SOLUTION: two trains running in opposite direction with speeds 30 km/hr and 50km/hr crosses each other in 15 sec.what is the length of the fastest train?

Question 477167: two trains running in opposite direction with speeds 30 km/hr and 50km/hr crosses each other in 15 sec.what is the length of the fastest train?
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
This is an interesting problem which can be nearly solved by some thoughtful analysis and a little algebra. However, the answer is slightly surprising.
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Let's begin by noticing that the problem has two different sets of units for time. The speed has units of hours and the time for the trains to pass is in units of seconds. Let's just find a conversion that will make everything in the same units.
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Start by recognizing that a kilometer is equal to 1000 meters and an hour is equal to 3600 seconds (60 minutes and each minute has 60 seconds so 60 * 60 = 3600). Therefore, we can say that a kilometer per hour is equal to 1000 meters in 3600 seconds as follows:
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1 km/hr = 1000/3600 m/sec
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Then, you can use a calculator to divide the 1000 by 3600 to get to four decimal places:
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1 km/hr = 0.2778 m/sec
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So we can convert speed in km/hr to speed in m/sec by multiplying by 0.2778
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The speeds we are given for the two trains are 30 km per hour and 50 km per hour. Multiply each of these speeds by 0.2778 and you get:
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30 km/hr = 0.2778*30 m/sec = 8.334 m/sec
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and
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50 km/hr = 0.2778*50 m/sec = 13.890 m/sec
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Now for some creative thinking. Since the trains are going in opposite directions, their speed in relation to each other is the total of their two speeds. Or the relative speed is 8.334 m/sec plus 13.890 m/sec = 22.224 m/sec.
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In other words, we don't really change the problem if one of the trains is stopped (zero speed) and the other train approaches it on a parallel track at the combined speed of 80 km/hr which is at 22.224 m/sec.
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So park one of the trains on a track. Then run the other train at it on the set of parallel tracks at a speed of 22.224 m/sec. When the nose of the moving train is at the same point as the nose of the parked train the clock starts. The problem says that 15 seconds later the tails of the two trains will be side by side and the distance from the nose of the parked train to the nose of the moving train will be the same as the combined length of the two trains. Keep in mind that we haven't specified which train is parked and which one is moving. It doesn't matter as long as the moving train is going at 22.224 m/sec relative to the parked train.
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That being the case, we can get the combined length of the two trains by finding out how far the moving train traveled in 15 seconds. To do that, just multiply its speed in meters per second by the number of seconds as follows:
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22.224 m/sec * 15 sec = 333.36 meters
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So the combined length of the two trains is 333.36 meters.
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Make sure that you understand this. It is one way of analyzing the problem.
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Once you agree with this approach, you can find the lengths of both trains as follows. Return to the concept that one train is moving at 8.334 m/sec. In 15 seconds the distance it travels is:
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8.334 m/sec * 15 sec = 125.01 meters
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At the same time the other train (the faster one) is traveling in the opposite direction at the rate of 13.890 m/sec. The distance it travels in 15 seconds is:
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13.890 m/sec * 15 sec = 208.35 meters
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Note that this combined distance is 125.01 + 208.35 = 333.36 meters and this is the same result as we got by assuming one train was parked and the other was moving toward it at 22.224 m/sec (which is 80 km/hr). From that analysis we knew that the sum of the lengths of the two trains was 333.36 meters.
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So we have been able to get this much of the answer ... one train is 125.01 meters long, and the other train is 208.35 meters long.
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But the problem doesn't ask for the lengths of the two trains. It asks for the length of the faster train. And here comes the trick. It doesn't make any difference which train is the faster as long as the relative speed difference is 8.334 plus 13.890 m/sec for a total relative speed difference of 22.224 m/s. Another way of looking at it is if the shorter train is moving at 8.334 m/sec and the longer train is moving at 13.890 m/sec their tails will meet in 15 seconds after their noses meet. Similarly, if the shorter train is running at a speed of 13.890 m/sec and the longer train is moving at a speed of 8.334 m/sec their tails will still meet at 15 seconds after their noses meet.
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Therefore, you don't have enough information to entirely solve the problem. You only have enough to find the lengths of the two trains. After that, you can say that the faster train may be 125.01 meters in length or it could be 208.35 meters in length, which ever you choose it to be.
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It will probably be helpful for you to make some sketches to convince yourself that this analysis is correct. Show the positions of the two trains a the time their noses meet. Then show their positions 15 seconds later when their tails are both at the same point.
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Hopefully this has been of some help to you. It may be a little confusing at first, but you can think about it more and maybe it will become a little clearer. At least it provides some guidance on what to expect as an answer.

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