How Does Relativistic Speed Affect the Force Exerted by a Train on Rails?

[jartsa]

I came up with this problem, which is non-trivial at least for me:

Train accelerates with constant proper acceleration. When the speed of the train relative to rails is zero, rails feel the train exerting 1000N force on the rails.

What force do rails feel the train exerting on the rails when the speed of the train is 0.87 c, relative to the rails?

 

[PeterDonis]

The quick answer is 500N, i.e., the force felt by the train (which by assumption is constant, since its proper acceleration is constant) divided by the gamma factor corresponding to the train's velocity relative to the rails (which is 2 for a speed of 0.87 c).

The easiest way I know of to understand why this is the answer is to observe that the force felt by the train is equal to the rate of change of the train's momentum in its momentarily comoving inertial frame, i.e., with respect to the train's proper time. In the rest frame of the rails, the train's proper time runs slow by the gamma factor; so a change in the train's momentum that takes 1 unit of the train's proper time will take gamma units of the rails' time. That is, the rate of change with respect to the rails will be the rate of change with respect to the train, divided by gamma.

 

[HallsofIvy]

But the force the train is exerting on the rail is perpendicular to the direction of motion. I don't see why gamma should come into this.

 

[PeterDonis]

the force the train is exerting on the rail is perpendicular to the direction of motion

The train's weight is, but that is not how I was interpreting the OP's question. I was interpreting the OP's question as asking about the force that is accelerating the train, which must be in the horizontal direction, i.e., parallel to the rails, since the train is accelerating horizontally.

Of course that raises the obvious question: how, if at all, does the train's weight change as it accelerates? This has been discussed on previous PF threads, but it's been a while; IIRC the answer we arrived at was that, somewhat counterintuitively, the train's weight increases by the factor gamma. I'll see if I can find any of those previous threads.

 

[Herman Trivilino]

But the force the train is exerting on the rail is perpendicular to the direction of motion.

No, it isn't. There's a component of that contact force parallel to the motion. That's the one being discussed because that's the one responsible for the train's acceleration.

 

[pervect]

I think the problem is pretty straightforward using the concept of proper velocity. My impression from previous discussions is that Jartsa doesn't use this concept. But I'll go through an analysis anyway as I'm getting different answers than some of the other posters to this thread.

We solve the problem by considering the conservation of momentum.

To solve the problem, we introduce the time coordinate t for time in the track frame, and the position coordinate x for position in the track frame. We introduce x mainly to define the proper velocity $u$, which is the rate of change of proper distance (x) with respect to proper time $\tau$, $u = dx/d\tau$. u is one component of the 4-velocity - but for this post I will only talk about the proper velocity and not the 4-velocity. We'll also introduce the non-proper velocity v = dx/dt, though we won't use it very much.

The proper acceleration $\alpha$ is $\alpha = du/d\tau$ by the definition of proper acceleration.

We can write the momentum of the train in the track frame as $p = m u = \gamma m v$, $\gamma = 1/\sqrt{1-(v/c)^2}$. The 3-force on the train is dp/dt. We know that in the track frame, the 3-force on the track is equal and opposite to the 3-force on the train, due to the conservation of momentum. Any momentum gained by the train must come from the track, this exchange of momentum is what is meant by the concept of force.

Thus we can write the 3-force f on the track as:
$$
f = \frac{d \,(m u)}{dt} = m \frac{du}{dt} = m \frac{du}{d\tau} \frac{d\tau}{dt} = \frac{m \alpha}{\gamma}
$$
where $\alpha$ is the proper acceleration as we noted previously.

If one wants to find $u(\tau)$ or $v(t)$ for constant proper acceleration, the solution is well known and I won't go through it here unless someone has a question. The well-known solution (see for instance http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html)is that $u = \sinh \alpha \tau$, which implies $\gamma(\tau) = \cosh \alpha \tau$.

Note that the sci.physics.faq webpage just gives the solution, it doesn't derive it. My approach to deriving the solution would be based on using 4-velocitie (as in MTW's textbook). I've found through experience that people don't seem to be willing to learn about 4-velocities in order to answer their questions, so I won't bother to go through such a soultion unless someone is actually interested in it.

 

[Drakkith]

But the force the train is exerting on the rail is perpendicular to the direction of motion. I don't see why gamma should come into this.

Is it? I thought the force mentioned was the force accelerating the train, which is parallel to the direction of motion, right?

 

[jartsa]

Thank you guys. So force felt by rails decreases as gamma increases. In the rail frame that decreasing force is the force that is accelerating the train (or maybe not, as we will find out later).

I guess we are all assuming the composition of the train stays constant. No change of amount of fuel occurs. One way to achieve that is a device that shoots small fuel pellets at the train at speed close to the train's speed. Overhead electric wires I don't consider good for this purpose, it's a bit like aiming a laser beam at the train.

So at very high speed it's the pellet shooter that is doing almost all the work when a train is accelerated.

The pellet shooter also exerts a large reaction force on the ground it's standing on.

I would guess the recoil force is 500N when the speed of the train is 0.87 c.

As rails are fixed on the ground and pellet shooter is fixed on the ground, ground feels a 1000N force when the speed of the train is zero and also when the speed of the train is 0.87 c. So every second the ground gains the same amount of momentum, and the train gains the same amount of opposite momentum.

 

[PeterDonis]

I guess we are all assuming the composition of the train stays constant.

Yes, that was assumed. One way to realize this would be to put an electrified third rail between the two rails the train runs on, and transfer power to the train that way.

Overhead electric wires I don't consider good for this purpose, it's a bit like aiming a laser beam at the train.

I'm not sure I understand what you mean by this. Overhead wires might have trouble making good contact, but a third rail, as I described above, should work fine.

I would guess the recoil force is 500N when the speed of the train is 0.87 c.

500N is the force felt by the rail because of the train pushing on it in order to accelerate. By Newton's Third Law, 500N is also the force the rail exerts back on the train, evaluated in the rail frame.

 

[jartsa]

I'm not sure I understand what you mean by this. Overhead wires might have trouble making good contact, but a third rail, as I described above, should work fine.

I suspect that the energy might push the train. Basically a power plant that delivers energy to a train feels a force away from the train, and the train feels a force away from the power plant, assuming straight power lines. Even if you don't buy this, just to be really sure, let's deliver the energy as pellets. And let's say pellets are fully converted to energy in the train, so that I can say that it's just pure energy that is shot at the train by a pellet shooter.

500N is the force felt by the rail because of the train pushing on it in order to accelerate. By Newton's Third Law, 500N is also the force the rail exerts back on the train, evaluated in the rail frame.

Ah but it doesn't work like that at all. At very high speeds trains accelerate by accelerating some energy in the backwards direction. In the train frame what the energy of the fuel does is to increase the kinetic energy of the ground, and the force felt on the train is the result of accelerating said energy from train speed to ground speed, like a rocket.

Seriously. I have thought about it, and something like that seems to be happening.

Increasing the kinetic energy of the ground involves pushing the ground, but the pushing consists mostly of pushing the kinetic energy, ground will not feel any of that part of the pushing.

1 MW photon rocket at speed 0.99 c, in ground frame: Pushes itself with a small force, pushes the ground that the photons hit with a very small force.

1 MW train at speed 0.99 c, in rail frame: Pushes itself with small force, pushes the rails with a very small force.

 

[m4r35n357]

Note that the sci.physics.faq webpage just gives the solution, it doesn't derive it. My approach to deriving the solution would be based on using 4-velocitie (as in MTW's textbook). I've found through experience that people don't seem to be willing to learn about 4-velocities in order to answer their questions, so I won't bother to go through such a soultion unless someone is actually interested in it.

The simplest self-contained derivation of the constant acceleration equations I have seen can be found here. If you know a shorter way, I for one would be interested to see it.

 

[DrGreg]

I disagree with the answer of 500 N. I reckon that (surprisingly) the answer is 1000 N -- the force in the rail frame remains constant if we assume the train's mass remains constant.

For a particle undergoing constant proper acceleration $\alpha$ in a straight line, we have, in inertial coordinates in units where $c = 1$,$$\begin{align*}
t &= \frac{1}{\alpha} \sinh \alpha \tau \\
x &= \frac{1}{\alpha} \cosh \alpha \tau \\
\gamma &= dt/d\tau = \cosh \alpha \tau \\
\text{celerity} &= dx/d\tau = \sinh \alpha \tau \\
v &= \frac{dx}{dt} = \frac{dx/d\tau}{dt/d\tau} = \tanh \alpha \tau \\
p &= \gamma m v = m \sinh \alpha \tau \\
f &= \frac{dp}{dt} = \frac{dp/d\tau}{dt/d\tau} = \frac{ m \alpha \cosh \alpha \tau}{ \cosh \alpha \tau} = m \alpha
\end{align*}$$assuming $m$ is constant. (That assumption would be wrong if the train is burning fuel carried by the train.)

The proper acceleration \alpha is \alpha = du/d\tau by the definition of proper acceleration.

That's one of the reasons why I don't like calling $dx/d\tau$ "proper velocity" and would rather use the alternative name "celerity". It's not true that $(d/d\tau) (\text{celerity})$ is proper acceleration, as the above calculation shows. (What is true is that proper acceleration is the magnitude of the 4-force, but for objects not at rest in the coordinate system, the 4-force has a temporal component as well as spatial components.)

 

[DrGreg]

The simplest self-contained derivation of the constant acceleration equations I have seen can be found here. If you know a shorter way, I for one would be interested to see it.

Long ago, I once gave a derivation in this thread.

(There's a typo in equation 5: delete the symbol $v_0$.)

 

[m4r35n357]

Long ago, I once gave a derivation in this thread.

(There's a typo in equation 5: delete the symbol $v_0$.)

Thanks, still trying to work my way through it . . .

I can't help feeling (units) that the $dx$ term equation 5 should also be divided by c rather than $c^2$.

Is there another typo re. eqns. 7 & 8 (I suspect you must mean substituting 5 & 7 into 4 at the start of the second post (rather than 2 & 3) but I don't see the pattern of what you are doing here yet).

 

[PeterDonis]

I reckon that (surprisingly) the answer is 1000 N

Hm, you're right.

Just to be sure, I'll first write down what the proper acceleration 4-vector looks like. It has components:

$$
\frac{d\gamma}{d\tau} = \alpha \sinh \alpha \tau = \alpha \gamma v
$$
$$
\frac{d\gamma v}{d\tau} = \alpha \cosh \alpha \tau = \alpha \gamma
$$

The magnitude of this vector is clearly $\alpha$, so it is indeed the proper acceleration. And multiplying it by $m$ gives the 4-force, whose components are $dE / d\tau$ and $dp / d\tau$. Then dividing by $\cosh \alpha \tau = \gamma = dt / d\tau$ gives $dE / dt$ and $dp / dt$, the latter of which is the 3-force in the rail frame, as derived by DrGreg.

So where did my reasoning in post #2 go wrong? Well, I said:

the force felt by the train is equal to the rate of change of the train's momentum in its momentarily comoving inertial frame, i.e., with respect to the train's proper time

I then adjusted the "rate of change" part by $\gamma$ to account for the change from the train's proper time to the time in the rail frame. But I forgot to take into account that the train's momentum is also frame-dependent! In the MCIF, the train's momentum after a small impulse has been applied is smaller by a factor of $\gamma$ than it is in the rail frame. So a given impulse applied over a time that is shorter by $\gamma$ gives a result that is smaller by $\gamma$; the two factors of $\gamma$ cancel, leaving the force the same.

So what does change between the train frame and the rail frame? Obviously what changes is $dE / dt$, i.e., the rate of change of the train's energy. In the train's MCIF, the 4-force is purely spatial; it doesn't change the train's energy at all (only its momentum). But in the rail frame, as DrGreg pointed out, the 4-force has a temporal component:

$$
\frac{dE}{dt} = m \alpha v
$$

This is a manifestation of what used to be thought of as "relativistic mass"--the force applied to the train, as seen in the rail frame, goes more and more into increasing the train's mass (a better word might be "inertia" here, since it's really "resistance to increase in velocity" that is going up) instead of its velocity, as its velocity gets closer to the speed of light. Another way of seeing this is to calculate the train's 3-acceleration in the rail frame:

$$
\frac{dv}{dt} = \frac{dv / d\tau}{dt / d\tau} = \frac{\alpha / \cosh^2 \alpha \tau}{\cosh \alpha \tau} = \frac{\alpha}{\gamma^3}
$$

This obviously goes to zero as $v$ approaches $1$ (and $\gamma$ increases without bound).

Sorry to beat this to death, but I wanted to make up for having given the wrong answer earlier.

 

[PeterDonis]

I suspect that the energy might push the train.

Of course it does; applying a force to the train takes energy. We have been assuming that the energy is stored in the rails, or something attached to the rails, so that it is at rest in the rail frame. It then gets converted, as seen in the rail frame, to the energy of the train, at the rate $dE / dt$ which I gave in my previous post. Note that $dE / dt = f v$ in the train frame, where $f$ is the 3-force $dp / dt$; this is exactly what we would expect for the power required from whatever energy source is pushing the train.

 

[DrGreg]

Thanks, still trying to work my way through it . . .

I can't help feeling (units) that the $dx$ term equation 5 should also be divided by c rather than $c^2$.

Oops, yes you are right. It's $(v_0/c)$ that should be deleted!

Is there another typo re. eqns. 7 & 8 (I suspect you must mean substituting 5 & 7 into 4 at the start of the second post (rather than 2 & 3) but I don't see the pattern of what you are doing here yet).

Actually, you substitute (2) and (3) into (5) and (6). (4) isn't used at all at this step.

P.S. Just noticed I previously spotted most of these typos in post #28 of that same thread!)

 

[m4r35n357]

Oops, yes you are right. It's $(v_0/c)$ that should be deleted!

Actually, you substitute (2) and (3) into (5) and (6). (4) isn't used at all at this step.

P.S. Just noticed I previously spotted most of these typos in post #28 of that same thread!)

Cheers, I should be OK with the rest now, might go and read that whole thread too ;)

 

[pervect]

assumption would be wrong if the train is burning fuel carried by the train.)

That's one of the reasons why I don't like calling $dx/d\tau$ "proper velocity" and would rather use the alternative name "celerity". It's not true that $(d/d\tau) (\text{celerity})$ is proper acceleration, as the above calculation shows. (What is true is that proper acceleration is the magnitude of the 4-force, but for objects not at rest in the coordinate system, the 4-force has a temporal component as well as spatial components.)

I can't argue with your derivation, 1000 Newtons it is.

 

[A.T.]

I can't argue with your derivation, 1000 Newtons it is.

Aside from the derivations, isn't it there a simple argument for this: The forward force, that a strain gauge would measure at the axle, and the proper acceleration an accelerometer on the train would measure, are both frame invariant and proportional to each other.

 

[PeterDonis]

The forward force, that a strain gauge would measure at the axle

Is the strain gauge at rest relative to the axle, or relative to the rails?

If it's the former, yes, the force and proper acceleration are obviously proportional (and the constant of proportionality is the rest mass $m$ of the train). But the force that's being asked about is the latter--what a strain gauge at rest relative to the rails would measure. That is a different invariant from the proper acceleration/strain gauge at rest relative to the axle invariant, because the two strain gauges are in different states of motion. DrGreg's derivation shows why these two different invariants have the same numerical value.

 

[A.T.]

Is the strain gauge at rest relative to the axle, or relative to the rails?

Isn't the force on the track equal but opposite to the force on the train, per Newton's 3rd?

 

[PeterDonis]

Isn't the force on the track equal but opposite to the force on the train, per Newton's 3rd?

In a given inertial frame, yes. But that in itself doesn't mean the magnitude of the 3-force in one frame has to be the same as the magnitude of the 3-force in a different frame. DrGreg's derivation shows that, in this case, it is.

 

[A.T.]

In a given inertial frame, yes. But that in itself doesn't mean the magnitude of the 3-force in one frame has to be the same as the magnitude of the 3-force in a different frame.

If two objects exert equal but opposite forces on each other, don't the instantaneous inertial rest frames of the objects have to agree on the magnitude of those forces? If not, how would you for example determine which of the two frames measures the higher magnitudes, given that all inertial frames are equivalent?

 

[PeterDonis]

how would you for example determine which of the two frames measures higher magnitudes, given that all inertial frames are equivalent?

The 4-force points in a particular spacelike direction; that direction picks out a preferred frame in which the 4-force is purely spatial (no time component). In DrGreg's derivation, that frame is the MCIF of the train.

"All inertial frames are equivalent" requires that the laws of physics be the same when expressed in all inertial frames. The law in question, Newton's third law, says that the magnitudes of the forces--train on rail and rail on train--have to be equal in the frame in which the law is expressed. That law, in itself, does not require that the magnitudes in one frame be the same as the magnitudes in another; the magnitudes could change when we transformed from one frame to another, as long as they both changed by the same amount, so they were still equal in the new frame.

To infer, just from "all inertial frames are equivalent", that the magnitudes of the forces must be the same in all frames, we would have to show that there is nothing that breaks the symmetry between all frames, in the particular scenario being analyzed. But that is not the case here: as above, the 4-force has to point in some particular spacelike direction, and that breaks the symmetry.

I think the reason that this breaking of the symmetry does not make the magnitudes of the forces in different frames unequal, in this particular case, is that the preferred frame--the train's MCIF--is not fixed; it changes as the train accelerates. So the spacetime direction of the 4-force changes.

 

[DrGreg]

For a particle undergoing constant proper acceleration $\alpha$ in a straight line, we have, in inertial coordinates in units where $c = 1$,$$\begin{align*}
t &= \frac{1}{\alpha} \sinh \alpha \tau \\
x &= \frac{1}{\alpha} \cosh \alpha \tau \\
\gamma &= dt/d\tau = \cosh \alpha \tau \\
\text{celerity} &= dx/d\tau = \sinh \alpha \tau \\
v &= \frac{dx}{dt} = \frac{dx/d\tau}{dt/d\tau} = \tanh \alpha \tau \\
p &= \gamma m v = m \sinh \alpha \tau \\
f &= \frac{dp}{dt} = \frac{dp/d\tau}{dt/d\tau} = \frac{ m \alpha \cosh \alpha \tau}{ \cosh \alpha \tau} = m \alpha
\end{align*}$$assuming $m$ is constant. (That assumption would be wrong if the train is burning fuel carried by the train.)

A few comments on this

1. The constancy of the proper acceleration was hardly used. My proof still applies if you delete the first two equations for $x$ and $t$, replace $\alpha \tau$ by $\varphi$ (rapidity) throughout, and observe that $d\varphi/d\tau$ is the proper acceleration $\alpha$, an invariant. What this shows is that for a particle of constant mass accelerating in a straight line, all inertial frames agree on the magnitude of the 3-force acting on the particle (even if the force varies over time).
2. It seems to be "just coincidence" that this is so; there doesn't seem to be any obvious reason for it, it's just something that drops out of the maths.
3. The result may well be false if the mass isn't constant or if the motion isn't along a straight line.
4. I can't claim credit for this result; I've seen it before (but it was so long ago that I can't remember if my method of proof was used).

 

[MikeLizzi]

A few comments on this

1. The constancy of the proper acceleration was hardly used. My proof still applies if you delete the first two equations for $x$ and $t$, replace $\alpha \tau$ by $\varphi$ (rapidity) throughout, and observe that $d\varphi/d\tau$ is the proper acceleration $\alpha$, an invariant. What this shows is that for a particle of constant mass accelerating in a straight line, all inertial frames agree on the magnitude of the 3-force acting on the particle (even if the force varies over time).
2. It seems to be "just coincidence" that this is so; there doesn't seem to be any obvious reason for it, it's just something that drops out of the maths.
3. The result may well be false if the mass isn't constant or if the motion isn't along a straight line.
4. I can't claim credit for this result; I've seen it before (but it was so long ago that I can't remember if my method of proof was used).

Oh good. From the book "Basic Relativity" by Richard A Mould (Chapter 8 section 8.2)
"...at every instant of time the force acting on the particle in the S' system will be f' = f..."
I have been strugglling for a long time trying to find out how the author got that.
Thanks.

 

[jartsa]

In the spirit of the first post, let's say a fully tanked train starts accelerating from zero speed, train's proper acceleration is a, and force on the rails is F in the rail frame.

Later the train is coasting at constant speed 0.87 c, and somehow the train has been tanked full again. Now the train again starts accelerating with proper acceleration a, and we ask what is the force on the rails in rail frame.

The right answer is F/gamma, that's my opinion, I don't know if I could somehow calculate that answer or not.

A passenger feels the same force pushing her at both speeds. Let us transform that force to rail frame: It's just the same force in the rail frame. So the whole train is pushed by force F at both speeds, in both frames. Correction: almost the whole train is pushed by force F, the fuel that is burned is a somewhat problematic thing in this regard.

 

[DrGreg]

In the spirit of the first post, let's say a fully tanked train starts accelerating from zero speed, train's proper acceleration is a, and force on the rails is F in the rail frame.

Later the train is coasting at constant speed 0.87 c, and somehow the train has been tanked full again. Now the train again starts accelerating with proper acceleration a, and we ask what is the force on the rails in rail frame.

The right answer is F/gamma, that's my opinion, I don't know if I could somehow calculate that answer or not.

A passenger feels the same force pushing her at both speeds. Let us transform that force to rail frame: It's just the same force in the rail frame. So the whole train is pushed by force F at both speeds, in both frames. Correction: almost the whole train is pushed by force F, the fuel that is burned is a somewhat problematic thing in this regard.

If the fuel consumption is negligibly small, my analysis still applies and the answer is $F$.

To include the fuel consumption, a slight modification to my method gives an answer $$
f = m \alpha + \frac{dm}{d\tau} \tanh \alpha \tau = m \alpha + v \frac{dm}{d\tau} = m \alpha + \gamma v \frac{dm}{dt}
$$
Initially, when $v=0$, $f=m\alpha=F$.

 

[PeroK]

Oh good. From the book "Basic Relativity" by Richard A Mould (Chapter 8 section 8.2)
"...at every instant of time the force acting on the particle in the S' system will be f' = f..."
I have been strugglling for a long time trying to find out how the author got that.
Thanks.

You can also crank this out from the velocity and acceleration transformations for motion in 1D:

$v = \frac{v'+ V}{1 + Vv'/c^2}$

$a = \frac{a'}{\gamma^3 (1 + Vv'/c^2)^3}$

Where the unprimed frame is the ground frame and the primed frame is a frame moving to the right at velocity $V$. And if we let the primed frame be the frame where the train is instantaneously at rest ($V = v, \ v' = 0$):

$a = \frac{a'}{\gamma^3}$

The force in the ground frame is:

$F = \frac{dp}{dt} = \gamma^3 ma = ma' = F'$

That's for any force $F$ or $F'$ and, in particular, where there is constant proper acceleration.

 

[PeroK]

If two objects exert equal but opposite forces on each other, don't the instantaneous inertial rest frames of the objects have to agree on the magnitude of those forces? If not, how would you for example determine which of the two frames measures the higher magnitudes, given that all inertial frames are equivalent?

In general, for two dimensional motion, the forces on a particle transform according to:

$F'_x = \frac{F_x - V/c^2(\vec{v}.\vec{F})}{1 - v_xV/c^2}, \ F'_y = \frac{F_y}{\gamma(1 - v_xV/c^2)}$

And, if we take the primed frame to be the frame in which the particle is instantaneously at rest, the velocity to be $(v, 0, 0)$ and the force to be $F = (F_x, F_y, 0)$, then:

$F'_x = \frac{F_x(1 - v^2/c^2)}{1 - v^2/c^2} = F_x, \ F'_y = \frac{F_y}{\gamma(1 - v^2/c^2)} = \gamma F_y$

The particle measures a larger force in the y-direction in its frame than is measured in the ground frame.

 

[jartsa]

Let's study what effect a short acceleration of train has on the momentum of rails which have rest mass m:

Train frame : in time t, rails gain momentum: gamma m delta v

Rail frame: in time gamma t, rails gain momentum: m delta v

Force on rails in the rail frame is quarter of the force on rails in the train frame, when gamma is 2.

Edit: delta v refers to velocity change of rails. Oh but delta v is not the same in both frames ... well that brings the "quarter" back to my previous guess "half" , I hope.

Oh and "train frame" means the frame where the train was at rest when the acceleration started. And "rail frame" means the frame where the rails were at rest when the acceleration started.

 

[jartsa]

A more rigorous version of the previous post, this may be as rigorous as I can get:

Let's calculate momentum change of rails that have rest mass $m$ and undergo a small velocity change $\Delta v$, in rail frame, when a train accelerates for a short time $t$, in train frame.

I will use $k = \frac{1}{\gamma}$

Frame where train was at rest originally, or "train frame":
because $\Delta v$ is small, we can say that in this frame rails change velocity by $k \Delta v$
$$ \Delta p = \gamma m * k \Delta v $$Frame where rails are at rest originally, or "rail frame":
$$ \Delta p' = m \Delta v $$Now let us calculate forces:

Frame where train was at rest originally, or "train frame":
$$ F=\frac{ \Delta p}{t} = \frac{\gamma m * k \Delta v}{t} = \frac{m \Delta v}{t}$$

Frame where rails were at rest originally, or "rail frame":
$$ F'=\frac{ \Delta p'}{\gamma t} = \frac{ m \Delta v}{\gamma t} $$

Comparing the forces in the two frames:
$$F/F' = \gamma $$

 

[PeterDonis]

Let's study what effect a short acceleration of train has on the momentum of rails

This seems to me to be a different version of the scenario than everyone else is analyzing. DrGreg's analysis implicitly assumes that the rails are at rest in a fixed inertial frame. That implies either that the rails have infinite rest mass (or at least a rest mass so much larger than the train's that the total impulse added by the train's acceleration does not produce any appreciable change in velocity--this would be the case if the rails were attached to a planet the size of Earth, for example), or that some other force is being exerted on the rails that cancels the reaction force of the train on the rails (for example, the rails could be attached to a rocket whose engine is providing the force).

 

[jartsa]

This seems to me to be a different version of the scenario than everyone else is analyzing. DrGreg's analysis implicitly assumes that the rails are at rest in a fixed inertial frame. That implies either that the rails have infinite rest mass (or at least a rest mass so much larger than the train's that the total impulse added by the train's acceleration does not produce any appreciable change in velocity--this would be the case if the rails were attached to a planet the size of Earth, for example), or that some other force is being exerted on the rails that cancels the reaction force of the train on the rails (for example, the rails could be attached to a rocket whose engine is providing the force).

I managed to avoid any calculation of the momentum of the train. Maybe that was the key to not making the same error as everybody else.

We want to know the force on the rails, so we calculate the momentum change of the rails, which rails are really massive in my scenario.

 

[Wee-Lamm]

I came up with this problem, which is non-trivial at least for me:
Train accelerates with constant proper acceleration. When the speed of the train relative to rails is zero, rails feel the train exerting 1000N force on the rails.
What force do rails feel the train exerting on the rails when the speed of the train is 0.87 c, relative to the rails?

Perhaps you asked the question in the wrong order?

I read that you are saying the train weighs 1000N while at rest, and want to know how much it will weigh once it reaches a speed of 0.87c.

If I did translate your question correctly, I suggest the rails would still measure a downward pressure from the train at 1000N.

 

[PeterDonis]

We want to know the force on the rails, so we calculate the momentum change of the rails

I don't think you've done that correctly (I'll respond to that in a separate post), but in any case that wasn't my point. My point was that you are using different assumptions from everyone else. So you can't compare your answer to everyone else's answer; you're comparing apples and oranges.

 

[PeterDonis]

because $\Delta v$ is small, we can say that in this frame rails change velocity by kΔv

No, you can't. The constraint is not that the ratio of velocities is $\gamma$. The constraint is that momentum has to be conserved when calculated in any particular frame. So in the rail frame, the momentum change of the rails must be equal and opposite to the momentum change of the train.

So if the rails have rest mass $M$ and the train has rest mass $m$ (note that it's a good idea not to use the same symbol for both rest masses, as you did), and the rails change velocity by $\Delta V$ in the rail frame (the frame where the rails are at rest before the velocity change), and the train changes velocity by $\Delta v$, then momentum conservation gives $M \Delta V = \gamma' m \left( v + \Delta v \right) - \gamma m v$, where $\gamma$ is the gamma factor for velocity $v$ and $\gamma'$ is the gamma factor for velocity $v + \Delta v$, where $v$ is the train's initial velocity in this frame. So the rails will have final velocity $- \Delta V$ and the train will have final velocity $v + \Delta v$.

Expanding out the momentum change of the train and assuming $\Delta v << v$, we get

$$
m \left( v + \Delta v \right) \left[ 1 - \left( v + \Delta v \right)^2 \right]^{- \frac{1}{2}} - m v \left( 1 - v^2 \right)^{- \frac{1}{2}} = \frac{1}{2} m \left( 2 v^2 \Delta v + v \Delta v^2 + \Delta v + v^2 \Delta v + 2 v \Delta v ^2 + \Delta v^3 \right) \approx \frac{1}{2} m \Delta v \left( 1 + 3 v^2 \right)
$$

So we have $\Delta V = \frac{m}{M} \Delta v \frac{1}{2} \left( 1 + 3 v^2 \right)$.

If we then transform into the train frame (the frame in which the train is at rest before the velocity change), then each of these final velocities gets transformed using the velocity addition formula. This gives, for the train:

$$
\frac{\Delta v}{1 - v \Delta v}
$$

For the rails:

$$
\frac{v + \Delta V}{1 + v \Delta V} = \frac{v + \frac{m}{M} \Delta v \frac{1}{2} \left( 1 + 3 v^2 \right)}{1 + \frac{m}{M} v \Delta v \frac{1}{2} \left( 1 + 3 v^2 \right)}
$$

If we assume that $m << M$ and $\Delta v << 1$, then we are left with a simple change in speed of $\Delta v$ for the train and zero for the rails in both frames. But in any case, the relationship between the two velocity changes is certainly not a simple factor of $\gamma$.

 

[jartsa]

I disagree with the answer of 500 N. I reckon that (surprisingly) the answer is 1000 N -- the force in the rail frame remains constant if we assume the train's mass remains constant.

For a particle undergoing constant proper acceleration $\alpha$ in a straight line, we have, in inertial coordinates in units where $c = 1$,$$\begin{align*}
t &= \frac{1}{\alpha} \sinh \alpha \tau \\
x &= \frac{1}{\alpha} \cosh \alpha \tau \\
\gamma &= dt/d\tau = \cosh \alpha \tau \\
\text{celerity} &= dx/d\tau = \sinh \alpha \tau \\
v &= \frac{dx}{dt} = \frac{dx/d\tau}{dt/d\tau} = \tanh \alpha \tau \\
p &= \gamma m v = m \sinh \alpha \tau \\
f &= \frac{dp}{dt} = \frac{dp/d\tau}{dt/d\tau} = \frac{ m \alpha \cosh \alpha \tau}{ \cosh \alpha \tau} = m \alpha
\end{align*}$$assuming $m$ is constant. (That assumption would be wrong if the train is burning fuel carried by the train.)

That's one of the reasons why I don't like calling $dx/d\tau$ "proper velocity" and would rather use the alternative name "celerity". It's not true that $(d/d\tau) (\text{celerity})$ is proper acceleration, as the above calculation shows. (What is true is that proper acceleration is the magnitude of the 4-force, but for objects not at rest in the coordinate system, the 4-force has a temporal component as well as spatial components.)

That's a force on the train in the rail frame, not a force on the rails in the rail frame.

What is the force on the rails in the rail frame when the train picks up its fuel from the ground? (fuel energy=mc²)

It's 1000N. 500 N is needed for accelerating the train, and 500 N is needed for accelerating the fuel from 0 to 0.87 c. If for a while the train stops picking up fuel, the force on the rails decreases to 500 N in rail frame.

How do I know the force is 1000 N in that case where I claim it's 1000 N?

Well that situation is equivalent to a situation where superman stands on the rails and pushes the train, because in both scenarios train's rest mass does not change and train's kinetic energy changes the same way. And it was calculated by DrGreg that superman exerts 1000 N force on the train and 1000 N force on the rails.

 

[Hornbein]

Am I correct in thinking that y'all are assuming a flat Earth?

 

[jartsa]

Am I correct in thinking that y'all are assuming a flat Earth?

Well I assume a flat Earth.

 

[jartsa]

A hot wheel rolls on a cool surface. The wheel loses heat and mass to the surface. How is momentum conserved in this case?

 

[jartsa]

A hot wheel rolls on a cool surface. The wheel loses heat and mass to the surface. How is momentum conserved in this case?

The less massive wheel rolls faster, that's how momentum is conserved. But the surface feels no force.

So if we reheat the wheel every now and then, without changing its speed, the wheel keeps accelerating while the force on the surface is always zero.

This is maybe somehow related to my original question which concerned force felt by a rail when a train is accelerating. Post #1:

Train accelerates with constant proper acceleration. When the speed of the train relative to rails is zero, rails feel the train exerting 1000N force on the rails.

What force do rails feel the train exerting on the rails when the speed of the train is 0.87 c, relative to the rails?

That question is missing the important information about the energy source of the train. So we can answer: "the force on the rails is zero, with suitable design, ha ha"