the spot at a speed of 12 m/s. A Security guard started chasing him 20 minutes after the thief started running and caught him in the next 20 minutes. What is the speed (in m/s) of the Security guard?

Solution: 
Given that,
Thief's speed = 12 m/s
Security guard starts 20 minutes or 1200 s later and catches thief in next 20 minutes or 1200 s

Now,
Distance covered by thief before guard starts,
d₁ = speed × time = 12 × 1200 = 14400 m ; [20 minutes = 1200 s]
So, thief has a 14400 m head start.

And
Guard catches thief in 20 minutes = 1200 s
d₂ = 12 × 1200 s = 14400 m 
∴ Guard covers total distance = head start + distance thief runs during chase = 14400 + 14400 = 28800 m
and Time taken by guard = 1200 s

∴ Speed of guard = Distance/Time = 28800/1200 = 24 m

Let the speed of the current be x kmph
Then speed of the boat in still water = 5x

∴ Downstream speed
= (5x + x) = 6x km/hr.
Upstream speed
= (5x - x)
= 4x km/hr

Now,
According to the question,
1.1 × 6x = 13.2 km
6.6x = 13.2
x = 13.2/6.6
x = 2 km/hr.

Upstream speed
= 4x = 4 × 2 = 8 km/hr

∴ 312 minutes
= 312/60 hours
= 5(1/5) hours

∴ Required Distance travelled upstream
= Speed × Time
= 8 × (26/5)
= 41.6 km

Average speed = (2 × 3 × 2)/(2 + 3)
= 12/5 km/h
Total time taken = 5 hours
Distance travelled = (12/5) × 5
= 12 km
Therefore, distance between his house and office
= 12/2
= 6 km.

Question: Excluding stoppages, the speed of a bus is 80 kmph and including stoppages, it is 60 kmph. For how many minutes does the bus stop per hour?

Solution:
স্টপেজ ছাড়া বাসের গতিবেগ = 80 কিমি/ঘন্টা
স্টপেজ সহ বাসের গতিবেগ = 60 কিমি/ঘন্টা

প্রতি ঘন্টায় বাসটি যে দূরত্ব কম অতিক্রম করে = (80 - 60) কিমি = 20 কিমি।

এই 20 কিমি দূরত্ব অতিক্রম করতে বাসটির যে সময় লাগতো, সেই সময়টুকুই বাসটি থেমে থাকে।

গতিবেগ 80 কিমি/ঘন্টা হলে 20 কিমি যেতে সময় লাগে:
সময় = দূরত্ব/গতিবেগ
= 20 কিমি/80 কিমি/ঘন্টা
= 1/4 ঘন্টা
= (1/4 × 60) মিনিট
= 15 মিনিট

সুতরাং, প্রতি ঘন্টায় বাসটি 15 মিনিট থামে।

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
⇒ 4 x + 9 (9 − x) = 61
⇒ 4 x + 81 − 9 x = 61
⇒ 5 x = 20
⇒ x = 4
Distance travelled on foot = 4x
= 4 × 4
= 16 km

এখানে, 12 মিনিট = 12/60 = 1/5 ঘন্টা
স্রোতের অনুকূলে নৌকার বেগ = 15 + 3 = 18 kmph
অর্থাৎ নৌকাটি 1 ঘন্টা বা 60 মিনিটে যায় = 18 km
নৌকাটি 12 মিনিটে যায় = (18 × 12)/60
= 3.6 km

Distance covered by man in first 2 hrs = 2 × 20 = 40 km
Remaining distance = (100 - 40) = 60 km
Time to cover the remaining distance = (60 /10) kmh = 6 hrs
∴ Required average speed = 100 / (2 + 6) = 12.5 kmh

Question: A 180 m long train takes 18 seconds to pass a pole. How long will it take to pass a 330 m long platform?

Solution:
আমরা জানি,
কোনো ট্রেন একটি খুঁটিকে অতিক্রম করলে ট্রেনটি তার নিজের দৈর্ঘ্যকে অতিক্রম করে।
আবার,
কোনো প্ল্যাটফর্মকে অতিক্রম করতে হলে ট্রেনকে নিজের দৈর্ঘ্য এবং প্ল্যাটফর্মের দৈর্ঘ্যের সমান দূরত্ব অতিক্রম করতে হয়।

∴ প্ল্যাটফর্ম অতিক্রম করলে ট্রেনের অতিক্রান্ত দূরত্ব = (ট্রেনের দৈর্ঘ্য + প্ল্যাটফর্মের দৈর্ঘ্য) = (180 + 330) মিটার = 510 মিটার 

এখন,
180 মিটার অতিক্রম করে = 18 সেকেন্ডে
∴ 1 মিটার অতিক্রম করে = 18/180 সেকেন্ডে
∴ 510 মিটার অতিক্রম করে = (18 × 510)/180 = 51 সেকেন্ডে

Question: In a 500 m race, the speeds of two runners, A and B are in the ratio 5 : 6. If A is given a start of 100m, by how many meters does A win the race?

Solution:
Total race length = 500 meters.
A is given a start of 100 meters, so A runs 500 - 100 = 400 meters.

Speed ratio A : B = 5 : 6.

Let, B runs = X meter

Therefore,
400/X = 5/6
⇒ X = (6 × 400)/5
∴ X = 480m

Remaining distance for B = 500 - 480 = 20 meters.
Therefore, A wins by 20 meters.

Question: A man travels from A to B at a speed of 36 km/hr in 74 minutes and he travels a distance from B to C with a speed of 45 km/hr in 111 minutes. Find the average speed of whole journey.

Solution:
Given that,
A man travels from A to B at a speed of 36 km/hr in 74 minutes and he travels a distance from B to C with a speed of 45 km/hr in 111 minutes. 

We know,
Average speed = Total distance/Total time taken

Now,
Time taken = 74 min : 111 min   [given]
Ratio of Time taken = 2 : 3

∴ Average Speed = {(36 × 2) + (45 × 3)}/(2 + 3) = 207/5
 = 41.4 km/hr

∴ So the average speed of whole journey is 41.4 km/h