Problems on Trains (Time, Distance and Speed)

Speed of train will help us to calculate different types of problems on motion.

CASE A. When a train passes a stationary object:

Let x be the length of the train which includes the engine also. When the end of the train passes the object, then the engine of the train has to move a distance equal to the train.

Then the time taken by the train to pass the stationary object = length of the train/speed of the train

CASE B. When a train passes a stationary object having some length:

When the end of the train passes the stationary object having some length, then the engine of the train has to move a distance equal to the sum of the length of the train and stationary object.

Then the time taken by the train to pass the stationary object = (length of the train + length of the stationary object)/speed of the train

Speed of Train

Problems to calculate the motion or speed of train:

1. Find the time taken by a train 150 m long, running at a speed of 90 km/hr in crossing the pole.

Solution:

Length of the train = 150m

Speed of the train = 90 km/hr

= 90 × 5/18 m/sec

= 25m/sec

Therefore, time taken by the train to cross the pole = 150 m/25m/sec = 6 seconds.

2. A train 340 m long is running at a speed of 45 km/hr. what time will it take to cross a 160 m long tunnel?

Solution:

Length of the train = 340 m

Length of the tunnel = 160m

Therefore, length of the train + length of the tunnel = (340 + 160) m = 500m

Speed of the train = 45 km/hr

Speed of the train = 45 × 5/18 m/sec

= 25/2 m/sec

= 12.5 m/sec

Therefore, time taken by the train to cross the tunnel = 500 m/12.5 m/sec.

= 40 seconds.

3. A train is running at a speed of 90 km/hr. if it crosses a pole in just 10 second, what is the length of the train?

Solution:

Speed of the train = 90 km/hr

Speed of the train = 90 × 5/18 m/sec = 25 m/sec

Time taken by the train to cross the pole = 10 seconds

Therefore, length of the train = 25 m/sec × 10 sec = 250 m

4. A train 280 m long crosses the bridge 170 m in 22.5 seconds. Find the speed of the train in km/hr

Solution:

Length of the train = 280 m

Length of the bridge = 170 m

Therefore, length of the train + length of the bridge = 280 m + 170 m = 450 m

Time taken by the train to cross the bridge = 22.5 sec = 45/2 sec.

Therefore, speed of train = 450 m/22.5 m/sec = 450/45/2 m/sec = 20 m/sec

To convert the speed from m/sec to km/hr, multiply by 18/5

Therefore, speed of the train = 20 × 18/5 km/hr = 72 km/hr.

Relationship between Speed, Distance and Time

What is the relationship between speed, distance and time?

Here we will learn the mathematical relation between the speed, distance and time. The speed of a moving body is the distance traveled by it in unit time.

If the distance is in km and time is in hours, then the speed is km/hr.

If the distance is in m and the time is in seconds, then the speed is m/sec.

For example:

1. A car travels 75 km in 1 hour; we say that the speed of the car is 75 kilometers per hour, i.e., 75 km/hr.

2. Maria runs 360 m in 90 seconds; we say that the speed of Maria is 360/90 m/sec = 4 m/sec.

3. If a car travels 60 km in 1 hour; we say that the speed of the car is 60 kilometers per hour, i.e., 60 km/hr.

Speed can be uniform or variable.

Uniform speed: If an object covers the same distance in the same intervals of time, then the speed of the object is said to be uniform.

For example:

A car covers 70 km in 1st hour, 70 km in 2nd hour, 70 km in 3rd and so on. We say that the car is moving with a uniform speed of 70 km/hr.

Variable speed: If an object covers different distance in the same interval of time, then the speed of the object is said to be variable.

For example:

If a car covers 59 km in the first hour, 64 km in the second hour and 57 km in the third hour. We say that the car is moving with a variable speed.

Average Speed: If a moving body travels d

_{1}

, d

_{2}

, d

_{3}

, …. d

_{n}

meters with different speeds V

_{1}

, V

_{2}

, ……, V

_{n}

m/sec in time t

_{1}

, t

_{2}

, …., t

_{n}

seconds respectively.

Then the average speed of the body =

$\frac{t o t a l d i s t a n c e t r a v e l e d}{t o t a l t i m e t a k e n} = \frac{d_{1} + d_{2} + d_{3} + . . . + d_{n}}{t_{1} + t_{2} + t_{3} + . . . + t_{n}}$

The relationship between speed, distance and time are connected by the following relation:

Speed = Distance/Time

Time = Distance/Speed

Distance = Speed × Time.

Conversion of Units of Speed

Here we will learn the conversion of units of speed

(i) Conversion of km/hr into m/sec

(ii) Conversion of m/sec to km/hr

How to convert km/hr into m/sec?

km/hr = 1 km/1 hr = 1 × 1000 m/60 × 60 sec = 1000/3600 m/sec = 5/18 m/sec

So, we say that to convert km/hr into m/sec, we multiply by 5/18.

Solved examples on conversion of km/hr into m/sec:

1. Convert 72 km/hr to m/sec

Solution:

72 km/hr

= 72 × 5/18 m/sec

= 20 m/sec

2. The speed of the bicycle is 90 km/hr, what is its speed in m/sec?

Solution:

Speed of bicycle = 90 km/hr

To convert the speed into m/sec, we multiply by 5/18

Therefore, speed of bicycle = 90 × 5/18 m/sec = 25 m/sec

3. A car covers a distance of 100 km in first two hours, 120 km in next 1 hour and 32 km in next 1/2 hour. Convert the speed into m/sec.

Solution:

Total distance covered by the car = (100 + 120 + 32) km = 252km

Total time taken = (2 + 1 + 1/2) hr = (4 + 2 + 1)/2 = 7/2 hr

Therefore, average speed = Distance covered/Time taken

= 252 km/(7/2) hr

= 252/1 × 2/7 km/hr

= 72 km/hr

To convert the speed into m/sec, we multiply by 5/18

Therefore, 72 km/hr = 72 × 5/18 = 20 m/sec

How to convert m/sec to km/hr?

m/sec = 1m/1sec = (1/1000) km/(1/60 × 60) hr = 60 × 60 /1000 km/hr = 18/5 km/hr

So, we say that to convert m/sec into km/hr, we multiply by 18/5

Solved examples on conversion of km/hr into m/sec:

1. Convert 40 m/sec to km/hr

Solution:

40m/sec

= 40 × 18/5

= 144 km/hr

2. The speed of a cyclist is 4 m/sec. find the speed in km/hr.

Solution:

Speed of a cyclist = 4 m/sec

To convert the speed in km/hr, we multiply by 18/5

Therefore, speed of the cyclist = 4 × 18/5 km/hr

= 72/5 km/hr

= 14.4 km/hr

Hence, we have learn the conversion of units of speed from km/hr to m/sec and m/sec to km/hr.

Problems on Calculating Speed

Here we will learn to solve different types of problems on calculating speed.

We know, the speed of a moving body is the distance traveled by it in unit time.

Formula to find out speed = distance/time

Word problems on calculating speed:

1. A man walks 20 km in 4 hours. Find his speed.

Solution:

Distance covered = 20 km

Time taken = 4 hours

We know, speed = distance/time

= 20/4 km/hr

Therefore, speed = 5 km/hr

2. A car covers a distance of 450 m in 1 minute whereas a train covers 69 km in 45 minutes. Find the ratio of their speeds.

Solution:

Speed of car = Distance covered/Time taken = 450/60 m/sec = 15/2

= 15/2 × 18/5 km/hr

= 27 km/hr

Distance covered by train = 69 km

Time taken = 45 min = 45/60 hr = 3/4 hr

Therefore, speed of trains = 69/(3/4) km/hr

= 69/1 × 4/3 km/hr

= 92 km/hr

Therefore, ratio of their speed i.e., speed of car/speed of train = 27/92 = 27 : 92

3. Kate travels a distance of 9 km from her house to the school by auto-rickshaw at 18 km/hr and returns on rickshaw at 15 km/hr. Find the average speed for the whole journey.

Solution:

Time taken by Kate to reach school = distance/speed = 9/18 hr = 1/2 hr

Time taken by Kate to reach house to school = 9/15 = 3/5 hr

Total time of journey = (1/2 + 3/5) hr

Total time of journey = (5 + 6)/10 = 11/10 hr

Total distance covered = (9 + 9) km = 18 km

Therefore, average speed for the whole journey = distance/speed = 18/(11/10) km/hr

= 18/1 × 10/11 = (18 × 10)/(1 × 11) km/hr

= 180/11 km/hr

= 16.3 km/hr (approximately).

Problems on Calculating Distance

Here we will learn to solve different types of problems on calculating distance.

We know, the formula to find out distance = speed × time

Word problems on calculating distance:

1. A train moves at a speed of 80 km/hr. How far will it travel in 36 minutes?

Solution:

Using the unitary method;

In 60 minutes, distance covered = 80 km

In 1 minute, distance covered = 80/60 km

Speed = 80 km/hr

Time = 36 minutes or 36/60 hours

We know, formula of distance = speed × time

= 80 × 36/60

= 48 km

Therefore, the train will travel 48 km in 36 minutes.

2. A student goes to school at the rate of 7 ½ km/hr and reaches 10 minutes late. If he travels at the speed of 10 km/hr he is 10 minutes early. What is the distance to the school?

Solution:

Let the distance to school be 1 km, then time taken to cover 1 km at the rate of 7 1/2 km/hr

= Distance /speed = 1/(15/2) = 2/15 hr = 2/15 × 60 minutes = 8 minutes

Time taken to cover 1 km at the rate of 10 km/hr

= Distance /speed = 1/10 hr = 1/10 × 60 minutes = 6 minutes

Therefore, difference in time taken = (8 – 6) minutes = 2 minutes

But actual difference in time is 20 minutes

When the difference in time is 2 minutes, distance to school = 1 km

When the difference in time is 1 minute, distance to school = 1/2 km

When the difference in time is 20 minutes, distance to school = 1/2 × 20 km

Therefore, the distance to school is 10 km.

3. Two persons jog and covers the same distance at the speed of 6 km/hr and 4 km/hr. Find the distance covered by each one of them when one takes 10 minutes longer than the other.

Solution:

Let the required distance be x km

Time taken to cover x km at 6 km/hr = x/6 hr

Time taken to cover x km at 4 km/hr = x/4 hr

According to the question, x/4 – x/6 = 10/60

⇒ x/4 – x/6 = 1/6

⇒ 3x – 2x /12 = 1/6

⇒ x/12 = 1/6

⇒ x = 12/6

Therefore, the required distance is 2 km.

Problems on Calculating Time

Here we will learn to solve different types of problems on calculating time.

We know, the formula to find out time = distance/speed

Word problems on calculating time:

1. A car travel 60 km in 30 minutes. In how much time will it cover 100 km?

Solution:

Using the unitary method;

Time taken to cover 60 km = 90 minutes

Time taken to cover 1 km = 90/60 minutes

Time taken to cover 100 km = 90/60 × 100 = 150 minutes

Formula of speed = distance/time

= 60 km/(3/2) hr [given 1 hour 30 min = 1 30/60 = 1 ½ hours = 3/2 hours]

= 60/1 × 2/3 km/hr = 40 km/hr

Now, using the formula of time = distance/speed = 100 km/40 km/hr = 5/2 hours

= 5/2 × 60 minutes, (Since 1 hour = 60 minutes)

= 150 minutes

2. Victor covers 210 km by car at a speed of 70 km/hr. find the time taken to cover this distance.

Solution:

Using the unitary method;

70 km is covered in 1 hour.

1 km is covered in 1/70 hours.

210 km is covered in 1/70 × 210 hours = 3 hours

Given: speed = 70 km/hr, distance covered = 210 km

Time taken = Distance/ Speed = 210/70 hours = 3 hours

3. A train covers a distance of 36 km in 15 minutes. Find the time taken by it to cover the same distance if its speed is decreased by 9 km/hr.

Solution:

Distance covered by train = 36 km

Time taken = 15 minutes = 15/60 hr = ¼ hr.

Therefore speed of train = Distance covered/time taken

= 36/(1/4) km/hr

= 36/1 × 4/1

= 144 km/hr

Reduced speed = 144 – 9 = 135

Therefore, required time = distance covered/speed

= 36/135 × 60 minutes

= 16 minutes

4. A man is walking at a speed of 6 km per hour. After every km, he takes rest for 2 minutes. How much time will it take to cover a distance of 4 km?

Solution:

Rest time = Number of rests × time of each rest

= 3 × 2 minutes

= 6 minutes

Total time to cover 4 km = 4/6 × 60 + 6 minutes

= (40 + 6) minutes.

Two Objects Move in Same Direction

If two objects move in same direction at different speeds

If speed of 1st object = x km/hr and

Speed of 2nd object = y km/hr

Therefore, their relative speed = (x – y) km/hr [x > y], then

Time after which the two objects meet = distance / relative speed = d km/ (x – y) km/hr

We know, speed of one object with respect to another is called relative speed.

If time after which they meet is given, i.e., time = t hrs.

Then, distance covered in ‘t’ hours = time × relative speed = t hours × (x – y) km/hr

Now we will learn to calculate when two object move in same direction at different speeds.

Solved examples:

Two athletes are running from the same place at the speed of 6 km/hr and 4 km/hr. find the distance between them after 10 minutes if they move in the same direction.

Solution:

When they move in same direction,

Their relative speed = (6 – 4) km/hr = 2 km/hr

Time taken = 10 minutes

Distance covered = speed × time

= (2 × 10/60) km

= 1/3 km

= 1/3 × 1000 m

= 333.3 m

Two Objects Move in Opposite Direction

If the two objects move in opposite direction at different speeds
If the speed of 1st object = x km/hr and
Speed of 2nd object = y km/hr
Therefore, their relative speed = (x + y) km/hr
We know, speed of one object with respect to another is called relative speed.

If distance between them = d km, then
Time after which the two objects meet = d km/(x + y) km/hr
If time after which they meet is given i.e., time = t hours
Then distance covered in ‘t’ hours = time × relative speed = t hours × (x + y) km/hr
Now we will learn to calculate when two object move in opposite direction at different speeds.
Solved examples:
Two athletes are running from the same place at the speed of 6 km/hr and 4 km/hr. find the distance between them after 10 minutes if they move in the opposite direction.
Solution:
When they move in opposite direction,
Their relative speed = (4 + 6) km/hr = 40 km/hr
Time taken = 10 minutes
Distance covered = speed × time
                        = 10 × 10/60 km

                        = 5/3 km
                        = 5/3 × 1000 m
                        = 5000/3 m
                        = 1666.6 m

Train Passes a Moving Object in the Same Direction

When train passes a moving object in the same direction
Let length of train = l meters and speed of train = x km/hr

Speed of the object = y km/hr and relative speed = (x – y) km/hr
Then, time taken by the train to pass the moving object = l meters/(x – y) km/hr
Now we will learn to calculate when train passes a moving object in the same direction.
Solved example:
A train 165 m long is running at the speed of 60 km/hr. In what time will it pass a man who is running at the speed of 6 km/hr in the same direction in which the train is moving?
Solution:
Man moving in the same direction of the train
Speed of train relative to the man = (60 – 6) km/hr

                                              = 54 km/hr
                                              = (54 × 5/18) m/sec
                                              = 15 m/sec
Time taken by the train to cross a man = distance/speed
                                                     = length of train/speed of train relative to man
                                                     = 165 m/15 m/sec
                                                     = 11 sec

Train Passes a Moving Object in the Opposite Direction

When train passes a moving object in the opposite direction

Let length of train = l meters and speed of train = x km/hr

Speed of the object = y km/hr and relative speed = (x + y) km/hr

Then, time taken by the train to pass the moving object = l meters/(x + y) km/hr

Now we will learn to calculate when train passes a moving object in the opposite direction.

Solved example:

A train 165 m long is running at the speed of 60 km/hr. In what time will it pass a man who is running at the speed of 6 km/hr in the opposite direction in which the train is moving?

Solution:

Man moving in the opposite direction in which the train is moving.

Speed of train relative to the man = (60 + 6) km/hr

= 66 km/hr

= 66 × 5/18 m/sec

= 55/3 m/sec

Time taken by the train to cross a man = length of train/speed of train relative to man

= 165 m/(55/3) m/sec

= (165 × 3/55) sec

= 9 sec

Train Passes through a Pole

When the train passes through a pole or tower or light-post or tree or a stationary object

If length of train = x meters and speed of the train = y km/hr, then time taken by the train to pass the stationary object = length of the train/speed of the train = x meters/y km/hr.

Note: Change km/hr into m/sec.

Solved examples to calculate when the train passes through a pole or a stationary object.

1. A train 150 m long is running at a uniform speed of 54 km/hr. How much time will it take to cross a pole?

Solution:

Speed of train = 54 km/hr

= 54 × 5/18 m/sec

= 15 m/sec

Length of the train = 150 meters

Therefore, time taken by the train to cross a pole = length of train/speed of train

= 150/15 sec

= 10 sec

Thus, train takes 10 seconds to cross the pole.

2. Find the time taken by a train 300 m long, running at a speed of 54 km/hr in crossing the pole.

Solution:

Length of the train = 300 m

Speed of the train = 54 km/hr

= 54 × 5/18 m/sec

= 15 m/sec

Therefore, time taken by the train to cross the pole = 300 m/15 m/sec

= 20 seconds.

3. A train is running at a speed of 45 km/hr. It crosses a tower in 8 seconds. Find the length of the train.

Solution:

Speed of train = 45 km/hr

= 45 × 5/18 m/sec

= 25/2 m/sec

Time takes to cross the tower = 8 seconds

Length of the train = speed × time

= 25/2 × 8 m

= 100 m

4. A train is running at a speed of 126 km/hr. if it crosses a pole in just 7 second, what is the length of the train?

Solution:

Speed of the train = 126 km/hr

Speed of the train = 126 × 5/18 m/sec = 35 m/sec

Time taken by the train to cross the pole = 7 seconds

Therefore, length of the train = 35 m/sec × 7 sec = 245 m

Train Passes through a Bridge

When the train passes through a bridge or platform or tunnel or a stationary object having some length

If length of train = x meters and length of the stationery object = y meters.

Also, speed of the train is z km/hr, then time taken by the train to pass the stationary object having length y meters.

= (length of the train + length of stationary object)/speed of the train

= (x meters + y meters)/z km/hr

Note: Change km/hr to m/sec.

Solved examples to calculate when the train passes through a bridge or a stationary object having some length.

1. A train 175 m long crosses a bridge which is 125 m long in 80 seconds. What is the speed of the train?

Solution:

Length of the train = 175 m.

Length of the bridge = 225 m

Distance covered by the train to cross the bridge = (175 + 225) m

= 400 m

Time taken by the train to cross the bridge = 80 seconds

Speed = distance/time

= 400/80 m/sec

= 5 m/sec.

2. A train 220 m long is running at a speed of 36 km/hr. What time will it take to cross a 110 m long tunnel?

Solution:

Length of the train = 220 m

Length of the tunnel = 110 m

Therefore, length of the train + length of the tunnel = (220 + 110) m = 330m

Speed of the train = 36 km/hr

Speed of the train = 36 × 5/18 m/sec = 10 m/sec

Therefore, time taken by the train to cross the tunnel = 330 m/10 m/sec.

= 33 seconds.

3. Find the time taken by 150 m long train passes through a bridge which is 100 m long, running at a speed of 72 km/hr.

Solution:

Speed of train = 72 km/hr = 72 × 5/18 m/sec = 20 m/sec

In order to cross a bridge of length 100 m, the train will have to cover a distance = (150 + 100) m = 250 m

Thus, speed = 20 m/sec and distance = 250 m

Time = distance/speed

= 250m/20 m/sec

= 25/2 sec

= 12.5 sec.

4. A 90 m long train is running at a speed of 54 km/hr. If it takes 30 seconds to cross a platform, find the length of the platform.

Solution:

Speed of the train = 54 km/hr = 54 × 5/18 m/sec = 15 m/sec

Time taken to cross the bridge = 30 sec

Distance covered by train to cross the platform = speed × time

= (15 × 30) m

= 450 m

To cross the platform, train covers a distance = length of train + length of platform

450 m = 90 m + length of platform

Therefore, length of platform = (450 – 90) m = 360 m

Two Trains Passes in the Same Direction

The concept of two trains passes in the same direction.

When two train passes a moving object (having some length) in the same direction.

Let length of faster train be l meters and length of slower train be m meters

Speed of faster train be x km/hr and speed of slower train be y km/hr

Relative speed = (x – y) km/hr

Then, time taken by the faster train to pass the slower train = (l + m) meters/(x – y) km/hr

Note: Change km/hr to m/sec

Now we will learn to calculate when two trains running on parallel tracks (having some length) in the same direction.

Solved examples when two trains passes (having some length) in the same direction:

1. Two trains 130 m and 140 m long are running on parallel tracks in the same direction with a speed of 68 km/hr and 50 km/hr. How long will it take to clear off each other from the moment they meet?

Solution:

Relative speed of trains = (68 – 50) km/hr

= 18 km/hr

= 18 × 5/18 m/sec

= 5 m/sec

Time taken by the train to clear off each other = sum of length of trains/relative speed of trains

= (130 + 140)/5 sec

= 270/5 sec

= 54 sec

2. The two trains are running on parallel tracks in the same direction at 70 km/hr and 50 km/hr respectively. The faster train passes a man 27 second faster than the slower train. Find the length of the faster train.

Solution:

Relative speed of the trains = (70 – 50) km/hr

= 20 km/hr

= 20 × 5/18 m/sec

= 50/9 m/sec

Length of the faster train = relative speed × time taken by train to pass

= 50/9 × 27 m

` = 150 m

Two Trains Passes in the Opposite Direction

The concept of two trains passes in the opposite direction.

When two train passes a moving object (having some length) in the opposite direction

Let length of faster train be l meters and length of slower train be m meters

Let the speed of faster train be x km/hr

Relative speed = (x + y) km/hr.

Then, time taken by the faster train to pass the slower train = (l + m) meters/(x + y) km/hr

Now we will learn to calculate when two trains running on parallel tracks (having some length) in the opposite direction.

Solved examples when two trains passes (having some length) in the opposite direction:

1. Two trains of length 150 m and 170 m respectively are running at the speed of 40 km/hr and 32 km/hr on parallel tracks in opposite directions. In what time will they cross each other?

Solution:

Relative speed of train = (40 + 32) km/hr

= 72 km/hr

= 72 × 5/18 m/sec

= 20 m/sec

Time taken by the two trains to cross each other = sum of length of trains/relative speed of trains

= (150 + 170)/20 sec

= 320/20 sec

= 16 sec

Therefore, the two trains crossed each other in 16 seconds.

2. Two trains 163 m and 187 m long are running on parallel tracks in the opposite directions with a speed of 47 km/hr and 43 km/hr in. How long will it take to cross each other?

Solution:

Relative speed of train = (47 + 43) km/hr

= 90 km/hr

= 90 × 5/18 m/sec

= 25 m/sec

Time taken by the two trains to cross each other = sum of length of trains/relative speed of trains

= (163 + 187)/25 sec

= 350/25 sec

= 14 sec

Therefore, the two trains crossed each other in 14 seconds.

Problems on Trains (Time, Distance and Speed)

CASE A. When a train passes a stationary object:

CASE B. When a train passes a stationary object having some length:

Problems on Calculating Distance

Problems on Calculating Time

Two Objects Move in Same Direction

Two Objects Move in Opposite Direction

Train Passes a Moving Object in the Same Direction

Train Passes a Moving Object in the Opposite Direction

Train Passes through a Pole

Train Passes through a Bridge

Two Trains Passes in the Same Direction

Two Trains Passes in the Opposite Direction

Practice Set

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