Conversion: It includes conversion of kilometer per hour (kmph) into meter per second (mps) or vice -versa.
Distance formula: D = S x T where,

D ⇒ Distance S ⇒ Speed T ⇒ Time

Relativity: It is a broad term. It describes about objects moving in either direction or in the same direction, speed of the objects, and time taken by the objects, etc.

Example 1: A train 100 m long is running at the speed of 30 km/hr. find the time taken by in to pass a man standing near the railway line.
Solution:

Speed of the train = (30 x [latex]\frac{5}{8}[/latex]) m/sec = ([latex]\frac{25}{3}[/latex]) m/sec
Distance moved in passing the standing man = 100 m.
Required time taken = ([latex]\frac{100}{\frac{25}{3}}[/latex]) = (100 x [latex]\frac{3}{25}[/latex]) sec = 12 sec.

Example 2: A train is moving at a speed of 132 km/hr. If the length of the train is 110 metres, how long will it take to cross a railway platform 165 metre long?
Solution:

Speed of the train = (132 x [latex]\frac{5}{18}[/latex]) m/sec = ([latex]\frac{110}{3}[/latex]) m/sec
Distance covered in passing the platform = (110+165) m = 275 m.
Time taken = (275 x [latex]\frac{3}{110}[/latex]) sec = ([latex]\frac{15}{2}[/latex]) sec = 7[latex]\frac{1}{2}[/latex] sec

Example 1: A man is standing on a railway bridge which is 180 m long. He finds that a train crosses the bridge in 20 seconds but himself in 8 seconds. Find the length of the train and its speed.
Solution:

Let the length of the train be x metres.
Then, the train covers x metres in 8 seconds and ([latex]x[/latex] + 180) metres in 20 seconds.
∴ [latex]\frac{x}{8}[/latex] = [latex]\frac{(x + 180)}{20}[/latex] ⇔ 20[latex]x[/latex] = 8([latex]x[/latex] + 180) ⇔ [latex]x[/latex] = 120.
∴ Length of the train = 120 m.
Speed of the train = ([latex]\frac{120}{8}[/latex]) m/sec = m/sec = (15 x [latex]\frac{18}{5}[/latex]) kmph = 54 kmph.

Example 2: A man sitting in a train which is travelling at 50 kmph observes that a goods train, travelling in opposite direction, takes 9 seconds to pass him. if the goods train is 280 m long, find its speed.
Solution:

Relative speed = ([latex]\frac{280}{9}[/latex]) m/sec = ([latex]\frac{280}{9}[/latex] x [latex]\frac{18}{5}[/latex]) kmph = 112 kmph.
∴ Speed of goods train = (112 - 50) kmph = 62 kmph.

1. [latex]x[/latex] km/hr = [latex]x[/latex] x [latex]\frac{5}{18}[/latex] m/s
2. [latex]x[/latex] m/s = [latex]x[/latex] x [latex]\frac{18}{5}[/latex] km/hr
3. Time taken by a train of length [latex]y[/latex] meters to pass a pole or a standing man or a signal post or an object of negligible width would be equal to the time taken by the train to cover [latex]y[/latex] meters which is primarily the length of the train.
4. Time taken by a train of length [latex]y[/latex] meters to pass a stationary object of length [latex]b[/latex] meters is the time taken by the train to cover ([latex]y + b[/latex]) meters.
5. Suppose two trains or two bodies are moving in the same direction at [latex]x[/latex] m/s and [latex]y[/latex] m/s, where [latex]x[/latex] > [latex]y[/latex], then their Relative speed = ([latex]x[/latex] - [latex]y[/latex]) m/s
6. Suppose two trains or two bodies are moving in the opposite direction at [latex]x[/latex] m/s and [latex]y[/latex] m/s, then their Relative speed = ([latex]x[/latex] + [latex]y[/latex]) m/s
7. If two trains of length [latex]a[/latex] meters and [latex]b[/latex] meters are moving in the opposite direction at [latex]x[/latex] m/s [latex]y[/latex] m/s, then the time taken by the faster train to cross the slower train = [latex]\frac{(a + b)}{(x + y)}[/latex] sec.
8. If two trains of length [latex]a[/latex] meters and [latex]b[/latex] meters are moving in the same direction at [latex]x[/latex] m/s [latex]y[/latex] m/s, then the time taken by the faster train to cross the slower train = [latex]\frac{(a + b)}{(x - y)}[/latex] sec.
9. If the two trains (or bodies) start at the same time from points A and B towards each other and after crossing they take [latex]x[/latex] and [latex]y[/latex] sec in reaching B and A respectively, then (A's speed) : (B's speed) = [latex]\sqrt{y}[/latex] : [latex]\sqrt{x}[/latex]

1. A train of length 200 m is running at the speed of 20 km/hr. Find the time taken by the train to pass a man standing near the railway line?
Solution:

Given,
Distance moved in passing the standing man = 200 m
Speed = 20 kmph
Convert kmph to mps
Speed = 20 x [latex]\frac{5}{18}[/latex] = [latex]\frac{50}{9}[/latex]
So, Required time = [latex]\frac{distance}{speed}[/latex]
⇒ Time = [latex]\frac{200}{\frac{50}{9}}[/latex]
⇒ Time = [latex]\frac{200 * 9}{50}[/latex]
⇒ Time = 36 sec
Therefore, the time taken by it to pass a man standing near the railway line = 36 sec

2. A man is standing on a railway bridge which is 200m long. He finds that a train crosses the bridge in 10 seconds but himself in 6 seconds. Find the length of the train and its speed?
Solution:

Let,
The length of the train = [latex]x[/latex] m
Then, the train covers [latex]x[/latex] m in 6 seconds and [latex]x + 200[/latex] m in 10 seconds.
So, Length of the train is
[latex]\frac{x}{6}[/latex] = [latex]\frac{x + 200}{10}[/latex]
⇒ 10 x [latex]x[/latex] = 6([latex]x[/latex] + 200)
⇒ 10 x [latex]x[/latex] - 6 x [latex]x[/latex] = 1200
⇒ 4 x [latex]x[/latex] = 1200
⇒ [latex]x[/latex] = 300
⇒ Length of train = 300 m
Then, Speed of the train = [latex]\frac{300}{6}[/latex] m/s = 50 m/s
⇒ 50 x [latex]\frac{18}{5}[/latex] km/hr
⇒ 180 km/hr
Therefore, length of train = 300m and
Speed of train = 180 km/hr.

3. A train 320 m long is running with speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the opposite direction in which the train is going?
Solution:

Given,
Relative Speed of the train and man = (60 + 6) = 66 kmph
Converting it into m/s, i.e.
⇒ 66 x [latex]\frac{5}{18}[/latex]
⇒ [latex]\frac{55}{3}[/latex] m/s
Time taken by the train to cross the man is
= time taken by it to cover 320 m at [latex]\frac{55}{3}[/latex] m/s
= 320 x [latex]\frac{3}{55}[/latex]
= 17.45 sec
Therefore, time taken by the train to cross the man is 17.45 sec

4. Two trains 200 m and 320 m long are running in the same direction with speeds of 72 km/hr and 54 km/hr. In how much time will the first train cross the second?
Solution:

Relative Speed of the trains = (72 - 54) km/hr = 18 km/hr
Convert it into m/sec, i.e.
⇒ 18 x [latex]\frac{5}{18}[/latex]
⇒ 5 m/s
Time taken by the trains to cross each other is
= time taken to cover (200 + 320)m at 5 m/s
= [latex]\frac{520}{5}[/latex]
= 104
Therefore, time taken by the trains to cross each other is 104 sec.

5. A man sitting in a train which is travelling at 60 kmph observes that a goods train, travelling in opposite direction, takes 9 seconds to pass him. If the goods train is 300 m long, find its speed?
Solution:

Given,
The goods train travels 60 kmph
Relative speed = [latex]\frac{300}{9}[/latex] m/s
Convert it into kmph, i.e.
[latex]\frac{300}{9}[/latex] x [latex]\frac{18}{5}[/latex] = 120kmph
Now, Speed of goods train is
= (120 - 60) kmph
= 60 kmph
Therefore, Speed of goods train = 60 kmph