## Speed, Time and Distance

#### Speed, Time and Distance

1. A train overtakes two person walking at 2 km per hr. and 4 km per hr. respectively and
passes completely them in 9 sec. and 10 sec. respectively. What is the length of the train?
1. 65 metres
2. 60 metres
3. 55 metres
4. 50 metres

1. Let the length of the train be x km and its speed y km per hr.
Case I : When it passes the man walking at 2 km per hr. in the same direction
Relative speed of train = (y – 2) km per hr.

 ∴ x = 9 seconds y − 2

 = 9 = 1 hour    ...(i) 3600 400

Case II : When the train crosses the man walking at 4 km per hr. in the same direction.
Relative speed of train= (y – 4) km per hr.
 ∴ x = 10 sec. y − 4

 ⇒ x = 10 hrs. y − 4 3600

 ⇒ x = 1 hrs.    ...(ii) y − 4 360

On dividing equation (i) by (ii),
we have
 y − 4 = 1/400 = 360 = 9 y − 2 1/360 400 10

⇒  10y – 40 = 9y – 18
⇒  10y – 9y = 40 – 18
⇒  y = 22 km per hr.
∴  From equaton (i), we have
⇒
 x = 1 22 − 2 400

 ⇒ x = 1 km 20

 = 1000 = 50 metres. 20

##### Correct Option: D

Let the length of the train be x km and its speed y km per hr.
Case I : When it passes the man walking at 2 km per hr. in the same direction
Relative speed of train = (y – 2) km per hr.

 ∴ x = 9 seconds y − 2

 = 9 = 1 hour    ...(i) 3600 400

Case II : When the train crosses the man walking at 4 km per hr. in the same direction.
Relative speed of train= (y – 4) km per hr.
 ∴ x = 10 sec. y − 4

 ⇒ x = 10 hrs. y − 4 3600

 ⇒ x = 1 hrs.    ...(ii) y − 4 360

On dividing equation (i) by (ii),
we have
 y − 4 = 1/400 = 360 = 9 y − 2 1/360 400 10

⇒  10y – 40 = 9y – 18
⇒  10y – 9y = 40 – 18
⇒  y = 22 km per hr.
∴  From equaton (i), we have
⇒
 x = 1 22 − 2 400

 ⇒ x = 1 km 20

 = 1000 = 50 metres. 20

1. A train travelling at the rate of 60 km per hr, while inside a tunnel, meets another train of half its length travelling at 90 km per hr.
 and passes completely in 4 1 seconds. 2
Find the length of the tunnel if the first train passes completely
 through it in 4 minutes 37 1 seconds. 2
1. 5 km
2. 3.5 km
3. 4.5 km
4. 6 km

1. Trains are running in opposite direction.
∴  Relative speed of the two trains
= 90 + 60 = 150 km per hr.

 Distance travelled in 4 1 seconds 2

with speed of 150 km per hr.
 = 150 × 5 m per sec. 18

 = 150 × 5 × 9 18 2

 = 375 metres 2

Let the length of the first train be x metres.
 Then the length of the second train be x metres 2

 ∴   x + x = 375 2 2

 ⇒ 3x = 375 2 2

⇒  3x = 375
⇒  x = 125 metres
Hence, the length of the first
train = 125 metres
Speed of the first train = 60 km per hr.
 = 60 × 5 m per sec. 18

 = 50 m per sec. 3

Time taken by the first train to cross the tunnel = 4 minutes
 and 37 1 sec. 2

 = 240 + 75 sec. = 480 + 75 2 2

 = 555 sec. 2

 Speed of first train = 50 m per sec. 3

 ∴  Distance covered by it in 555 sec. 2

 = 50 × 555 = 4625 metres 3 2

Hence, length of tunnel
= 4625 – 125 = 4500 metres
= 4.5 km

##### Correct Option: C

Trains are running in opposite direction.
∴  Relative speed of the two trains
= 90 + 60 = 150 km per hr.

 Distance travelled in 4 1 seconds 2

with speed of 150 km per hr.
 = 150 × 5 m per sec. 18

 = 150 × 5 × 9 18 2

 = 375 metres 2

Let the length of the first train be x metres.
 Then the length of the second train be x metres 2

 ∴   x + x = 375 2 2

 ⇒ 3x = 375 2 2

⇒  3x = 375
⇒  x = 125 metres
Hence, the length of the first
train = 125 metres
Speed of the first train = 60 km per hr.
 = 60 × 5 m per sec. 18

 = 50 m per sec. 3

Time taken by the first train to cross the tunnel = 4 minutes
 and 37 1 sec. 2

 = 240 + 75 sec. = 480 + 75 2 2

 = 555 sec. 2

 Speed of first train = 50 m per sec. 3

 ∴  Distance covered by it in 555 sec. 2

 = 50 × 555 = 4625 metres 3 2

Hence, length of tunnel
= 4625 – 125 = 4500 metres
= 4.5 km

1. Two trains 200 metres and 175 metres long are running on parallel lines.
 They take 7 1 seconds.when running in opposite directions 2
 and 37 1 seconds when running in the same direction 2
to pass each other. Find their speeds in km per hour.
1. 118 kmph ; 75 kmph
2. 108 kmph ; 72 kmph
3. 120 kmph ; 75 kmph
4. 125 kmph ; 80 kmph

1. Let the speed of the train be x metre per sec. and y metre per sec. respectively.
Sum of the length of the trains = 200 + 175 = 375 metres
Case : I
When the trains are moving in opposite directions
Relative speed = (x + y) m per sec.
In this case the time taken by the trains to cross each other

 = 375 sec. x + y

 ∴ 375 = 15 x + y 2

⇒  x + y = 50     ...(i)
Case : II
When the trains are moving in the same direction.
Relative speed = (x – y) m per sec.
In this case, the time taken by the trains to cross each other
 = 375 sec. x − y

 ∴ 375 = 75 x − y 2

⇒  x – y = 10     ...(ii)
Now, x + y = 50
 x – y = 10 _________ 2x = 60

⇒  x = 30
Putting this value in equation (i),
we have
y = 50 – 30 = 20
∴  Speed of trains = 30 m per sec.
 = 30 × 18 = 108 km per hr. 5

 and 20 m per sec. = 20 × 18 = 72 km per hr. 5

##### Correct Option: B

Let the speed of the train be x metre per sec. and y metre per sec. respectively.
Sum of the length of the trains = 200 + 175 = 375 metres
Case : I
When the trains are moving in opposite directions
Relative speed = (x + y) m per sec.
In this case the time taken by the trains to cross each other

 = 375 sec. x + y

 ∴ 375 = 15 x + y 2

⇒  x + y = 50     ...(i)
Case : II
When the trains are moving in the same direction.
Relative speed = (x – y) m per sec.
In this case, the time taken by the trains to cross each other
 = 375 sec. x − y

 ∴ 375 = 75 x − y 2

⇒  x – y = 10     ...(ii)
Now, x + y = 50
 x – y = 10 _________ 2x = 60

⇒  x = 30
Putting this value in equation (i),
we have
y = 50 – 30 = 20
∴  Speed of trains = 30 m per sec.
 = 30 × 18 = 108 km per hr. 5

 and 20 m per sec. = 20 × 18 = 72 km per hr. 5

1. A train running at 25 km per hour take 18 seconds to pass a platform. Next, it takes 12 seconds to pass a man walking at the rate of 5 km per hr. in the same direction. Find the length of the platform.
1. 25 metres
2. 20 metres
3. 24 metres
4. 28 metres

1. Let the length of train be x metres and the length of platform be y metres.

 Speed of the train = 25 × 5 m/sec 18

 = 125 m per sec. 18

Time taken by train to pass the platform
 = (x + y) × 18 sec. 125

 ∴  (x + y) × 18 = 18 125

or,  x + y = 125    ...(i)
Speed of train relative to man = (25 + 5) km per hr.
 = 30 × 5 m per sec. 18

 = 25 m per sec. 3

Time taken by the train to pass the man
 = x × 3 sec. 25

 = 3x sec. 25

 ∴ 3x = 12 25

 ⇒  x = 25 × 12 = 100 metres 3

Putting x = 100 in equation (i),
we get, y = 25 metres.
∴  Length of train = 100 metres
and length of the platform = 25 metres.

##### Correct Option: A

Let the length of train be x metres and the length of platform be y metres.

 Speed of the train = 25 × 5 m/sec 18

 = 125 m per sec. 18

Time taken by train to pass the platform
 = (x + y) × 18 sec. 125

 ∴  (x + y) × 18 = 18 125

or,  x + y = 125    ...(i)
Speed of train relative to man = (25 + 5) km per hr.
 = 30 × 5 m per sec. 18

 = 25 m per sec. 3

Time taken by the train to pass the man
 = x × 3 sec. 25

 = 3x sec. 25

 ∴ 3x = 12 25

 ⇒  x = 25 × 12 = 100 metres 3

Putting x = 100 in equation (i),
we get, y = 25 metres.
∴  Length of train = 100 metres
and length of the platform = 25 metres.

1. Two places A and B are 162 kms apart. A train leaves A for B and at the same time another train leaves B for A. The two trains meet at the end of 6 hours. If the train travelling from A to B travels 8 km per hr. faster than the other, find the speed of the faster train.
1. 16.5 kmph
2. 16 kmph
3. 17 kmph
4. 17.5 kmph

1. Both trains meet after 6 hours.
∴  The relative speed of two trains

 = 162 = 27 km per hr. 6

The speed of the slower train starting from B
 = 27 − 8 = 19 =9 1 km per hr. 2 2 2

∴  The speed of the faster train
 = 9 1 + 8 = 17 1 km per hr. 2 2

##### Correct Option: D

Both trains meet after 6 hours.
∴  The relative speed of two trains

 = 162 = 27 km per hr. 6

The speed of the slower train starting from B
 = 27 − 8 = 19 =9 1 km per hr. 2 2 2

∴  The speed of the faster train
 = 9 1 + 8 = 17 1 km per hr. 2 2