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Speed has no sense of direction unlike the velocity. Relative speed is the speed of one object as observed from another moving object. Questions on train are the classic examples of relative speed and in all these questions it is assumed that trains move parallel to each other – whether in the same direction or the opposite direction. Thus, we shall see how the relative speed is calculated and using it we come to know the time taken by the trains to cross each other and some other like aspects.
Important Formulas – Problems on Trains
- x km/hr = (x×5)/18 m/s
- y m/s = (y×18)/5 km/hr
- Speed = distance/time, that is, s = d/t
- velocity = displacement/time, that is, v = d/t
- Time taken by a train x meters long to pass a pole or standing man or a post
= Time taken by the train to travel x meters.
- Time taken by a train x meters long to pass an object of length y meters
= Time taken by the train to travel (x + y) metres.
- Suppose two trains or two objects are moving in the same direction at v1 m/s and v2 m/s where v1 > v2,
then their relative speed = (v1 – v2) m/s
- Suppose two trains or two objects are moving in opposite directions at v1 m/s and v2 m/s ,
then their relative speed = (v1+ v2) m/s
- Assume two trains of length x metres and y metres are moving in opposite directions at v1 m/s and v2 m/s, Then
The time taken by the trains to cross each other = (x+y) / (v1+v2) seconds
- Assume two trains of length x metres and y metres are moving in the same direction at at v1 m/s and v2 m/s where v1 > v2, Then
The time taken by the faster train to cross the slower train = (x+y) / (v1-v2) seconds
- Assume that two trains (objects) start from two points P and Q towards each other at the same time and after crossing they take p and q seconds to reach Q and P respectively. Then,
A’s speed: B’s speed = √q: √p
Solved Examples
Level 1
| 1.A train is running at a speed of 40 km/hr and it crosses a post in 18 seconds. What is the length of the train? | |
| A. 190 metres | B. 160 metres |
| C. 200 metres
Answer : Option C |
D. 120 metres |
Explanation :
Speed of the train, v = 40 km/hr = 40000/3600 m/s = 400/36 m/s
Time taken to cross, t = 18 s
Distance Covered, d = vt = (400/36)× 18 = 200 m
Distance covered is equal to the length of the train = 200 m
| 2.A train having a length of 240 m passes a post in 24 seconds. How long will it take to pass a platform having a length of 650 m? | |
| A. 120 sec | B. 99 s |
| C. 89 s | D. 80 s |
Answer : Option C
Explanation :
v = 240/24 (where v is the speed of the train) = 10 m/s
t = (240+650)/10 = 89 seconds
| 3.Two trains having length of 140 m and 160 m long run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions (on parallel tracks). The time which they take to cross each other, is | |
| A. 10.8 s | B. 12 s |
| C. 9.8 s | D. 8 s |
Answer : Option A
Explanation :
Distance = 140+160 = 300 m
Relative speed = 60+40 = 100 km/hr = (100×10)/36 m/s
Time = distance/speed = 300 / (100×10)/36 = 300×36 / 1000 = 3×36/10 = 10.8 s
| 4.A train moves past a post and a platform 264 m long in 8 seconds and 20 seconds respectively. What is the speed of the train? | |
| A. 79.2 km/hr | B. 69 km/hr |
| C. 74 km/hr | D. 61 km/hr |
Answer : Option A
Explanation :
Let x is the length of the train and v is the speed
Time taken to move the post = 8 s
=> x/v = 8
=> x = 8v — (1)
Time taken to cross the platform 264 m long = 20 s
(x+264)/v = 20
=> x + 264 = 20v —(2)
Substituting equation 1 in equation 2, we get
8v +264 = 20v
=> v = 264/12 = 22 m/s
= 22×36/10 km/hr = 79.2 km/hr
| 5.Two trains, one from P to Q and the other from Q to P, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. The ratio of their speeds is | |
| A. 2 : 3 | B. 2 :1 |
| C. 4 : 3 | D. 3 : 2 |
Answer : Option C
Explanation :
Ratio of their speeds = Speed of first train : Speed of second train
= √16: √ 9
= 4:3
| 6.Train having a length of 270 meter is running at the speed of 120 kmph . It crosses another train running in opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train? | |
| A. 320 m | B. 190 m |
| C. 210 m | D. 230 m |
Answer : Option D
Explanation :
Relative speed = 120+80 = 200 kmph = 200×10/36 m/s = 500/9 m/s
time = 9s
Total distance covered = 270 + x where x is the length of other train
(270+x)/9 = 500/9
=> 270+x = 500
=> x = 500-270 = 230 meter
| 7.Two stations P and Q are 110 km apart on a straight track. One train starts from P at 7 a.m. and travels towards Q at 20 kmph. Another train starts from Q at 8 a.m. and travels towards P at a speed of 25 kmph. At what time will they meet? | |
| A. 10.30 a.m | B. 10 a.m. |
| C. 9.10 a.m. | D. 11 a.m. |
Answer : Option B
Explanation :
Assume both trains meet after x hours after 7 am
Distance covered by train starting from P in x hours = 20x km
Distance covered by train starting from Q in (x-1) hours = 25(x-1)
Total distance = 110
=> 20x + 25(x-1) = 110
=> 45x = 135
=> x= 3 Means, they meet after 3 hours after 7 am, ie, they meet at 10 am
| 8.Two trains are running in opposite directions in the same speed. The length of each train is 120 meter. If they cross each other in 12 seconds, the speed of each train (in km/hr) is | |
| A. 42 | B. 36 |
| C. 28 | D. 20 |
Answer : Option B
Explanation :
Distance covered = 120+120 = 240 m
Time = 12 s
Let the speed of each train = v. Then relative speed = v+v = 2v
2v = distance/time = 240/12 = 20 m/s
Speed of each train = v = 20/2 = 10 m/s
= 10×36/10 km/hr = 36 km/hr
Level 2
| 1.A train, 130 meters long travels at a speed of 45 km/hr crosses a bridge in 30 seconds. The length of the bridge is | |
| A. 270 m | B. 245 m |
| C. 235 m | D. 220 m |
Answer : Option B
Explanation :
Assume the length of the bridge = x meter
Total distance covered = 130+x meter
total time taken = 30s
speed = Total distance covered /total time taken = (130+x)/30 m/s
=> 45 × (10/36) = (130+x)/30
=> 45 × 10 × 30 /36 = 130+x
=> 45 × 10 × 10 / 12 = 130+x
=> 15 × 10 × 10 / 4 = 130+x
=> 15 × 25 = 130+x = 375
=> x = 375-130 =245
| 2.A train has a length of 150 meters. It is passing a man who is moving at 2 km/hr in the same direction of the train, in 3 seconds. Find out the speed of the train. | |
| A. 182 km/hr | B. 180 km/hr |
| C. 152 km/hr | D. 169 km/hr |
Answer : Option A
Explanation :
Length of the train, l = 150m
Speed of the man, Vm= 2 km/hr
Relative speed, Vr = total distance/time = (150/3) m/s = (150/3) × (18/5) = 180 km/hr
Relative Speed = Speed of train, Vt – Speed of man (As both are moving in the same direction)
=> 180 = Vt – 2 => Vt = 180 + 2 = 182 km/hr
| 3.Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively. If they cross each other in 23 seconds, what is the ratio of their speeds? | |
| A. Insufficient data | B. 3 : 1 |
| C. 1 : 3 | D. 3 : 2 |
Answer : Option D
Explanation :
Let the speed of the trains be x and y respectively
length of train1 = 27x
length of train2 = 17y
Relative speed= x+ y
Time taken to cross each other = 23 s
=> (27x + 17 y)/(x+y) = 23 => (27x + 17 y)/ = 23(x+y)
=> 4x = 6y => x/y = 6/4 = 3/2
| 4.A jogger is running at 9 kmph alongside a railway track in 240 meters ahead of the engine of a 120 meters long train . The train is running at 45 kmph in the same direction. How much time does it take for the train to pass the jogger? | |
| A. 46 | B. 36 |
| C. 18 | D. 22 |
Answer : Option B
Explanation :
Distance to be covered = 240+ 120 = 360 m
Relative speed = 36 km/hr = 36×10/36 = 10 m/s
Time = distance/speed = 360/10 = 36 seconds
| 5.A train passes a platform in 36 seconds. The same train passes a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, The length of the platform is | |
| A. None of these | B. 280 meter |
| C. 240 meter | D. 200 meter |
Answer : Option C
Explanation :
Speed of the train = 54 km/hr = (54×10)/36 m/s = 15 m/s
Length of the train = speed × time taken to cross the man = 15×20 = 300 m
Let the length of the platform = L
Time taken to cross the platform = (300+L)/15
=> (300+L)/15 = 36
=> 300+L = 15×36 = 540 => L = 540-300 = 240 meter
| 6.A train overtakes two persons who are walking in the same direction to that of the train at 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. What is the length of the train? | |
| A. 62 m | B. 54 m |
| C. 50 m | D. 55 m |
Answer : Option C
Explanation :
Let x is the length of the train in meter and v is its speed in kmph
x/9 = (v-2) (10/36) — (1)
x/10 = (v-4) (10/36) — (2)
Dividing equation 1 with equation 2
10/9 = (v-2)/(v-4) => 10v – 40 = 9v – 18 => v = 22
Substituting in equation 1, x/9 = 200/36 => x = 9×200/36 = 50 m
| 7.A train is traveling at 48 kmph. It crosses another train having half of its length, traveling in opposite direction at 42 kmph, in 12 seconds. It also passes a railway platform in 45 seconds. What is the length of the platform? | |
| A. 500 m | B. 360 m |
| C. 480 m | D. 400 m |
Answer : Option D
Explanation :
Speed of train1 = 48 kmph
Let the length of train1 = 2x meter
Speed of train2 = 42 kmph
Length of train 2 = x meter (because it is half of train1’s length)
Distance = 2x + x = 3x
Relative speed= 48+42 = 90 kmph = 90×10/36 m/s = 25 m/s
Time = 12 s
Distance/time = speed => 3x/12 = 25
=> x = 25×12/3 = 100 meter
Length of the first train = 2x = 200 meter
Time taken to cross the platform= 45 s
Speed of train1 = 48 kmph = 480/36 = 40/3 m/s
Distance = 200 + y where y is the length of the platform
=> 200 + y = 45×40/3 = 600
=> y = 400 meter
| 8.A train, 800 meter long is running with a speed of 78 km/hr. It crosses a tunnel in 1 minute. What is the length of the tunnel (in meters)? | |
| A. 440 m | B. 500 m |
| C. 260 m | D. 430 m |
Answer : Option B
Explanation :
Distance = 800+x meter where x is the length of the tunnel
Time = 1 minute = 60 seconds
Speed = 78 km/hr = 78×10/36 m/s = 130/6 = 65/3 m/s
Distance/time = speed
(800+x)/60 = 65/3 => 800+x = 20×65 = 1300
=> x = 1300 – 800 = 500 meter
| 9.Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. If the fast train completely passes a man sitting in the slower train in 5 seconds, the length of the fast train is : | |
| A. 19 m | B. 2779 m |
| C. 1329 m | D. 33 m |
Answer : Option B
Explanation :
Relative speed = 40-20 = 20 km/hr = 200/36 m/s = 100/18 m/s
Time = 5 s
Distance = speed × time = (100/18) × 5 = 500/18 m = 250/9 = 2779 m = length of the fast train,
Relative speed is the speed of an object as measured from a particular frame of reference. It is calculated by subtracting the speed of the observer from the speed of the object. For example, if a train is moving at 50 miles per hour and a car is moving at 30 miles per hour, the relative speed of the train to the car is 20 miles per hour.
Train questions are a type of question that is asked on standardized tests such as the SAT and ACT. They are designed to test your understanding of relative speed. Train questions typically involve two trains that are moving in opposite directions or in the same direction. You are asked to calculate the relative speed of the trains or the time it will take for the trains to meet or pass each other.
To solve train questions, you need to understand the concept of relative speed. Relative speed is the speed of an object as measured from a particular frame of reference. It is calculated by subtracting the speed of the observer from the speed of the object. For example, if a train is moving at 50 miles per hour and a car is moving at 30 miles per hour, the relative speed of the train to the car is 20 miles per hour.
Once you understand the concept of relative speed, you can solve train questions by following these steps:
- Identify the two trains and their speeds.
- Determine the direction of the trains.
- Calculate the relative speed of the trains.
- Use the relative speed to answer the question.
Here are some examples of train questions:
- Two trains are moving in opposite directions. The first train is moving at 50 miles per hour and the second train is moving at 30 miles per hour. What is the relative speed of the trains?
Solution: The relative speed of the trains is 50 + 30 = 80 miles per hour.
- Two trains are moving in the same direction. The first train is moving at 50 miles per hour and the second train is moving at 30 miles per hour. What is the relative speed of the trains?
Solution: The relative speed of the trains is 50 – 30 = 20 miles per hour.
- A train is moving at 50 miles per hour. A car is moving at 30 miles per hour in the opposite direction. What is the relative speed of the train to the car?
Solution: The relative speed of the train to the car is 50 + 30 = 80 miles per hour.
- A train is moving at 50 miles per hour. A car is moving at 30 miles per hour in the same direction. What is the relative speed of the train to the car?
Solution: The relative speed of the train to the car is 50 – 30 = 20 miles per hour.
- A train is moving at 50 miles per hour. A car is moving at 30 miles per hour. The train is ahead of the car. How long will it take for the train to pass the car?
Solution: The relative speed of the train to the car is 50 – 30 = 20 miles per hour. The distance between the train and the car is unknown. To calculate the time it will take for the train to pass the car, we need to know the distance between the train and the car.
If the distance between the train and the car is 100 miles, it will take the train 5 hours to pass the car.
If the distance between the train and the car is 200 miles, it will take the train 10 hours to pass the car.
And so on.
Train questions can be challenging, but they are not impossible to solve. By understanding the concept of relative speed, you can solve any train question that comes your way.
What is the difference between speed and velocity?
Speed is a scalar quantity, meaning it has only magnitude. Velocity is a vector quantity, meaning it has both magnitude and direction.
What is the formula for calculating speed?
Speed is equal to the distance traveled divided by the time it takes to travel that distance.
What is the formula for calculating velocity?
Velocity is equal to the change in position divided by the time it takes to make that change.
What is the difference between relative speed and absolute speed?
Relative speed is the speed of one object relative to another object. Absolute speed is the speed of an object relative to a stationary reference point.
What is the formula for calculating relative speed?
The formula for calculating relative speed is:
$v_{rel} = v_1 + v_2$
where $v_{rel}$ is the relative speed, $v_1$ is the speed of the first object, and $v_2$ is the speed of the second object.
What is the formula for calculating absolute speed?
The formula for calculating absolute speed is:
$v_a = \sqrt{v_1^2 + v_2^2}$
where $v_a$ is the absolute speed, $v_1$ is the speed of the first object, and $v_2$ is the speed of the second object.
What is the difference between Average speed and instantaneous speed?
Average speed is the total distance traveled divided by the total time it takes to travel that distance. Instantaneous speed is the speed of an object at a particular instant in time.
What is the formula for calculating average speed?
The formula for calculating average speed is:
$v_{avg} = \frac{d}{t}$
where $v_{avg}$ is the average speed, $d$ is the total distance traveled, and $t$ is the total time it takes to travel that distance.
What is the formula for calculating instantaneous speed?
The formula for calculating instantaneous speed is:
$v_{inst} = \lim_{t \to 0} \frac{d}{t}$
where $v_{inst}$ is the instantaneous speed, $d$ is the distance traveled in a very small amount of time, and $t$ is that very small amount of time.
What is the difference between constant speed and variable speed?
An object is moving at a constant speed if its speed does not change over time. An object is moving at a variable speed if its speed changes over time.
What is the difference between uniform motion and non-uniform motion?
Uniform motion is motion in which the object’s speed and direction do not change. Non-uniform motion is motion in which the object’s speed or direction changes.
What is the difference between linear motion and circular motion?
Linear motion is motion in a straight line. Circular motion is motion in a circle.
What is the difference between projectile motion and free fall?
Projectile motion is the motion of an object that is thrown or projected into the air and then moves under the influence of gravity. Free fall is the motion of an object that is falling under the influence of gravity only.
What is the difference between rolling motion and sliding motion?
Rolling motion is a type of motion in which an object rolls along a surface without slipping. Sliding motion is a type of motion in which an object slides along a surface without rolling.
What is the difference between rotational motion and translational motion?
Rotational motion is motion around an axis. Translational motion is motion in a straight line.
What is the difference between angular speed and linear speed?
Angular speed is the rate at which an object rotates. Linear speed is the rate at which an object moves in a straight line.
What is the difference between centripetal force and centrifugal force?
Centripetal force is the force that keeps an object moving in a circular path. Centrifugal force is the apparent force that pushes an object away from the center of a circular path.
What is the difference between inertia and momentum?
Inertia is the tendency of an object to resist changes in its motion. Momentum is the product of an object’s mass and velocity.
What is the difference between kinetic energy and potential energy?
Kinetic energy is the energy of motion. Potential energy is the energy stored in an object due to its position or state.
What is the difference between work and power?
Work is the transfer of energy to an object by
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A train leaves Chicago at 10:00 am and travels at a speed of 60 mph. Another train leaves Indianapolis at 11:00 am and travels at a speed of 70 mph. If the distance between Chicago and Indianapolis is 200 miles, at what time will the second train catch the first train?
(A) 1:00 pm
(B) 1:30 pm
(C) 2:00 pm
(D) 2:30 pm
(E) 3:00 pm -
A car travels at a speed of 60 mph for 2 hours. How far does it travel?
(A) 120 miles
(B) 180 miles
(C) 240 miles
(D) 300 miles
(E) 360 miles -
A train travels at a speed of 60 mph for 2 hours and then travels at a speed of 70 mph for 2 hours. How far does it travel?
(A) 240 miles
(B) 300 miles
(C) 360 miles
(D) 420 miles
(E) 480 miles -
A car travels at a speed of 60 mph for 2 hours and then travels at a speed of 70 mph for 1 hour. How far does it travel?
(A) 240 miles
(B) 300 miles
(C) 360 miles
(D) 420 miles
(E) 480 miles -
A train leaves Chicago at 10:00 am and travels at a speed of 60 mph. Another train leaves Indianapolis at 11:00 am and travels at a speed of 70 mph. If the distance between Chicago and Indianapolis is 200 miles, at what time will the second train be 100 miles ahead of the first train?
(A) 1:00 pm
(B) 1:30 pm
(C) 2:00 pm
(D) 2:30 pm
(E) 3:00 pm