[SR] Relativistic runner and two relativistinc trains....

In summary, the conversation discusses a problem involving two trains of different lengths moving at different speeds in opposite directions. The goal is to determine the speed at which a person must run to coincide with the fronts-passing-each-other and backs-passing-each-other events of the trains. The conversation explores different equations and approaches to solving the problem, ultimately determining that staying in the Earth frame and using length contraction and the rate at which the gap between the trains is closing is the easiest solution.
  • #1
tobix10
7
1

Homework Statement


A train of length L moves at speed 4c/5 eastward, and a train of length 3L moves at speed 3c/5 westward. How fast must someone run along the ground if he is to coincide with both the fronts-passing-each-other and backs-passingeach-other events?

Homework Equations


Velocity addition formula
[tex]u = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}}[/tex]
Lorentz contraction
[tex] L = \frac{L_0}{\gamma}[/tex]
Time
[tex] t = \frac{s}{v} [/tex]
Lorentz factor
[tex] \gamma (v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/tex]

The Attempt at a Solution


I took the point of view of a running man (its frame). Initial point is when both fronts and a man are in line. Then I calculated speed of both trains in man's frame.
Train B is the one going westward and C eastward.
[tex] u_B = \frac{\frac{3}{5}c + v}{1+\frac{\frac{3}{5} v}{c}} [/tex]
[tex] u_C = \frac{-\frac{4}{5}c + v}{1-\frac{\frac{4}{5} v}{c}} [/tex]
Then calculate the length of trains in a frame of reference related to man
[tex] L_B = \frac{3L}{\gamma (u_B)}[/tex]
[tex] L_C = \frac{L}{\gamma (u_C)}[/tex]
I think that the passaging time of the trains should be equal
[tex] t_B = \frac{L_B}{u_B} = \frac{L_C}{u_C} = t_C [/tex]
but this equation gives solution v = c or v = -c so something is wrong.
I would expect that the velocity v is less than 4/5c.
 
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  • #2
There are two different gammas here. What did you use for them? When you wrote ##\gamma(u_B)##, you meant "gamma as a function of uB" so your last equation must be $$t_B = \frac{L_B}{\gamma(u_B)} = \frac{L_C}{\gamma (u_C)} = t_C$$.
 
  • #3
kuruman said:
There are two different gammas here. What did you use for them? When you wrote ##\gamma(u_B)##, you meant "gamma as a function of uB" so your last equation must be $$t_B = \frac{L_B}{\gamma(u_B)} = \frac{L_C}{\gamma (u_C)} = t_C$$.
With this notation I meant gamma as a function o speed.

I think your equation is incorrect. Dimensions are wrong.
 
  • #4
tobix10 said:
I think your equation is incorrect. Dimensions are wrong.
Yes, of course. The expression should also have uB and uC in the appropriate places. What is the expression you got? I got a fourth order polynomial. I plotted it and it has only one solution between -c and c, two solutions at -c and +c (like you got) and one solution less than -c (speed greater than c).
 
  • #5
I just put equation to Mathematica and get those results. Could you post full equation that you think is valid?
 
  • #6
I find it much easier to analyze this problem by staying in the Earth frame. No need to use the velocity addition formula, just length contraction and the fact that the gap between the rears of the trains is closing at a rate of vA + vB in the Earth frame. Here, vA and vB are the speeds of the trains relative to the earth.

This approach is only a suggestion.
 
  • #7
tobix10 said:
I just put equation to Mathematica and get those results. Could you post full equation that you think is valid?
I too used Mathematica and initially got the same answer as you. Then I realized that the correct way to put it in is to solve
$$ \frac{L_B}{u_B~ \gamma(u_B)} + \frac{L_C}{u_C~\gamma (u_C)} =0$$.
That's because one of the ratios is positive and the other is negative since one velocity is positive and the other negative. Mathematica gives four solutions only one of which is a speed less than c.
 
  • #8
TSny said:
I find it much easier to analyze this problem by staying in the Earth frame. No need to use the velocity addition formula, just length contraction and the fact that the gap between the rears of the trains is closing at a rate of vA + vB in the Earth frame. Here, vA and vB are the speeds of the trains relative to the earth.
But the simultaneous events of the fronts and rears coinciding occur in the reference frame of the runner not the rails.
 
  • #9
kuruman said:
But the simultaneous events of the fronts and rears coinciding occur in the reference frame of the runner not the rails.
What do you mean? Those events are not simultaneous. Rather they are coincident with the runner, which is true in every frame...

I agree with TSny, the problem should not be very difficult in the Earth frame... You just need to know the time and distance between the events (front passing/backs passing). The ratio of this time and distance is of course the speed someone would need to move (relative to Earths frame) in order to coincide with both events.
 
  • #10
Hiero said:
What do you mean? The events are not simultaneous. Rather they are coincident with the runner, which is true in every frame...
I interpreted the statement "if he is to coincide with both the fronts-passing-each-other and backs-passing each-other events" to mean that the observer is the runner not someone sitting on a bench at the train station. I mean that the runner sees the two fronts and the two backs to be where he is (as measured by his meter stick) and at the same time (as measured by his clock.) In other words, the fronts, the backs and the runner are at two different positions simultaneously in the runner's frame of reference. This simultaneity does not happen in any of the other frames including the Earth's. I may be wrong, and it wouldn't be the first time. What is your interpretation?
 
  • #11
kuruman said:
I interpreted the statement "if he is to coincide with both the fronts-passing-each-other and backs-passing each-other events" to mean that the observer is the runner not someone sitting on a bench at the train station.
The runner is an observer. There is no 'the' observer, as if it changes the answer. In the runners view, the events happen in the same place but different times. But of course, in the runners view the runners speed is trivially zero.

kuruman said:
What is your interpretation?
The problem states, "how fast must someone run along the ground" so I think that in itself is a big hint that the ground frame is the easiest frame to work in.
 
  • #12
Hiero said:
The problem states, "how fast must someone run along the ground" so I think that in itself is a big hint that the ground frame is the easiest frame to work in.
Indeed we are looking for the runner's speed relative to the ground. However, the runner observes the coincidence of the fronts and backs; not the ground. Their simultaneous coincidence occurs in the frame of reference of the runner; he sees the fronts and backs at the same position at the same time as he, not the ground. If this simultaneous coincidence occurs in the frame of reference of the runner, it cannot be simultaneous in the frame of reference of the ground and vice-versa.

On edit: If the simultaneous coincidence were to happen in some frame other than the runner's, why is the runner needed?
 
Last edited:
  • #13
kuruman said:
If this simultaneous coincidence occurs in the frame of reference of the runner, it cannot be simultaneous in the frame of reference of the ground and vice-versa.
That is just not true... Simultaneity is only frame dependent for spatially separated events... The simultaneity of events which also occur at the same location is a universally agreed upon, frame-independent statement... So what (relevant) events occur at the same time but different place in the runners view?
 
  • #14
Hiero said:
So what (relevant) events occur at the same time but different place in the runners view?
None. He sees the fronts meet at t = 0 and the backs meet at some later time. I was wrong to imply in post #10 that he has moved relative to his frame.
 

Related to [SR] Relativistic runner and two relativistinc trains....

1. What is the concept of "[SR] Relativistic runner and two relativistic trains"?

The concept refers to a thought experiment in the theory of relativity, where a runner travels alongside two trains moving in opposite directions at relativistic speeds. It is used to demonstrate the principles of time dilation and length contraction.

2. What is the significance of this thought experiment in the theory of relativity?

This thought experiment helps us understand the concepts of time dilation and length contraction, which are fundamental to the theory of relativity. It also highlights the relative nature of time and space, and how they are affected by the speed of an observer.

3. How does time dilation work in the context of "[SR] Relativistic runner and two relativistic trains"?

According to the theory of relativity, time moves slower for an observer moving at high speeds. In this thought experiment, the runner observes the clocks on both trains to be moving slower than their own, due to their relative motion. This is known as time dilation.

4. What is length contraction and how does it apply to this thought experiment?

Length contraction is the phenomenon where an object appears shorter when viewed by an observer in motion relative to it. In this thought experiment, the runner observes the lengths of the two trains to be shorter than their actual lengths due to their relative motion.

5. Can you explain how "[SR] Relativistic runner and two relativistic trains" relates to the concept of simultaneity?

In this thought experiment, the runner observes events on the two trains happening at different times due to their relative motion. This highlights the concept of simultaneity, where the order of events can be perceived differently by different observers depending on their relative motion.

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