Time and Speed - Train and Boat

Distance covered by B in 9 sec.
=(100/45)x 9m= 20 m.
A beats B by 20 metres.

Question: A train passes a telegraph post and a 300 m long bridge in 10 seconds and 25 seconds, respectively. Find the speed of the train in km/h.

Solution:
Let,
Length of the train = x metres
Speed of the train = y m/sec

Time to pass the telegraph post:
x/y = 10
⇒ x = 10y

Time to pass the bridge:
(x + 300)/25 = y
⇒ x + 300 = 25y
⇒ 10y + 300 = 25y
⇒ 25y - 10y = 300
⇒ 15y = 300
⇒ y = 20 m/sec

∴ Speed of the train = 20 × (18/5) km/h
= 72 km/h

So, the speed of the train is 72 km/h.

Let the speed of the train be v km/hr.
(v - 2) : (v - 4) = 10 : 9 [speed and time are inversely proportional]
⇒ 9v - 18 = 10v - 40
⇒ v = 22 km/hr.
Length of the train
= (22 - 2) × (5/18) × 9
= 50 meter.

Question: Pavel was cycling at 30 km/h and was passed by Ratan who was cycling at 40 km/h. If Ratan cycles for c minutes at his speed and then stops, how long, in minutes, will it take for Pavel to reach him?

Solution: 
পাভেলের গতি = 30 কিমি/ঘণ্টা = 30/60 = 0.5 কি. মি./মিনিট।
রতনের গতি = 40 কিমি/ঘণ্টা = 40/60 = 2/3 কি. মি./মিনিট।

এখন, c মিনিটে
পাভেলের দূরত্ব = 0.5c কি. মি.
রতনের দূরত্ব = (2/​3) × c = 2c/3 কি. মি.

∴ পাভেলের এই দূরত্ব অতিক্রম করতে সময় লাগবে = দূরত্ব/গতি
= (2c/3)/0.5
= 4c/3 মিনিট

Question: An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 5/3 hours, it must travel at a speed of-

Solution: 
Given that,
Speed = 240 kmph 
Time = 5 hours 

We know,
 Distance = Speed × Time 
= 240 × 5
∴ Distance = 1200 km

Again,
Given: 
Distance = 1200 km 
New Time = 5/3 hours 
 
 New Speed = Distance/New Time
= 1200/(5/3)
= (1200 × 3)/5
= 720 kmph
 
 ∴ The aeroplane must travel at a speed of 720 kmph.

Let's take the length of each train = x,
total length of both trains = 2x,
Crossing time = 30 sec.
Relative speed = 90 - 60 = 30 km/hr
= 30 × 5/18
= 25/3 m/sec.
∴ Total length = Time × Relative speed
⇒ 2x = 30 × 25/3
⇒ 2x = 10 × 25
= 125 m.

V = (2 × 40 × 60)/(40 + 60) = 48kmph

Alternative method:
Average speed is total distance travelled divided by total time taken.

Let the distance travelled is 2x or half of distance is x.

The first half of distance x is travelled with speed of 40km/hr, so time taken is x/40 hr.

Similarly time taken to travel the remaining x km is at speed of 60km/hr, so time taken for this 2nd half of distance is x/60.

Now, the total time taken to travel 2x distance is = x/40 + x/60 hr
= 3x/120 + 2x/120hr
= 5x/120 hr
= x/24 hr.

The average speed of car is 2x/(x/24) or 2 × 24 km/hr or 48km/hr.

Velocity of the stream = 4 km/hr.
The speed of the boat in still water is 14 km/hr.
Speed downstream = 14 + 4 = 18 km/hr.
Speed upstream = 10 km/hr.
Let the distance between A and B be x km.
Time taken to travel downstream from A to B + Time taken to travel upstream from B to C(mid of A and B)
= 38 hours.
⇒ x/18 + (x/2)/10 = 38
⇒ x/18 + x/20 = 38
⇒ 19x/180 = 38
⇒ x/180 = 2
⇒ x = 360 km.

Question: A train 240 meters long passes a pole in 15 seconds. How long will it take to pass through a platform 300 meters long?

Solution: 
Train length = 240 meters
Time to pass a pole = 15 seconds

Speed = (240/15) m/s = 16 m/s

Total distance to be covered (train + platform) = 240 + 300 = 540 meters
So, Time = 540/16
= 33.75 seconds

Question: A pair of trains set off at the same moment from opposite ends: one from Rajshahi to Dhaka and the other from Dhaka to Rajshahi. After passing each other, one takes 9 hours and the other 16 hours to complete their trips. Find the speed ratio of the two trains.
(একটি ট্রেন রাজশাহী থেকে ঢাকার দিকে এবং আরেকটি ঢাকা থেকে রাজশাহীর দিকে একই সময়ে যাত্রা শুরু করে। তারা যখন মিলিত হয়, তখন দেখা যায় একটির গন্তব্যে পৌঁছাতে ৯ ঘণ্টা এবং অন্যটির ১৬ ঘণ্টা লাগে। এই অনুযায়ী, তাদের গতি অনুপাতে কত?)

Solution: 
দুটি ট্রেন একই সময়ে যাত্রা শুরু করেছে - একটি রাজশাহী থেকে ঢাকা, অন্যটি ঢাকা থেকে রাজশাহী।

পথে এক সময় তারা একে অপরকে অতিক্রম করেছে।
সেই অতিক্রম করার পর, এক ট্রেন ৯ ঘণ্টা, অন্যটি ১৬ ঘণ্টা সময় নিয়ে নিজ নিজ গন্তব্যে পৌঁছেছে।

ধরা যাক:
ট্রেন A যাচ্ছে রাজশাহী থেকে ঢাকা (গতিবেগ = v₁)
ট্রেন B যাচ্ছে ঢাকা থেকে রাজশাহী (গতিবেগ = v₂)

তারা যখন মাঝ পথে দেখা করে (ধরা যাক M পয়েন্টে), তখন তারা একই সময় নিয়ে সেই পয়েন্টে পৌঁছায় (যেহেতু একসাথে শুরু করেছে)।

মিলনের পর:

ট্রেন A বাকি পথ যেতে ৯ ঘণ্টা নেয়, দূরত্ব = 9 × v₁
ট্রেন B বাকি পথ যেতে ১৬ ঘণ্টা নেয়, দূরত্ব = 16 × v₂
এই দূরত্ব দুটি সমান নয়, কিন্তু এদের অনুপাতই বলে দেবে গতির অনুপাত।

দুটি ট্রেন একসাথে শুরু করে এক পয়েন্টে দেখা করলে, যে ট্রেনটি অতিক্রমের পরে কম সময় নেয়, তার গতি বেশি।

গতির অনুপাত নির্ণয়ের সূত্র:
গতির অনুপাত = √(দ্বিতীয় ট্রেনের সময় / প্রথম ট্রেনের সময়)
অর্থাৎ:
v1/v2 = √(16/9) = 4/3

দুটি ট্রেনের গতির অনুপাত = ৪ : ৩

Question: A train 150 meters long takes 40 seconds to cross a 350-meter-long bridge. How much time will the train take to cross a 250-meter-long platform?

Solution:
Length of train = 150 m
Length of bridge = 350 m
∴ Total distance to cross bridge = 150 + 350 = 500 m
Time taken = 40 seconds
∴ Speed of train = Total distance/Time
= 500/40 = 12.5 m/s

Length of platform = 250 m
∴ Total distance to cross platform = 150 + 250 = 400 m

∴ Time taken = Total distance/Speed
= 400/12.5 seconds
= 32 seconds

Relative speed
= (80 + 55) km/hr
= 135 km/hr.
= 135 × (5/18) m/sec
= (75/2) m/sec.

Distance covered = (120 + 90 + 90)
= 300 m.

∴ Required time = {300 × (2/75)}
= 8 sec.

Question: A man rowed 3 miles upstream in 90 minutes. If the river flowed with current of 2 miles per hour, how long did the man's return trip take?

Solution:
Let,
The velocity of the boat is x mph
and the stream be y mph
In upstream,
In 90 minutes, he goes 3 miles
In 60 minutes, he goes (3 × 60)/90 = 2 miles

ATQ,
x - y = 2
⇒ x - 2 = 2 [stream's velocity = 2 mph]
⇒ x = 4

So, velocity in downstream = x + y = 2 + 4 = 6 mph
He goes 6 miles in 1 h
He goes 3 miles in = 3/6
= [(1/2) × 60] minutes
= 30 minutes

Let the speeds of the faster and slower trains be x m/sec and y m/sec respectively.
Then,
240/(x - y) = 60
⇒ x - y = 4 ........(i)

And, 240/(x + y) = 3
⇒ x + y = 80 ........(ii)

Adding (i) and (ii), we get
2x = 84
⇒ x = 42
Putting x = 42 in (i), we get:
42 - y = 4
⇒ y = 38

Hence, speed of the faster train = 42 m/sec.

Question: A bus moves 600 meters in a minute, and a car travels 90 km in 60 minutes. How much faster is the car than the bus? 

Solution:
Speed of the bus = Distance/Time
= (600/1) meters/minute
= (600/1000)/(1/60) km/h
= (600 × 60)/1000 km/h
= 36 km/h

A car travels 90 km in 60 minutes
So, speed of the car = Distance/Time
= 90 km/h [60 minutes = 1 hour]

∴ Difference in speed between the car and the bus = (90 - 36) km/h = 54 km/h.

Question: A train can travel 25% faster than a car. Both start from point A at the same time and reach point B 100 kms away from A at the same time. On the way, however, the train lost about 15 minutes while stopping at the stations. The speed of the car is:

Solution:
ধরি, গাড়ির গতিবেগ = x কিমি/ঘন্টা।

যেহেতু ট্রেনের গতিবেগ গাড়ির গতিবেগের চেয়ে 25% বেশি, 
∴ ট্রেনের গতিবেগ = x + (x × 25/100)
= x + 0.25x = 1.25x কিমি/ঘন্টা

গাড়ির মোট সময় = দূরত্ব/গতিবেগ
= 100/x ঘন্টা

ট্রেনের মোট সময় (স্টপেজ ছাড়া) = দূরত্ব/গতিবেগ
= 100/(1.25x) ঘন্টা

ট্রেনটি স্টেশনে 15 মিনিট থেমেছিল।
15 মিনিট = 15/60 ঘন্টা = 1/4 ঘন্টা

যেহেতু ট্রেন এবং গাড়ি একই সময়ে গন্তব্যে পৌঁছায়, তাই গাড়ির মোট সময় এবং ট্রেনের স্টপেজ সহ মোট সময় সমান।

∴গাড়ির সময় = (ট্রেনের সময় + স্টপেজ সময়)
⇒ 100/x = 100/(1.25x) + 1/4
⇒ 100/x - 80/x = 1/4
⇒ 20/x = 1/4
⇒ x = 20 × 4
⇒ x = 80 কিমি/ঘন্টা

সুতরাং, গাড়িটির গতিবেগ হলো 80 কিমি/ঘন্টা।

Let, each train’s length x
Relative speed = 108 + 90 = 198 km/h = 198 × (5/18) m/s
Distance covered in 3 minutes = 198 × (5/18) × 3 × 60 = 9900 m
∴ length of a train = 9900/2 = 4950 m

Question: A 320 metre long train crosses a platform twice its length in 48 seconds. What is the speed of the train in km/hr?

Solution:
দেওয়া আছে,
ট্রেনটির দৈর্ঘ্য = 320 মিটার
প্ল্যাটফর্মটির দৈর্ঘ্য = 2 × 320 = 640 মিটার

অতিক্রান্ত মোট দূরত্ব = (ট্রেনের দৈর্ঘ্য + প্ল্যাটফর্মের দৈর্ঘ্য)
= (320 + 640) মিটার
= 960 মিটার

সময় লেগেছে = 48 সেকেন্ড

∴ ট্রেনটির গতিবেগ = দূরত্ব / সময়
= 960 / 48 মিটার/সেকেন্ড
= 20 মিটার/সেকেন্ড

এখন,
1 মিটার/সেকেন্ড = (1/1000)/(1/3600) কিমি/ঘন্টা
= 3.6 কিমি/ঘন্টা

1 মিটার/সেকেন্ড = 3.6 কিলোমিটার/ঘন্টা
∴ 20 মিটার/সেকেন্ড = 20 × 3.6 কিলোমিটার/ঘন্টা
= 72 কিলোমিটার/ঘন্টা

সুতরাং, ট্রেনটির গতিবেগ 72 কিমি/ঘন্টা।

Question: A boat takes 8 hours to travel 32 km upstream (against the current). If it were traveling downstream (with the current), it would take only 4 hours to cover the same distance. What is the speed of the current?

Solution:
দেওয়া আছে,
স্রোতের প্রতিকূলে 32 কিমি যেতে সময় লাগে 8 ঘণ্টা।
∴ প্রতিকূলে নৌকার গতিবেগ = 32/8 = 4 কিমি/ঘণ্টা।

আবার, স্রোতের অনুকূলে 32 কিমি যেতে সময় লাগে 4 ঘণ্টা।
∴ অনুকূলে নৌকার গতিবেগ = 32/4 = 8 কিমি/ঘণ্টা।

আমরা জানি,
স্রোতের গতিবেগ = (অনুকূলে গতিবেগ - প্রতিকূলে গতিবেগ) / 2
= (8 - 4)/2 কিমি/ঘণ্টা
= 4/2 কিমি/ঘণ্টা
= 2 কিমি/ঘণ্টা

সুতরাং, স্রোতের গতিবেগ 2 কিমি/ঘণ্টা।

Question: A train running at the speed of 90 km/h crosses a pole in 10 seconds. What is the length of the train?

Solution:
Speed = 90 km/h
= [90 × (5/18)] m/sec
= 25 m/sec

∴ Length of the train = (25 × 10) m
= 250 m

So, the length of the train is 250 meters.

Question: A train 360 m long passes a pole in 30 seconds. How long will it take to pass a platform 540 m long?

Solution: 
সমাধান:
ট্রেনটির মোট দূরত্ব অতিক্রম করতে হবে = (360 + 540) মিটার = 900 মিটার 

ট্রেনটি 360 মিটার অতিক্রম করতে সময় নেয় = 30 সেকেন্ড 
ট্রেনটি1 মিটার অতিক্রম করতে সময় নেয় = 30/360 সেকেন্ড 
ট্রেনটি 900 মিটার অতিক্রম করতে সময় নেয় = (30 × 900)/360 সেকেন্ড 
= 75 সেকেন্ড

Question: A swimmer covers 3 km against the current in 30 minutes and the same distance with the current in 15 minutes. What is the speed of the swimmer in still water?

Solution:
Given that,
Distance covered = 3 km
Time consumed = 30 minutes = 1/2 hour.
∴ speed of the swimmer against current (upstream) = 3/(1/2) = 6 km/hr

Again,
With the current,
Distance covered = 3 km
Time consumed = 15 minutes = 15/60 = 1/4 hr
∴ speed of the swimmer with current (downstream) = 3/(1/4) = 12 km/hr

let,
swimmer's speed in still water = x km/hr
speed of current = y km/hr

According to question,
x + y = 12 ------(1)
x - y = 6 -------(2)

(1) + (2) ⇒
2x = 18
⇒ x = 9  

∴ swimmer's speed in still water = 9 km/hr

Relative speed = 120 + 80
= 200 km/hr.
= 200 × (5/18)
= 500/9
Time = 9 seconds
Distance covered = (500/9) × 9
= 500 meter.
Length of the train = (500 - 270) meter.
= 230 meter.

Let the distance be x
Speed upstream = (40-10) = 30 kmph
Speed downstream = (40+10) = 50 kmph
Total time taken = 1 hr
⇒ x/50 + x/30 = 1
⇒ 8x/150 = 1
⇒ x = 150/8 = 18.75 km

ধরি,
একটি জাহাজ A speed - এ যাচ্ছে এবং অপরটি B speed এ যাচ্ছে।
নদীর প্রস্থ w
At time = t₁
A (t₁) = 750
B(t₁) =(w - 750)
⇒ B/A = (w - 750)/750
750B = (w - 750)A
At time = t₂
A(t₂) = w + 250
B(t₂) = w +(w - 250)
= 2w - 250
⇒ B/A = (2w - 250)/(w + 250)
(w + 250)B = (2w - 250)A .... Eq.[1]
750B = (w - 750/A
B = [(w - 750)/750]A
Plug into Eq. [1]
(w + 250)[(w - 750)/750]A = (2w - 250) A
(w + 250)(w - 750)/750 =750(2w -250)
w² - 500w - 187,500 = 1500w - 187500
w2 - 2000w = 0
w2 = 2000w
w = 2000 yards.

Question: A car reaches from City A to City B in 9 hours travelling at a speed of 40 km/hr. If its speed is increased by 20 km/hr, then the time of journey is reduced by-

Solution: 
Original speed = 40 km/hr
Time taken = 9 hours

∴ Distance between City A and City B = speed × time
= 40 × 9 = 360 km

∴ New speed = 40 + 20 = 60 km/hr

∴ New time taken = distance/new speed
= 360/60
= 6 hours

∴ Reduction in time = original time - new time
= 9 - 6
= 3 hours

So the time of journey is reduced by 3 hours.

Question: A train is moving at a speed of 132 km/hour. If the length of the train is 110 meters, how long will it take to cross a railway platform 165 meters long.

Solution: 
Given that,
Length of train = 110 m
Length of platform = 165 m
∴ Total distance to be covered = 110 + 165 = 275 meters
Speed of train = 132 km/h
= 132 × (1000/3600) m/s
= 132 × (5/18) m/s
= 110/3 m/s

Time taken = Distance/Speed
= 275/(110/3)
= (275 × 3)/110 
= 7.5 seconds

So the train will take 7.5 seconds to cross the 165 meter long railway platform.

Question: A bus travels 300 km in 5 hours. What is its average speed?

Solution:
ট্রেনটির অতিক্রান্ত দূরত্ব = 300 কি.মি.
মোট সময় = 5 ঘণ্টা

গড় গতিবেগ = 300/5 km/h
= 60 km/h

- 25 m/sec
- 1500 m/min = 1500/60 = 25 m/sec
- 90 km/hr = (90×1000)/3600 = 25 m/sec

Relative speed= 24-18= 6 km/hr
Time required by faster train to overtake slower train
= 27/6 hr
= 4(1/2) hr
∴ Distance between Q and R:
= 18×4(1/2)
= 81 km

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
= 20/1
= 20 km/hr
Upstream Speed = 20/2 = 10 km/hr

X + Y = 20 km/hr
and, X - Y = 10 km/hr

Adding them we get,
X + Y + X - Y = 30 km/hr
∴ X=15 km/hr = Speed of swimmer in still water

∴ Y = 20 - 15 = 5 km/hr = Speed of river.

Given speed = 63 km/hr = 63 × 5/18 = 35/2 m/s
Let the length of the bridge = x mts
Given time taken to cover the distance of (170 + x) mts is 30 sec.
We know speed = distance / time 
⇒ 35/2 = (170+x)/30
⇒ 340 + 2x = 1050
⇒ 2x = 710
∴ x = 355

Question: A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is:

Solution:
Let man's rate upstream be x kmph.
Then, his rate downstream = 2x kmph.

∴ (Speed in still water) : (Speed of stream) = {(2x + x)/2} : {(2x - x)/2}
= (3x/2) : (x/2)
= 3 : 1

Let the speed of first car = 7x
2nd car = 8x.

According to the question
8x = 200/5 = 40 [using speed = distance time]
x = 40/8 = 5

Speed of first car = 7x = 7 X 5 = 35 Km/hr

Speed in still water =6 km/hr.
Speed against the current =6/3 km/hr = 2 km/hr
Let the speed of the current be x km/hr
so, 6-x = 2
=> x = 4 km/hr.

Question: The average speed of a bus is half the speed of a train. The train covers 1000 km in 20 hours. How much distance will the bus cover in 48 minutes?

Solution:
ট্রেনের গতিবেগ = অতিক্রান্ত দূরত্ব/সময়
= 1000 কিমি/20 ঘন্টা
= 50 কিমি/ঘন্টা

এখন, বাসের গতিবেগ ট্রেনের গতিবেগের অর্ধেক।
∴ বাসের গতিবেগ = ট্রেনের গতিবেগ × 1/2
= 50 কিমি/ঘন্টা × 1/2
= 25 কিমি/ঘন্টা

∴ বাসের অতিক্রান্ত দূরত্ব = বাসের গতিবেগ × সময়
= 25 কিমি/ঘন্টা × 48 মিনিট
= 25 কিমি/ঘন্টা × (48/60) ঘন্টা
= 20 কিমি

সুতরাং, বাসটি 48 মিনিটে 20 কিমি দূরত্ব অতিক্রম করবে।

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

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

X+Y = 10 + 2 = 12 km/hr and X-Y = 10 - 2 = 8 km/hr
Let Time be T hours for downstream
Distance is same
∴ D = D
∴ 12 × T = 8 × (T + 4)
∴ T = 8 hours = Time for downstream
Distance = 12km/hr × 8 hours = 96 km

Question: A train takes 8 seconds to cross a pole and 18 seconds to cross a platform of length 120m. What is the length of the train?

Solution:
Let the length of the train be L meters.

Now,
Time to cross a pole = 8 s
Distance covered = length of train = L
∴ Speed = L/8 m/s

Again, 
Platform length = 120 m
Distance covered = L + 120
Time taken = 18 s
∴ Speed = (L + 120)/18 m/s

ATQ, 
L/8 = (L + 120)/18
⇒ 18L = 8L + 960
⇒ 18L - 8L = 960
⇒ 10L = 960
⇒ L = 960/10
∴ L = 96 m

So the length of the train is 96 meters.

The ratio of speeds of A, B, C = 6 : 3 : 1
The ratio of time taken by A, B, C = 1/6 : 1/3 : 1
To simplify it, we will multiply it by the LCM of ratio of speeds given.
Hence, the ratio of time taken by A, B, C = 1 : 2 : 6

[Speed is inversely proportional to time, meaning if speed increases, time decreases. So, ratio of time would be reciprocal of the ratio of speed given. ]

Time taken by C to covered given distance = 78 = 6 × 13
The ratio of time of A and C = 1 : 6
Thus, time taken by A = 13 min.

Alternative method:
Let C's speed be x metres/min
Let time taken by A be y min
Then B's speed = 3x metres/min
And, A's speed = 6x metres/min
Ratio of speed of A and C = Ratio of time by C and A
6x : x = 78 : y
6x/x = 78/y
y = 13 min

Let, distance from A to B = x km
ATQ, 
x/40 + x/30 + 1 = 15
Or, (3x + 4x)/120 = 15 - 1
Or, 7x = 14 × 120
Or, x = 240
∴ distance = 120 km

Let distance travelled by cat before dog catches it be D
We know, time for which Dog and Cat ran is same
∴ T = T
∴ D/5 = (D + 80)/7 [D = S x T]
∴ D = 200 m

The Distance covered by the motorcycle with speed 60km/hr in 3 hours = (60 x 3)
= 180 km
Now,
Speed = Distance/Time
Since the train covers the same 180 km in 3/2 hours (we can write 1 and half hours as 3/2hours)
Then the speed of the train = 180/ (3/2)
= 180 × (2/3)
= 120 km/hr.
Hence, the train travelled at the speed of 120km/hr to cross 180km in 1 and half hour.