Time and Speed - Train and Boat

Assume both trains meet x hours after 7 a.m.
Distance covered by train starting from P in x hours = 20 x km
Distance covered by train starting from Q in (x − 1) hours = 25 (x − 1) km
Total distance = 110 km
⇒ 20 x + 25 (x − 1) = 110
⇒ 45 x = 135
⇒ x = 3
Hence, they meet 3 hours after 7 a.m.
i.e., they meet at 10 a.m.

Alternative method:
Distance travelled by first train in 1 hour = 20 km
Therefore, at 8 a.m., both trains will be (110 − 20) = 90 km apart.
Since the relative speed is (20 + 25) = 45 kmph,
they will cover this distance in 90/ 45 = 2 hours.
I.e. They will meet at 10 a.m.

Let X be the distance, then
(x/5)−(x/8) = 3/2
x = 20km

speed of A : speed of B = 7/3:1 = 7:3
It means, in a race of 7 metres, A gains (7−3 ) = 4 metres.
If A needs to gain 80 metres, race should be of (7/4)×80 = 140 metres.

Question: A train passes by a lamp post at platform in 6 sec. and passes by the platform completely in 24 sec. If the length of the platform is 360 meter, then length of the train is?

Solution:
Let, the length of the train is x meter.
The train passing the lamp post at a speed of x/6 meter/sec
And, the train passing the platform at a speed of (360 + x)/24 meter/sec

∴ x/6 = (360 + x)/24
⇒ x = (360 + x)/4
⇒ 4x = 360 + x 
⇒ 3x = 360
⇒ x = 360/3
⇒ x = 120

∴ Length of the train is 120 meter.

Speed downstream = 15 km/hr
Rate of the current = 1(1/2) km/hr.
Speed in still water = {15 - 1(1/2)}
= 13(1/2) km/hr.
Rate against the current = {13(1/2) - 1(1/2)} km/hr
= 12 km/hr.

Question: The speed of a boat in still water is 15 km/h, and the speed of the current is 5 km/h. In how much time (in hours) will the boat travel a distance of 60 km upstream and the same distance downstream?

Solution: 
Given that, 
Speed of boat in still water = 15 km/h
Speed of current = 5 km/h
Distance (each way) = 60 km

∴ Downstream speed = 15+ 5 = 20 km/h
∴ Upstream speed = 15 - 5 = 10 km/h

∴ Time to travel 60 km downstream = 60/20 = 3 hours
∴ Time to travel 60 km upstream = 60/10 = 6 hours

∴ Total time = 3 + 6 = 9 hours

∴ The boat will take 9 hours in total.

Let,
Speed of boat = x and, Speed of stream = y
ATQ,
x - y = 24/6 = 4 ...... (i)
x + y = 24/4 = 6 ....... (ii)
From,
(ii) – (i),
x + y - x + y = 6 - 4
Or, 2y = 2
∴ y = 1
So, Speed of the stream = 1 km/hr

Question: A swimmer swims from a point P against the current for 6 min and then swims back along the current for next 6 min and reaches at a point Q. If the distance between P and Q is 120 m then the speed of the current (in km/h) is:

Solution: 
Given that, 
Swimmer swims 6 min against current and 6 min along current
Distance between P and Q = 120 m

Let,
Speed of swimmer in still water = u m/min
Speed of current = v m/min

Now,
Distance travelled upstream (against current), d₁ = (u - v) × 6
And distance travelled downstream (along current), d₂ = (u + v) × 6

Net displacement from starting point = 120 m
d₂ - d₁ = 120
⇒ 6(u + v) - 6(u - v) = 120
⇒ 6u + 6v - 6u + 6v = 120 
⇒ 12v = 120
⇒ v = 120/12
⇒ v = 10 m/min
⇒ v = (10 × 60)/1000
⇒ v = 600/1000
∴ v = 0.6 km/h

So the speed of the current is 0.6 km/h.

Let the speed one train be x and the speed of the second train be y
Length of the first train = Speed × Time = 28x
Length of second train = Speed × Time = 18y

So, {(28x + 18y)/(x + y)} = 26
⇒ 28x + 18y = 26x + 26y
⇒ 2x = 8y

Therefore,
x : y = 4 : 1.

Question: If 2^{x - 1} + 2^{x + 1} = 320, then 2^{x - 6} is equal to

Solution:

Question: What is the length of a train that takes 15 seconds to pass a pole when it runs at a speed of 54 km/h?

Solution:
আমরা জানি, ট্রেন যখন কোনো খুঁটি বা ব্যক্তিকে অতিক্রম করে, তখন ট্রেনটি তার নিজের দৈর্ঘ্য অতিক্রম করে।

ট্রেনের দৈর্ঘ্য = গতিবেগ × সময়

এখানে, ট্রেনের গতিবেগ = 54 km/h
= 54 × (1000/3600) m/s
= 15 m/s

সময় = 15 সেকেন্ড

∴ ট্রেনের দৈর্ঘ্য = গতিবেগ × সময়
= 15 × 15 মিটার
 = 225 মিটার

অতএব, ট্রেনটির দৈর্ঘ্য হলো 225 মিটার।

Question: Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 12 seconds respectively and they cross each other in 24 seconds. The ratio of their speeds is:

Solution:
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 = 12y metres.

ATQ,
(27x + 12y)/(x + y) = 24
⇒ 27x + 12y = 24x + 24y
⇒ 27x - 24x = 24y - 12y
⇒ 3x = 12y
⇒ x/y = 12/3
∴ x : y = 4 : 1

Total distance travelled in 12 hours = (35 + 37 + 39 + ..... upto 12 terms)
This is an A.P with first term, a=35, number of terms,
n = 12,
d = 2.
Required distance = n/2 {2a+(n-1)×2}
= 12/2[2 x 35+{12-1) x 2]
= 6(70+22)
= 552 kms.

Question: A man completes a journey in 8 hours. He travels the first half of the journey at the rate of 40 km/hr and the second half at the rate of 60 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)/40 ঘন্টা
= d/80 ঘন্টা

দ্বিতীয় অর্ধেক যাত্রায়, সময় = দূরত্ব/গতিবেগ
= (d/2)/60 ঘন্টা
= d/120 ঘন্টা

প্রশ্নমতে,
(d/80) + (d/120) = 8
⇒ (3d + 2d)/240 = 8
⇒ 5d/240 = 8
⇒ 5d = 8 × 240
⇒ 5d = 1920
⇒ d = 1920/5
⇒ d = 384 কিমি

∴ মোট যাত্রার দূরত্ব 384 কিলোমিটার।

Let the speed of the water in still water = x
Given that speed of the stream = 3 kmph
Speed downstream
= (x+3) kmph
Speed upstream
= (x−3) kmph
He travels a certain distance downstream in 4 hour and come back in 6 hour.
ie, distance travelled downstream in 4 hour = distance travelled upstream in 6 hour
since distance = speed × time,
we have
(x+3)4= (x−3)6
⇒ (x+3)2= (x−3)3
⇒ 2x+6= 3x−9
⇒ x= 6+9= 15 kmph

Question: A, B and C start together from the same place to walk round a circular path of length 12 km. A walks at the rate of 4 km/h, B 3 km/h and C 3/2 km/h. They will meet together at the starting place at the end of-

Solution:
Given that,
Circumference of circular path = 12 km
Speeds, A = 4 km/h
B = 3 km/h
C = 3/2 = 1.5 km/h

We know, 
Time = Distance/Speed

Than, 
Time for A = 12/4 = 3 hours
Time for B = 12/3 = 4 hours
Time for C = 12/1.5 = 8 hours

∴ Required time,
= LCM of 3, 4, 8.
= 24 hours.

∴ They will meet together at the starting place at the end of 24 hours.

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

Suppose B takes t hours to walk d km.
Then, A takes (t + 2) hours to walk d km.
With double speed, A will take = 1/2(t+2) hours.
∴ t − 1/2(t+2) = 1
⇒ {2t − (t+2)}/2 = 1
⇒ 2t − (t+2) = 2
⇒ t = 4
∴ B takes 4 hours to walk d km.

Speed of tiger = 40 m/min
Speed of deer = 20 m/min.
Relative speed = 40 - 20 = 20 m/min.
Initial difference in distance = 50 × 8 = 400 m

Time taken to catch = 400/20 = 20 min.
Distance traveled in 20 min,
= 20 × 40
= 800 m.

Question: How long does a train 1000 m long moving at a speed of 90 km/hr would take to pass through a 500 m long bridge?

Solution:
Here, the time taken by the train to pass the bridge completely would be the time it takes to cover 1000 + 500 = 1500 m at the speed of 90 km/hr
= 90 × (5/18)
= 25 m/sec

Therefore, time required = 1500/25
= 60 sec
= 1 minute

Question: A train was delayed by 30 minutes due to a technical snag. To cover the distance of 100 km on time, the driver had to increase its usual speed by 10 km/h. Find the usual speed of the train in km/h?

সমাধান:
ধরি, ট্রেনটির স্বাভাবিক গতিবেগ ছিল x কিমি/ঘন্টা।
তাহলে, বর্ধিত গতিবেগ হবে (x + 10) কিমি/ঘন্টা।

স্বাভাবিক গতিতে 100 কিমি যেতে সময় লাগে = 100/x ঘন্টা।
বর্ধিত গতিতে 100 কিমি যেতে সময় লাগে = 100/(x + 10) ঘন্টা।

সময়ের পার্থক্য 30 মিনিট বা 30/60 = 1/2 ঘন্টা

প্রশ্নমতে, 
(100/x) - {100/(x + 10)} = 1/2
⇒ {100(x + 10) - 100x}/x(x + 10) = 1/2
⇒ 1000/x(x + 10) = 1/2
⇒ (x² + 10x) = 2000
⇒ x² + 10x - 2000 = 0
⇒ x² + 50x - 40x - 2000 = 0
⇒ x(x + 50) - 40(x + 50) = 0
⇒ (x - 40)(x + 50) = 0

যেহেতু গতিবেগ ঋণাত্মক হতে পারে না, তাই x = 40
∴ ট্রেনটির স্বাভাবিক গতিবেগ হলো 40 কিমি/ঘন্টা।

Question: A passenger sitting by the window of a train notices that he can count 16 electric poles in one minute. If the poles are known to be 30 meters apart, at what speed is the train traveling in km/h?

সমাধান:
মোট গুনে দেখা বৈদ্যুতিক খুঁটির সংখ্যা = 16 টি।
খুঁটিগুলির মধ্যবর্তী গ্যাপের সংখ্যা = (16 - 1) = 15 টি।

দুটি খুঁটির মধ্যবর্তী দূরত্ব = 30 মিটার।
∴ 1 মিনিট বা 60 সেকেন্ডে মোট অতিক্রান্ত দূরত্ব = (30 × 15) মিটার = 450 মিটার।

ট্রেনটির গতিবেগ = দূরত্ব/সময়
= 450/60 মিটার/সেকেন্ড
= 45/6 মিটার/সেকেন্ড
= 15/2 মিটার/সেকেন্ড
= (15/2 × 18/5) কিমি/ঘন্টা
= (3 × 9) কিমি/ঘন্টা
= 27 কিমি/ঘন্টা।

∴ ট্রেনটির গতিবেগ হলো 27 কিমি/ঘন্টা।

Question: A man completes a journey of 120 km by a car. He covers 60 km at 30 km/h, 40 km at 40 km/h, and the remaining 20 km at 20 km/h. What is his average speed for the whole journey?

Solution:
A man completes a journey of 120 km by a car.
First case, 
Distance covered: 60 km at 30 km/h
∴ Time = 60 ÷ 30 = 2 hours

Second case,
40 km at 40 km/h
∴ Time = 40 ÷ 40 = 1 hour

Third case,
the remaining 20 km at 20 km/h
 ∴ Time = 20 ÷ 20 = 1 hour

∴ Total distance = 60 + 40 + 20 = 120 km
∴ Total time = 2 + 1 + 1 = 4 hours

∴ Average speed = Total distance ÷ Total time = 120 ÷ 4 = 30 km/h

Let,
Speed = x km/h
So,
40x - 40x × (14/15) = 5
Or, 40x - 112x/3 = 5
Or, 120x - 112x = 15
Or, x = 15/8
∴ distance covered = (15 × 40)/8 = 75 km

Question: Two trains with lengths 126 m and 119 m respectively are moving towards each other. Their speeds are 12 m/s and 23 m/s, respectively. What will be the time needed by the trains to cross each other?

Solution:
To cross each other completely, the two trains must cover a total distance equal to the sum of their lengths.
∴ Total distance to be covered = 126 + 119 = 245 meters

Since they are moving towards each other, their relative speed is the sum of their individual speeds.
∴ Relative speed = 12 + 23 = 35 m/s

∴ Time taken to cross each other = Total distance/Relative speed
= 245/35
= 7 seconds

∴ The trains will take 7 seconds to completely cross each other.

d = v × t
= (4 - 3) × 4
= 4 km

Relative speed = (120 + 80) km/hr
= (200×5/18) m/s
= 500/9 m/s
Let, The length of other train be x m
Then,
x + 270 = 500/9 × 9
Or, x = 500 - 270 = 230

Speed of the boat in still water = 12 km/hr.
Speed of the stream = 4 km/hr.
Speed downstream = (12 + 4)
= 16 km/hr.
Time is taken to travel 68 km downstream
= 68/16
=17/4
= 4.25 hrs.

Average speed = {(2 × 500 × 700)/(500 + 700)} km/hr
= (1750/3) km/hr
= 583(1/3) km/hr.
Hence, The average speed of the aeroplane for the entire journey is 583(1/3) km/hr

Question: A bus travels the first 150 km at 75 km/h and the next 150 km at 50 km/h. What is the average speed for the whole trip?

Solution: 
Total distance = 150 km + 150 km = 300 km

Now, 
Time for first part, 
Time = 150/75 = 2 hours

And time for second part, 
time = 150/50 = 3 hours

∴ Total time = 2 + 3 = 5 hours

Average speed = Total distance/Total time
= 300/5
= 60 km/h

So the average speed for the whole trip is 60 km/h.

Shortcut:
When equal distances are covered at speeds a and b,
Average speed = 2ab/(a + b)
= (2 × 75 × 50)/(75 + 50) 
= 60 km/h

Speed downstream = 1/(5/60)
= 60/5
= 12 km/hr
Speed upstream = 4/1
= 4 km/hr
velocity of the current = 1/2(12 - 4)
= (1/2) × 8
= 4 km/hr.

Question: A man completes a journey in 5 hours. He travels the first half of the journey at the rate of 20 km/hr and the second half at the rate of 30 km/hr. Find the total journey in km.

সমাধান:
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)/20 ঘন্টা
= D/40 ঘন্টা

দ্বিতীয় অর্ধেক যাত্রায়,
সময় = দূরত্ব/গতিবেগ
= (D/2)/30 ঘন্টা
= D/60 ঘন্টা

প্রশ্নমতে,
D/40 + D/60 = 5
⇒ (3D + 2D)/120 = 5
⇒ 3D + 2D = 5 × 120
⇒ 5D = 600
⇒ D = 600/5
⇒ D = 120 কিমি

সুতরাং, মোট যাত্রার দূরত্ব হলো 120 কিমি।

Question: With an average speed of 50 km/hr, a train reaches its destination in time. If it goes with an average speed of 40 km/hr, it is late by 24 min. The total journey is:

Solution:
Difference between timings = 24 min = 24/60 hr = 2/5 hr.
Let the length of the journey be x km.

Then,
(x/40) - (x/50) = 2/5
Or, (5x - 4x)/200 = 2/5
Or, x/200 = 2/5
Or, x = (25 × 200)
∴ x = 80 km.

∴ The total journey is 80 km.

Question: A monkey climbs a 70 m high pole. In first minute he climbs 8 m and slips down 4 in the next minute. How much time is required by it to reach the top?

সমাধান:
১ম মিনিটে উঠে ৮ মিটার
২য় মিনিটে নামে ৪ মিটার

∴  ২ মিনিটে উঠবে ৪ মিটার 
৩ মিনিটে উঠবে ৪ + ৮ = ১২ মিটার 
৪ মিনিটে উঠবে ১২ - ৪ = ৮ মিটার 

প্যাটার্ন অনুসারে, 
১, ৩, ৫, ৭তম,............ মিনিটে উঠবে যথাক্রমে ৮, ১২, ১৬, ২০...........মিটার
২, ৪, ৬, ৮তম.......... মিনিটে উঠবে যথাক্রমে ৪, ৮, ১২, ১৬......... মিটার

এভাবে, ৩০ মিনিটে উঠবে ৬০ মিটার
৩১ মিনিটে উঠবে ৬৮ মিটার
৩২ মিনিটে উঠবে = ৬৮ - ৪ = ৬৪ মিটার

∴ উঠা বাকি আছে = ৭০ - ৬৪ = ৬ মিটার

৮ মিটার উঠতে সময় লাগে ১ মিনিট বা ৬০ সেকেন্ড
৬ মিটার উঠতে সময় লাগে (৬০ × ৬)/৮ সেকেন্ড
= ৪৫ সেকেন্ড

∴ মোট সময় লাগবে = ৩২ মিনিট ৪৫ সেকেন্ড

[অপশনে ৩২ মিনিট ৪৫ সেকেন্ড না থাকায় সঠিক উত্তর হবে ঘ) None of the above]

===============================

প্রচলিত সমাধান:

যেহেতু ১ মিনিটে ৮ মিটার উঠে এবং শেষ মিনিটে এই দূরত্ব উঠবে সেহেতু এটা বিয়োগ করলে থাকে ৭০ - ৮ = ৬২ মিটার

১ম মিনিটে উঠে ৮ মিটার
২য় মিনিটে নামে ৪ মিটার

∴  ২ মিনিটে উঠবে ৪ মিটার 

অর্থাৎ ৪ মিটার উঠে = ২ মিনিটে
∴ ৬২ মিটার উঠে = (২ × ৬২)/৪
= ৩১ মিনিটে 

∴ খুঁটি বেয়ে উঠতে সময় লাগবে = ৩১ + ১ = ৩২ মিনিট

দেওয়া আছে, লোকটি স্রোতের অনুকূলে 4 ঘন্টায় যায় 18 km
অর্থাৎ, স্রোতের অনুকূলে 1 ঘন্টায় যায় 18/4 km = 4.5 km
এবং লোকটি স্রোতের প্রতিকূলে 12 ঘন্টায় যায় 18km
অর্থাৎ, লোকটি স্রোতের প্রতিকূলে 1 ঘন্টায় যায় 18/12 = 1.5km
ধরি, নৌকার বেগ x km/hr এবং স্রোতের বেগ y km/hr

প্রশ্নমতে,
x + y = 4.5 ... (i) এবং x - y = 1.5 ... (ii)
(i) থেকে (ii) বিয়োগ করে পাই,
 2y = 3
⇒ y = 1.5 km/hr

Let the speed of the stream be x km/hr
Upstream Speed = 15 + x
Downstream Speed = 15 - x

So,
{30/(15+x)} + {30/(15-x)} = 4(1/2)(4 hours 30 minutes)
⇒ {900/(225-x2)} = 9/2
⇒ 9x² = 225
⇒ x² = 25
⇒ x = 5.
Hence, the speed of the stream is 5 km/hr.