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Relative Velocity in One Dimension · Lesson 4 of 5

Crossing Lengths

13 min read2 ideas4 checks2 figures to playBy Asheshanand Acharya
What you will learn 0 / 2
A train passing a pole or a bridge
Look around
Waiting at the level crossing

At a level crossing near Kanpur the gate comes down, and a long goods train rolls past. You count the wagons, and the wait seems endless, although the train is not slow. A single engine at the same speed would clear the crossing in a second or two.

you will be able to calculate how long a train takes to pass a point, to cross a bridge, and to pass another train.

Follow one point of the train

In the first chapter a train of \(600\,\text{m}\) crossed a bridge of \(400\,\text{m}\), and the point-particle model failed. The length of the train was comparable with the bridge. The way out is to follow one point of the train, its front. The front moves at the speed of the train, like any point of a rigid body.

The train starts to cross when its front reaches the start of the bridge. It has crossed when its rear leaves the end of the bridge. By then the front has moved the length of the bridge plus the length of the train. To pass a pole, which has no length, the front moves only the length of the train.

A 600 m train passing a pole and crossing a 400 m bridgeA train 600 metres long moves right at 20 metres per second. The dashed arrow shows how far the front of the train has moved since it reached the pole or the start of the bridge. To pass a pole the front must move 600 metres, the length of the train, which takes 30 seconds. To cross the 400 metre bridge completely the front must move 1000 metres, the bridge plus the train, which takes 50 seconds.train — two coaches pulled by a locomotive, side view bridge 400 m time: 0.0 s front has moved 0 m of 1000 m
Fig 4.1The train of lesson 1 of the first chapter, now as a moving picture. A train \(600\,\text{m}\) long moves at \(20\,\text{m/s}\). Crossing the \(400\,\text{m}\) bridge, the front moves \(1000\,\text{m}\) in \(50\,\text{s}\). Choose "passing a pole": the front moves \(600\,\text{m}\) in \(30\,\text{s}\).

Watch the dashed arrow in Fig 4.1. It shows how far the front has moved since it reached the bridge. At \(20\,\text{s}\) the front is at the far end of the bridge, but most of the train is still behind it. Only at \(50\,\text{s}\) does the rear leave the bridge. With the pole, the rear passes the pole when the front is \(600\,\text{m}\) beyond it, after \(30\,\text{s}\).

RememberTime to pass a pole or cross a bridge
\[ t_{\text{pole}} = \dfrac{L}{v}, \qquad t_{\text{bridge}} = \dfrac{L + B}{v} \]
Holds whena train or any long body moving at constant speed \(v\) past a fixed point or a fixed length.
Symbols\(L\) length of the train, \(B\) length of the bridge, platform or tunnel, \(v\) speed of the train
Units\(L, B\) in m, \(v\) in m/s, \(t\) in s
Checkpoint · numericUnanswered

A train \(150\,\text{m}\) long moves at \(54\,\text{km/h}\). How many seconds does it take to pass a platform \(300\,\text{m}\) long completely?

s± 5 %
CheckpointUnanswered

A train \(200\,\text{m}\) long moves at \(20\,\text{m/s}\). A man walks beside the track at \(2\,\text{m/s}\) in the same direction as the train. How long does the train take to pass the man?

Two trains crossing each other

Sit in the other train

Now the bridge moves. Train A passes train B on the next track. Change to the frame of train B. In that frame B is at rest, like a bridge of length \(L_B\). Train A moves past it at the relative speed \(|v_A - v_B|\). The front of A must cover the length of B and then its own length. So the relative distance is \(L_A + L_B\).

Train A passing train B, seen from train BTrain A, 200 metres long, passes train B, 150 metres long, on the next track. The scene is drawn in the frame of train B, so B is at rest and the ground slides past. Train A moves at its speed relative to B and must cover 350 metres, the sum of the two lengths. Chips choose opposite directions, 35 metres per second relative and 10 seconds, or the same direction, 10 metres per second relative and 35 seconds.train — two coaches pulled by a locomotive, side view 35 m/s relative to B time: 0.0 s moved 0 m of 350 m
Fig 4.2Drawn in the frame of train B (\(150\,\text{m}\)), so B stands still and the ground slides. Train A (\(200\,\text{m}\)) covers \(350\,\text{m}\) relative to B. Opposite directions, \(35\,\text{m/s}\) relative: \(10\,\text{s}\). Same direction, \(10\,\text{m/s}\) relative: \(35\,\text{s}\).

The dotted lines in Fig 4.2 mark the two ends of train B. Crossing starts when the front of A reaches the first line. It ends when the rear of A passes the second line. Between the two, A has moved \(200 + 150 = 350\,\text{m}\) relative to B. The two chips change only the relative speed, and so only the time.

RememberTwo trains passing each other
\[ t = \dfrac{L_A + L_B}{|v_A - v_B|} \]
Holds whentwo trains on parallel tracks at constant velocities, signs given to both velocities with one convention.
Symbols\(L_A, L_B\) lengths of the trains, \(v_A, v_B\) their velocities, \(t\) the time from the fronts meeting to the rears parting
Units\(L\) in m, \(v\) in m/s, \(t\) in s
ExampleA passenger train and a goods train

A passenger train \(200\,\text{m}\) long runs at \(72\,\text{km/h}\). A goods train \(150\,\text{m}\) long runs at \(54\,\text{km/h}\) on the next track. How long do they take to pass each other (a) in opposite directions and (b) in the same direction?

Reveal one step at a time and finish the rest yourself.

A passenger at the window

A passenger sitting in train A is a point, not a length. For the passenger, the other train passes the window when it has moved its own length relative to the passenger. So a watch is enough to find the length of the other train: \(L_B = |v_A - v_B| \times t\).

Checkpoint · numericUnanswered

Priya's train moves at \(15\,\text{m/s}\). A train on the next track moves at \(20\,\text{m/s}\) in the opposite direction. It takes \(8\,\text{s}\) to pass the window where Priya sits. How long is that train, in metres?

m± 50 %
Checkpoint · in your own wordsUnanswered

Two trains of the same lengths pass each other twice: once running in opposite directions, once in the same direction at the same speeds. Explain why the second crossing takes longer, and say what stays the same.

In the world
Why railway signals care about length

Indian Railways divides a line into blocks, and a train may enter a block only when the train ahead has left it completely. "Left completely" means the rear of the train has passed the signal, so the front has already moved the length of the train beyond it. A long goods train keeps a block occupied longer than a short passenger train at the same speed. The time is \((L + B)/v\), the bridge formula of this lesson.

Exam traps
  1. Using only the bridge length for a train crossing a bridge.
    The rear must leave the bridge, so the front moves \(L + B\).
  2. Using the difference of the lengths when two trains pass.
    In the frame of one train, the other moves the sum of the two lengths.
  3. Treating a man walking beside the track as a pole.
    A moving man is a point that moves. Use the relative speed of the train with respect to the man.