Motion along a straight line - problem 1

Two trains, one 200 m long traveling at a speed of 30 km/h and the other 300 m long traveling at a speed of 60 km/h are headed towards each other, on parallel tracks. How long it will take from the moment they meet to the moment they finish passing?

Solution.
Read it carefully. Understand the problem? O.K. We denote the quantities from the problem as follows:
l1 = 200 m,  l2=300m,  v1=30 km/h,  v2 = 60 km/h. We can make the sketch of the situation described in the problem.

Fig M1-P1-1

We must calculate the time t the trains will need to move from situation denote as Beginning to the one denoted End. In other words, we must calculate the time needed to travel given distance. Generally time needed to travel distance S at speed v ist
t = S / v        [1]

But, what is this distance S and what is the speed v ? The distance S does not depend on the way the trains travel: simultaneously or first train 1, then train 2 or only one moves, the other not.
You can find S “experimentally”.
- Put two pencils of different length as the train are shown in the upper part of the Fig. M1-P1-1.
- Move slowly the “train” 1 to the right to the position as in lower part of this Figure and measure the displacement required for this translation.
You will find that this displacement is a sum of lengths of two pencils used to perform this “experiment”. So, with notation for our problem

S = l1 + l2 .        [2]

Then repeat this “experiment” but this time moving the “train” 2. You will get the same result. You can play further with these pencils and will find, that going from situation Beginning to the End, ALWAYS requires total displacement S = l1 + l2.
We have than quantity S required in Eq. 1.

But what about the speed from this equation? Each train moves at its own speed.
Think for a moment. If you travel by car at the speed 50 mi/h (about 80 km/h) and another car is traveling towards you at 40 mi/h, what is a speed you are approaching each other? Its simple, 50 mi/h + 40 mi/h = 90mi/h. With a few moments of thinking you will come to the conclusion that in our problem the resultant distance S is traveled at speed equal to the sum of the speeds of the individual trains, that is

v = v1+ v2        [3]

Substituting Eqs. [2] and [3] into Eq. [1] we get

t=( l1 + l2)/( v1+v2)      [4]


The Eq. [4] is an answer to our problem.

You should always solve the problem analytically first.

Now, according to the strategy of solving problems, described at the beginning of this TUTORIAL, we must check correctness of our solution. First we check dimension of the final result. On the basis of Eq. [4]

[t] = [km] / [km/h] = h

We got hours for time, so we can expect, that solution is correct, but remember - it is not 100% proof of correctness.And the final step, substituting numbers into the final Equation.

t = (0.2 km + 0.3 km) / (30 km/h + 60 km/h) = 0.5 / 90.0 h = 0.00556 h

It is rather very short time, so we express it in seconds

t =0.00556 h = 0.00556 * 3600 s = 20 s

For someone who travels by trains this results sounds reasonable. Not all problems will be discussed and analyzed with such details, it would require to much space, but some will be to show you the best way of approaching the task of solving problems.


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