A bell of a train ride! 🚆

Classical Mechanics Level 2

You're on a fast-moving train, and there's some really bad weather that does not let you look outside at all, as to measure the train's speed. You want to know the velocity of the train, so you use a tube (that has a 550 meters length ) made out of a perfect sound absorbing material (I mean, who leaves home without one? ), that goes on all the way to the other end of the train (and has a small opening on your side and a perfect sound reflecting screen on the other side).

You then grab a really loud bell (on your side) and hit it with all your strengths, while you measure the time it takes for the sound wave (the echo, especifically) to come back; which does so in 2 seconds .

What's the speed of the train?

Assume:

The speed of sound to be 343 meters per second
The speed of the train to be constant.
The ground outside the train to be v=0.

Also, I'm really sorry if there are any spelling mistakes or something like that (English isn't my native tongue)

Good luck!

68 m/s 254 m/s 343 m/s There's no way to know 207 m/s

1 solution

Emilio Vigliecca
Dec 22, 2019

Well, for starters we take the 2 seconds and the 550 meters and calculate with what speed, relative to ourselves (which is the same as relative to the train) the sound moved, so we know that in one second it moved 550 meters (constant speed, and you know it went all the way to the screen and back, so it moved 1100 meters in two seconds, which is 550 meters per second). Now we must apply the Relativity Principle, and knowing that relative to the ground the speed of sound is 343 m/s we can tell that we measured more speed due to having ourselves speed in our frame of reference relative to the ground, the speed of the train, so we know that, as we are moving towards the sound wave it's speed, relative to us, is greater. We know that it will be as greater as our speed, and we know how much greater it is because we know what is its speed relative to an observer standing still on the ground (343 m/s) so we just do 550 m/s - 343m/s to get 207 m/s

BONUS QUESTION: If you did the same but instead of using sound you used a laser, light , would you be able to figure out the speed of the train? Consider you could measure the times despite being practically unmeasurable.

For the answer of bonus question, we are unable to figure out the speed of train because light will soon get reflected much faster than in our initial case with the sound wave failing us to observe any delay as light speed is 10^6 times the speed of sound.

Pradeep Tripathi - 1 year, 5 months ago

You're right, but I wanted to take this to the fact that the speed of light is the same to all observers/frames of reference hence you could never use that the way you do here

Emilio Vigliecca - 1 year, 5 months ago

Doesn't this violate the Galilean principle of relativity? How would the speed of sound inside the train be affected by the motion of the train if it is not accelerating?

Tristan Goodman - 1 year, 5 months ago

The speed of sound behaves like the speed of any object, despite being the movement of mechanical energy through air, though that's not the case with light (everybody measures the same speed, mainly because in case of sound you can move relative to the air, but with light, as there's no actual ether or something you actually move relative to). I used the speed of sound because it does not accelerate and because it's a known number relative to the air, so you could actually do this. We tend to ignore this because in normal circumstances it's not perceivable at all.

Emilio Vigliecca - 1 year, 5 months ago

Yeah but if the train is moving at constant velocity, then the air inside should be in the same inertial frame. The principle of relativity implies that if you are inside a vehicle moving at constant velocity, there is no experiment you can do to determine you're speed relative to the ground outside the vehicle, without looking outside. In other words, I think the math of your reasoning works out, but you seem to be assuming that you are moving relative to the medium through which sound travels, which is not the case if you and the sound are inside a closed train car that is moving at constant velocity.

Tristan Goodman - 1 year, 5 months ago

A bell of a train ride! 🚆

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