Passing of a train II

Task number: 387

A passenger train of length \({L_1}=60\,\mathrm{m}\) travels at speed of \({v_1}=80\,\frac{km}{h}\) . How long does it take this train to pass a freight train of length \({L_2}=120\,\mathrm{m}\) traveling at a speed of \({v_2}=30\,\frac{km}{h}\)?

Assume the trains travel:

a) in the same direction.

b) in opposite directions.

Hint 1:
Before you start solving the problem, convert the speed of the trains into base units.
Solution of Hint 1:
Convert the speed of the trains into base units:

The passenger train: \(v_{1}\,=\,80\,\mathrm{\frac{km}{h}}\,\dot=\, 22.2\,\mathrm{\frac{m}{s}}\)

The freight train: \(v_{2}\,=\,30\,\mathrm{\frac{km}{h}}\,\dot=\, 8.3\,\mathrm{\frac{m}{s}}\).
Hint 2:
Draw a picture for both cases.

Imagine that you are the engine driver of the freight train. What is the velocity of the passenger train relative to you in both cases?
Solution of Hint 2:
a) The trains travel in the same direction.

We can imagine that from the view of the engine driver of the freight train his train is standing still and the passenger train travels along at the speed of
\[v_{a}\,=\,v_{1}-v_{2}\,=\,13.9\,\mathrm{\frac{m}{s}}\,.\]
b) The trains travel in opposite directions.

We can imagine that from the view of the engine driver of the freight train his train is standing still and the passenger train travels along at the speed of
\[v_{b}\,=\,v_{1}+v_{2}\,=\,30.5\,\mathrm{\frac{m}{s}}\,.\]
Hint 3:
What distance does the passenger train have to travel in order to pass the freight train in both cases?
Solution of Hint 3:
The train has to travel the same distance in both cases, as you can see in the pictures (the grey passenger train is shown in the place where it starts).

a) the same direction.

b) opposite directions.

In both cases the locomotive of the passenger train has to travel the distance:
\[L_{1}+L_{2}\]
Hint 4:
You know the distance which the trains have to travel to pass each other. And you also know the speed by which the trains are passing. Therefore you can easily find the time needed by the trains to pass each other.
Solution of Hint 4:
We distinguish the two cases when calculating the time. We use the equation for distance traveled during rectilinear motion: distance = speed ∙ time. From this equation we express the desired time: time = distance/speed.

a) The trains travel in the same direction:
\[t_{1}\,=\,\frac{L_{1}+L_{2}}{v_{a}}\,=\,\frac{60 + 120}{13.9}\,\mathrm{s} \,\dot= \,12.9\,\mathrm{s}.\]
b) The trains travel in opposite directions:
\[t_{2}\,=\,\frac{L_{1}+L_{2}}{v_{b}}\,=\,\frac{60 + 120}{30.5}\,\mathrm{s} \,\dot= \,5.9\,\mathrm{s}.\]
Solution a) the same direction
First of all we convert the speed of the trains to the base units:

The passenger train: \(v_{1}\,=\,80\,\mathrm{\frac{km}{h}}\,\dot=\, 22.2\,\mathrm{\frac{m}{s}}\)

The freight train: \(v_{2}\,=\,30\,\mathrm{\frac{km}{h}}\,\dot=\, 8.3\,\mathrm{\frac{m}{s}}\).

We draw a picture of the situation:

When the trains travel in the same direction the speed at which the trains move relative to each other is:
\[v_{a}\,=\,v_{1}-v_{2}\,=\,13.9\,\mathrm{\frac{m}{s}}\,.\]
The train has to travel the distance which you can see in the picture (the grey passenger train is shown in the place where it starts).

The locomotive of the passenger train has to travel the distance:
\[L_{1}+L_{2}\]
From the determined speed and distance we can now easily find the time needed by the trains to pass each other.
\[t_{1}\,=\,\frac{L_{1}+L_{2}}{v_{a}}\,=\,\frac{60 + 120}{13.9}\,\mathrm{s} \,\dot= \,12.9\,\mathrm{s}.\]
Solution b) opposite directions
First of all we convert the speed of the trains into base units:

The passenger train: \(v_{1}\,=\,80\,\mathrm{\frac{km}{h}}\,\dot=\, 22.2\,\mathrm{\frac{m}{s}}\)

The freight train: \(v_{2}\,=\,30\,\mathrm{\frac{km}{h}}\,\dot=\, 8.3\,\mathrm{\frac{m}{s}}\).

We draw a picture of the situation:

When the trains travel in opposite directions the speed by which the trains move relative to each other is:
\[v_{b}\,=\,v_{1}+v_{2}\,=\,30.5\,\mathrm{\frac{m}{s}}\,.\]
The train has to travel the distance which you can see in the picture (the grey passenger train is shown in the place where it starts).

The locomotive of the passenger train has to travel the distance:
\[L_{1}+L_{2}\]
From the determined speed and distance we can easily find the time needed by the trains to pass each other:
\[t_{1}\,=\,\frac{L_{1}+L_{2}}{v_{b}}\,=\,\frac{60 + 120}{30.5}\,\mathrm{s} \,\dot= \,5.9\,\mathrm{s}.\]
Answer
The passenger train passes the freight train in \( 12.9\,\mathrm{s}\) when the trains travel in the same direction.

When the trains travel in opposite directions, it takes the passenger train \( 5.9\,\mathrm{s}\).
Reference to a similar task
A similar but easier task about a train passing a stationary object is the task Passing of a train I.