Problems On Trains All Competence Aptitude

Given:

Lengths of two trains = 108 m & 112 m 

Speeds of two trains = 45 km/h & 54 km/h 

Formulas used:

Relative speed (Opposite direction) = Speed1 + Speed2 

Time = Distance/Relative Speed 

Calculation:

Distance to be covered by two trains = Length of two trains 

⇒ 108 m + 112 m = 220 m 

Time = 220 m/(45 + 54) km/h 

⇒ 220/ 99 × 5/18 

⇒ 220/(99 × 5/18)

⇒ 220/99 × 18/5             [1 hour = 3600 seconds, 1 km = 1000 m] 

⇒ 8 

To solve this problem, we need to find the relative speed of the two trains. When two objects are moving towards each other, their relative speed is the sum of their individual speeds.

Given:
- Length of each train = 125 meters
- Speed of one train = 65 km/hour
- Time taken to cross each other = 6 seconds

First, let's convert the speed of the train from km/hour to m/s, as the length of the train is given in meters:

Speed of one train = 65 km/hour = (65 * 1000) / 3600 m/s
                                 = 18.0556 m/s

Now, let's find the relative speed of the two trains:

Relative speed = Speed of first train + Speed of second train

Given that they cross each other in 6 seconds, we can use the formula:

Relative speed = Total distance / Time

The total distance covered when the two trains cross each other is the sum of their lengths.

Total distance = Length of first train + Length of second train
                           = 125 meters + 125 meters
                           = 250 meters

Now, plug in the values to find the relative speed:

Relative speed = 250 meters / 6 seconds
                      ≈ 41.67 m/s

Now, since the relative speed is the sum of the speeds of the two trains, and we know the speed of one train, we can find the speed of the other train:

Relative speed = Speed of first train + Speed of second train

41.67 m/s = 18.0556 m/s + Speed of second train

Speed of second train = 41.67 m/s - 18.0556 m/s
                                 = 23.6144 m/s

Now, let's convert the speed of the second train from m/s to km/hour:

Speed of second train = (23.6144 * 3600) / 1000 km/hour
                                   ≈ 85.01 km/hour

So, the speed of the second train is approximately 85 km/hour. Therefore, the correct answer is:

85 km/hour

Let's solve this problem step by step.

Given:
- Speed of the bus = 45 km/hr
- Distance covered by the bus in 30 seconds = 150 meters (as it overtakes the truck)
- Time taken = 30 seconds

First, let's convert the speed of the bus from km/hr to m/s since the distance is given in meters and time is given in seconds:

\(45 \text{ km/hr} = \dfrac{45 \times 1000}{3600} \text{ m/s} = \dfrac{45000}{3600} \text{ m/s} ≈ 12.5 \text{ m/s}\)

Now, let's denote the speed of the truck as \(v\) m/s.

Since the bus overtakes the truck, the relative speed between the bus and the truck will be the difference in their speeds.

Relative speed = Speed of bus - Speed of truck

\((12.5 \text{ m/s} - v) = \dfrac{150 \text{ meters}}{30 \text{ seconds}} = 5 \text{ m/s}\)

Now, let's solve for \(v\):

\(12.5 \text{ m/s} - v = 5 \text{ m/s}\)

\(v = 12.5 \text{ m/s} - 5 \text{ m/s}\)

\(v = 7.5 \text{ m/s}\)

Now, let's convert this speed from m/s to km/hr:

\(v = 7.5 \text{ m/s} \times \dfrac{3600}{1000} \text{ km/hr}\)

\(v = 27 \text{ km/hr}\)

So, the speed of the truck is 27 km/hr.

Therefore, the correct answer is:

27 km/hr

Let's break down the problem step by step.

Given:
- Speed of the train = 30 m/s
- Length of the platform = 600 meters
- Time taken to cross the platform = 30 seconds

We need to find the length of the train.

When the train crosses the platform, it covers the length of the train plus the length of the platform.

So, the total distance covered by the train in crossing the platform is the sum of the length of the train and the length of the platform.

Let's denote the length of the train as \(L\) meters.

Total distance covered by the train = \(L\) (length of the train) + 600 (length of the platform)

Now, using the formula: Distance = Speed × Time, we can calculate the distance covered by the train.

So, \(L + 600 = 30 \text{ m/s} \times 30 \text{ seconds}\)

\(L + 600 = 900 \text{ meters}\)

Now, solve for \(L\):

\(L = 900 \text{ meters} - 600 \text{ meters}\)

\(L = 300 \text{ meters}\)

So, the length of the train is 300 meters.

Therefore, the correct answer is:

300 meters.

To solve this problem, we need to consider the relative speed of the two trains and the combined length of both trains.

Given:
- Length of the first train = 110 meters
- Speed of the first train = 60 km/hr
- Length of the second train = 170 meters

First, let's convert the speed of the first train from km/hr to m/s since the length of the trains is given in meters.

\(60 \text{ km/hr} = \dfrac{60 \times 1000}{3600} \text{ m/s} = \dfrac{60000}{3600} \text{ m/s} = 16.67 \text{ m/s}\)

Now, let's find the relative speed of the two trains. Since they are moving in opposite directions, we need to add their speeds.

Relative speed = Speed of the first train + Speed of the second train

Relative speed = \(16.67 \text{ m/s} + 0 \text{ m/s}\) (the second train is standing still on the parallel track)

Relative speed = \(16.67 \text{ m/s}\)

Now, when two objects of length \(L_1\) and \(L_2\) are moving in opposite directions, they cover the sum of their lengths during the time they cross each other.

Total distance to be covered = Length of the first train + Length of the second train

Total distance = \(110 \text{ meters} + 170 \text{ meters} = 280 \text{ meters}\)

Now, we can use the formula: Time = Distance / Speed to find the time taken.

\(Time = \dfrac{Total \ distance}{Relative \ speed} = \dfrac{280 \text{ meters}}{16.67 \text{ m/s}}\)

\(Time ≈ 16.8 \text{ seconds}\)

So, it takes approximately 16.8 seconds for the first train to cross the second train.

Therefore, the correct answer is:

16.8 seconds.

To find the speed of the train in meters per second, we first need to find the total distance the train travels when it passes both the platform and its own length.

Given:
- Length of the train = 50 meters
- Length of the platform = 100 meters
- Time taken to pass both = 10 seconds

The train passes both its own length and the platform's length, so the total distance covered is the sum of the lengths of the train and the platform.

Total distance = Length of train + Length of platform

Total distance = 50 meters + 100 meters = 150 meters

Now, we use the formula: Speed = Distance / Time to find the speed of the train.

Speed = Total distance / Time

Speed = 150 meters / 10 seconds

Speed = 15 meters per second

So, the speed of the train in meters per second is 15.

Therefore, the correct answer is: 15.

To solve this problem, let's denote the length of the train as \( L \) meters.

When the train passes a bridge, it covers the length of the bridge and its own length. 

So, the time taken by the train to pass each bridge is equal to the time taken to cover its own length plus the length of the bridge, divided by the speed of the train.

Let's set up equations for both bridges:

For the first bridge:
Time taken to pass the first bridge = \( 100 \) seconds
Length of the first bridge = \( 500 \) meters

Time taken to pass the first bridge = \( \dfrac{500 + L}{\text{speed}} \)

For the second bridge:
Time taken to pass the second bridge = \( 60 \) seconds
Length of the second bridge = \( 250 \) meters

Time taken to pass the second bridge = \( \dfrac{250 + L}{\text{speed}} \)

Since the speed of the train is the same for both bridges, we can set up an equation using the given times:

\[ \dfrac{500 + L}{\text{speed}} = 100 \]

\[ \dfrac{250 + L}{\text{speed}} = 60 \]

We can solve these equations to find the value of \( L \), the length of the train.

First, let's simplify the equations:

\[ 500 + L = 100 \times \text{speed} \]

\[ 250 + L = 60 \times \text{speed} \]

Now, let's subtract the second equation from the first:

\[ (500 + L) - (250 + L) = 100 \times \text{speed} - 60 \times \text{speed} \]

\[ 500 - 250 = 100 \times \text{speed} - 60 \times \text{speed} \]

\[ 250 = 40 \times \text{speed} \]

\[ \text{speed} = \dfrac{250}{40} = 6.25 \, \text{m/s} \]

Now that we have the speed of the train, we can substitute it into one of the equations to solve for \( L \):

\[ \dfrac{500 + L}{6.25} = 100 \]

\[ 500 + L = 6.25 \times 100 \]

\[ 500 + L = 625 \]

\[ L = 625 - 500 = 125 \]

So, the length of the train is \( \boxed{125} \) meters.

To find the speed of the train, we need to consider the time it takes to pass the platform and the lamp post.

Let's denote:
- Length of the platform = 50 meters
- Time taken to pass the platform = 14 seconds
- Time taken to pass the lamp post = 10 seconds

When the train passes the platform, it covers the length of the platform plus its own length. Similarly, when it passes the lamp post, it covers only its own length.

Let the length of the train be \( L \) meters.

So, the speed of the train can be calculated using the formula: Speed = Distance / Time.

When passing the platform:\[ Speed = \dfrac{50 + L}{14} \]

When passing the lamp post:\[ Speed = \dfrac{L}{10} \]

Since the speed of the train is the same in both cases, we can equate the two expressions for speed:

\[ \dfrac{50 + L}{14} = \dfrac{L}{10} \]

To solve for \( L \), first, let's clear the denominators by multiplying both sides of the equation by \( 14 \times 10 = 140 \):

\[ 10(50 + L) = 14L \]

\[ 500 + 10L = 14L \]

Now, let's isolate \( L \):

\[ 4L = 500 \]

\[ L = \dfrac{500}{4} \]

\[ L = 125 \]

So, the length of the train is 125 meters.

Now, let's find the speed of the train using either of the original equations:

\[ Speed = \dfrac{50 + 125}{14} \]

\[ Speed = \dfrac{175}{14} \]

\[ Speed \approx 12.5 \text{ m/s} \]

Now, let's convert the speed from m/s to km/hr:

\[ Speed = 12.5 \times \dfrac{3600}{1000} \]

\[ Speed = 45 \text{ km/hr} \]

So, the speed of the train is 45 km/hr.

Therefore, the correct answer is: 45.

To find the speed of the train, we'll use the formula:

\[ \text{Speed} = \dfrac{\text{Total distance}}{\text{Time taken}} \]

First, let's calculate the total distance the train covers, which includes the length of the train and the length of the platform.

Given:
- Length of the train = 500 meters
- Length of the platform = 221 meters
- Time taken to pass the station (platform) = 35 seconds

Total distance covered by the train = Length of train + Length of platform

\[ \text{Total distance} = 500 \, \text{m} + 221 \, \text{m} = 721 \, \text{m} \]

Now, we can use the formula for speed:

\[ \text{Speed} = \dfrac{721 \, \text{m}}{35 \, \text{s}} \]

Now, let's convert the speed from meters per second to kilometers per hour:

\[ \text{Speed} = \dfrac{721 \, \text{m}}{35 \, \text{s}} \times \dfrac{3600 \, \text{s}}{1000 \, \text{m}} \]

\[ \text{Speed} = \dfrac{721 \times 3600}{35 \times 1000} \, \text{km/hr} \]

\[ \text{Speed} = \dfrac{2595600}{35 \times 1000} \, \text{km/hr} \]

\[ \text{Speed} \approx 74.16 \, \text{km/hr} \]

So, the speed of the train is approximately 74.16 km/hr.

Therefore, the correct answer is:

74.16 km/hr.

To find the speed of the train, we'll first calculate the total distance it covers, then use the formula for speed.

Given:
- Length of the train = 150 meters
- Length of the platform = 450 meters
- Time taken to cross the platform = 20 seconds

Total distance covered by the train = Length of train + Length of platform

Total distance = 150 meters + 450 meters = 600 meters

Now, we'll use the formula for speed:

\[ \text{Speed} = \dfrac{\text{Total distance}}{\text{Time taken}} \]

\[ \text{Speed} = \dfrac{600 \, \text{m}}{20 \, \text{s}} \]

Now, let's convert the speed from meters per second to kilometers per hour:

\[ \text{Speed} = \dfrac{600 \, \text{m}}{20 \, \text{s}} \times \dfrac{3600 \, \text{s}}{1000 \, \text{m}} \]

\[ \text{Speed} = \dfrac{600 \times 3600}{20 \times 1000} \, \text{km/hr} \]

\[ \text{Speed} = \dfrac{2160000}{20000} \, \text{km/hr} \]

\[ \text{Speed} = 108 \, \text{km/hr} \]

So, the speed of the train is \( 108 \, \text{km/hr} \).

Therefore, the correct answer is:

\( 108 \, \text{km/hr} \).