A logical problem for math whizzes

[DavidReishi]

We are spying on a train manufacturer’s collection of top-secret futuristic trains to determine some facts about how these high-speed trains are powered. We see from a posted sign that each train is composed of cars all of the same length. We are able to measure the front car of each train, and we find that the length of the cars differ slightly from train to train. Next, the manufacturer decides to test-drive the trains through a tunnel. Before test-driving each train, the total amount of energy powering the train is announced, which is slightly different for each train. It is also announced that all the trains travel at precisely the same speed. Upon each train being driven through the tunnel, we are able to count how many cars of the train exit the tunnel every 30 seconds, which of course is different for each train since they differ from each other in car length, but travel at the same speed. We find that, in comparing the trains, there is correspondence between their differences in total energy, and the difference in how many cars exit the tunnel every 30 seconds.

Since we could get no more information than this, the question to which we want an answer is simple. Is the total amount of energy powering the trains, which is slightly different for each train, necessarily a function of each train’s total length? If so, why? Or if not, why not?

 

[ProfuselyQuarky]

Is this homework or for fun . . . ?

 

[DavidReishi]

It's actually for fun. I created it. But I really want people to give me an answer.

 

[ProfuselyQuarky]

Hmm . . . I’m thinking no because, regardless of the energy that is powering each train, the speed of the trains are the same. I can imagine that if we were sitting down and looking at this scenario happen in front of us, it would look like an illusion, since the length of the cars and the amount of energy provided to each train is only “slight in difference, but that’s what I’m thinking. Am I missing something?

 

[DavidReishi]

It might be me who's missing something. Does it change things for you if we get rid of the "slight" differences and just make 'em differences? By the way, thanks for answering.

 

[ProfuselyQuarky]

It might be me who's missing something. Does it change things for you if we get rid of the "slight" differences and just make 'em differences? By the way, thanks for answering.

No, the word "slight" does not change anything. I was just saying that if we we looking at the event you describe in your puzzle, it'd be hard to tell what's happening. My guess to you initial question in the OP is no since regardless of the energy that is powering each train, the speed of the trains are the same. What's your solution?

 

[DavidReishi]

I'm no longer sure of the solution. I believed that the info makes it necessary that the total energy of the trains is proportional to their total length. But now I'm confused. Right, speed is a non-factor. But then must not the element of time in the trains' observed "frequency" (cars exiting the tunnel per 30 seconds) also be a non-factor? Thus making their total energies proprtional to their car-length?

 

[ProfuselyQuarky]

It does not make sense that the number of cars exiting the tunnel corresponds with the energy on each train. Speed and length of each car is all that matters and, since the speed for each train is the constant, only the length of the cars determines how many exit the tunnel during the 30 second intervals, correct?

Kind of like this . . . .

A kid on a tricycle is pedaling 10 revolutions for every second to reach a speed of 5 miles per hour. An adult is pedaling 2 revolutions every second to reach the same speed. The kid is obviously using more energy, but their speeds are the same. It doesn’t matter that the kid is using more energy. If they both kid and adult went through a tunnel, we’d probably see the adult ride our first because his bike is of greater length.

 

[DavidReishi]

I think the medium of trains is forcing you to think too concretely. When we think of trains and energy, we think of the energy pushing or pulling the train's movement. But look at light. Photons of certain wavelengths are known to contain specific quantities of energy. Yet Planck found that the total energy of each quanta of light is proportional to its frequency, i.e. how many wavelengths pass a certain point every second, even though the speed of light-waves at all wavelengths is assumed to be the same, c. To me, this seems to imply logically that, if we take each quanta as a wave, the total energy of a light-wave is proportional to its total length (i.e. even though light-waves aren't conceived as having definite lengths.) Do you agree with that logic?

 

[ProfuselyQuarky]

Er . . . this problem is becoming much more complicated than it appears on the surface. Yes, I agree that waves of certain wavelengths are known to contain specific quantities of energy, but . . . . I never considered the concept of waves and frequency to find it’s way into the problem. Trains aren’t waves, but you’re treating them as such.

Maybe I'm just plain wrong to not understand you argument?

 

[DavidReishi]

No, you're right. Trains aren't light-waves, and it's precisely their difference that screws up my problem. I thought trains were a good equivalent, since trains have cars (wavelengths), their frquency can be conceived (how many cars exit a tunnel per unit of time), and yet trains have definite length. But now I see that they're still too different to be used in the same problem.

But if we switch the problem to light-waves, do you see the logic? As far as frequency creeping into it, I share your sentiment. I can't seem to figure out why Planck made the total energy of a light quanta proportional to its frequency instead of its wavelength, when its frequency is simply wavelengths passing per second, and all light is assumed to travel at the same speed.

 

[ProfuselyQuarky]

I see your logic if the trains are switched to light waves. I see it entirely. The original problem was just a bad analogy. I think you have a knack at making riddles, though! Keep it up and post more!

 

[DavidReishi]

Spanks!