The Universe, a Metal Rod and gravity (Part 1)
While exploring a nearby interstellar neighborhood, you stumble upon a long metal rod made of aluminum. You use your tape measure to determine it’s 1 0 c m in diameter and exactly 1 lightyear in length.
Now, you decide to push the rod on one of its ends in the direction of the rod's length. Once you push the rod, you hop in your space ship and take off towards the other end.
Can you get to the other end before the other end starts moving?
Assume that your spaceship can travel at speeds up to, but not greater than, the speed of light c .
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
12 solutions
And assuming infinite rigidity of the rod?
Log in to reply
Unfortunately, we must be very careful with this assumption. Assuming infinite rigidity would also correlate to infinite elasticity of the object. Elasticity, however, is the factor that increases or decreases the speed of sound in an object. This is due to the fact that every individual molecule engages in simple harmonic motion when disturbed by a sound wave. When elasticity is increased, so does speed of the simple harmonic motion of the molecules. This means that the overall speed of sound in that object is increased. Thus, assuming infinite rigidity, which also correlates to infinite elasticity, would lead to an object where light can travel through it at an infinite speed which is impossible.
Log in to reply
Measuring the deflection caused by a man pushing who is on the order of 0.00000000000005% of the weight of the object is also physically impossible. Which physical impossibilities should we assume and which should we not?
As a fun thought: if our astronaut was somehow slammed into the rod at 100km/s and survived, once he began watching the rod at the other side he'd have to wait somewhere around 15 minutes (conservatively) for the motion he caused to be approximately equivalent to the expansion of the aluminum from his radiative heat. That assumes an emissivity of his suit is even less than polished metal, that the rod is pre-heated, and that only a 1 square foot area of the suit radiates heat.
I've used exactly this sort of thought experiment to demonstrate that infinite rigidity is impossible. If it were then you would have information travelling faster than light. I suspect that is the reason the rod was specified as being made of aluminium.
I literally have a million questions about this question.
I've never been great at physics so pardon my ignorance; say our astronaut is very strong and the initial force is greater than the speed of sound of the object---or is this an impossibility because the object would shatter or something?
Log in to reply
trainercase answered it. The reason your "push" travels at the speed of sound is because both your push and the transmission of sound are two examples of the same phenomena, a compression wave. Super simple explanation, when you push on one end, you are applying force which compresses the very end of the stick slightly. The molecules you have pushed push the next molecules in line, and those molecules push the next ones, and so on, so the 'push force' (the compression) acts as a wave propagating through the stick.
The reason the stick would be destroyed is because that compressive force becomes greater than the forces maintaining the structural stability of the stick.
This is also why you cannot have an indestructible material, because the force to break something in this way is necessarily equal to the internal forces holding it together, and so for something to withstand infinite external force it would need an infinite equal and opposite internal force (see Newton's third law).
Log in to reply
That's what I was kind of thinking, but now I can really "see" it in my head w/ your explanation. Thank you!
I don't see why the aluminum rod is not rigid. If it's rigid... then the rod moves as a fully integral unit. Therefore it is impossible to beat it anywhere anyway.
Log in to reply
The assumption that the rod is "fully rigid" works well for small velocities where Newtonian (or rather, non-relativistic) mechanics work well. It does not work so well in this case where you have a 1 lightyear long rod. If you were to assume full rigidity then the far end of the rod would "know" that you are applying a force on the close end immediately. This would mean that you have sent information/an effect to the far end of the rod faster than the speed of light which would take a full year to reach the far end.
In Lucas Chavez Meyles answer, he reasoned that the sound wave is equivalent to the information of "the push" traveling forward in the rod, but lets look at another case that limits it to below the speed of light. In a simplified picture, but still interesting, the internal dynamics of the rod are atoms and electromagnetic forces between atoms. The electromagnetic interaction is carried by electromagnetic waves (photons) through the medium. The photons themselves travel at the speed of light, but interact with nearby atoms (giving them a push) and are absorbed, and these atoms then send out new photons. This interaction takes time and thus even if you had a rod of a very nice material with high wave propagation speed it would still lose out to another photon outside the rod racing alongside it.
Edit: Usha Devi's response to Ryan Martin's question explain this second paragraph in a nice way.
Another fun thing to think about is: What happens if you grab the close end of the rod, hold it tight, and start spinning around in a circle? What will it look like?
this was my thought as well.
Log in to reply
Me too. For a popular problem this has to be one of the worst worded problems. What does ‘push the rod’ actually mean in terms of impact? Nothing can be assumed about the dispersal of energy along the rod due to its inertia, so it may not move at all!
If aluminum were a gas or plasma, with characteristics of compressibility at the level of force possible by an astronaut, if space was a confined area in which atoms only moved in a single direction, if hopping back into the spaceship could be done at a speed that is faster than light, if the spacecraft could exceed lightspeed (c) instantaneously, then the the answer could be correct. With any of the conditions not met the answer is no. If the object was in a non compressible state (solid or liquid) the instant force is applied all the inter-atomic forces connecting the particles would result in unitary acceleration and subsequently be moving. If the there is any delay bumbling along to re-enter the spacecraft or fumbling with door/ignition keys the object will be in motion at the velocity dictated by the force applied. If the spacecraft cannot exceed the speed of light then it cannot arrive at the end of that object prior to the force being applied when the effect of force and time have physical effect. (Although, if it did exceed c on the trip to the end of the object, it would continue to be too early to ever observe the movement of the object in perpetuity).
Log in to reply
If it were perfectly rigid, then it wouldn't conduct sound. Consider that your "push" is to hit it with a hammer -- the push is, literally, just the sound wave moving along it. That's what sound is, a compression wave that moves through an object. Even solid objects (because they conduct sound) can have compression waves. The wave moves through the object at the speed of sound -- but it is more correct to say that we define the speed of sound through the object as the speed that the compression wave moves through it.
I don't think we even need this. The answer is worded as : "Yes, the space ship could theoretically catch up with the moving rod". Even if this seems to have nothing to do with the question (we are asked, if we can reach the other end before (!) it starts moving, so catching up to it, which suggests that it's already moving, seems unrelated), we can still say that we will never push the rod hard enough to make it reach the speed of the light. So, as it will always be slower than the light, we can eventually catch up to it :-)
Thus, we can just choose the first answer because of inaccuracy in how it's phrased, without even worrying about all this stuff :-)
Log in to reply
No matter how hard you push the stick, the compression pulse should always travel at the same rate.
The question is "Can you get to the other end before the other end starts moving ?" Clearly, this explanation does not satisfy this part of the problem.
There is no instant acceleration from 0 to c tho' - since the statement about the ability to reach c does not state anything about it's rate of accelleration the answer to this question has to be based on assumptions to some degree. A third ("can't say") option or more info would be needed.
Log in to reply
You just assume you can reach c in the appropriate amount of time
It is not that the wave moves through the object at the speed of sound. It is that we define the speed of sound as the speed that the longitudinal wave moves through it. That is literally all that sound is.
So lets think small scale here cause I am having a hard time conceptualizing this. If I push a yard stick across the floor, the entire yardstick moves. I would never be able to catch the end of the yard stick because it moves simultaneously as the end I push. Why does this principle not apply to the same solid object albeit a extraordinarily long one? wouldnt a push of one end of a solid object move the other end simultaneously due to the density of the molecules?
Log in to reply
I think the issue with your argument is that the entire yardstick does NOT quite move as one, even though it appears to. If you think first of a yard-long stick of jelly (jello if you are from USA!), if you tap one end, does the other end move immediately, or would you see a shock wave transmit through it until the other end moved perhaps a second later? If you push it slowly rather than tap it, the same thing will happen. Replace the jelly with something stiffer (eg rubber), and the shock wave will probably move a lot quicker, but it will still take a finite time to get to the other end. You can make the yardstick of something much stiffer, eg wood, or titanium, but as nothing is infinitely stiff (which would imply it could could resist any force without deforming one iota), moving one end will not immediately move the other end as the individual molecules always have to transfer the motion onwards to the next, and so on. Another way of looking at it is that if you could push an astronomically-long rod and get the other end to move at the same time, then you would be able to transmit information faster than the speed of light, which is as impossible as moving a physical object faster than the speed of light.
Log in to reply
A compression wave would form in the aluminum and travel at the rate of approximately 16 miles per second. It more than 10,000 times slower than the speed of light. You would easily get to the end - a long time before it moved... frankly at that length, and he compression factor of the aluminum, if you pushed it a full meter, you probably wouldn't be able to even measure any movement at the other end.
I like your reasoning John. I do want to contest one piece which is your statement regarding the impossibility of communicating information at the speed of light. How do entangled particles fit into this? My understanding is that they do exactly that, where initiating change on one particle instantaneously changes the information of its partner regardless of the distance. Thank you
The first theoretical movement of the other end will not be due to the push, but due to gravity interacting between the rod and the traveler and ship. Those forces, based on current theory, propagate at the speed of light, and would affect the far end prior to any possible arrival time. In any case I don't think it will ever be possible to measure precisely enough to identify any motion at one end base on our activities at the other end given the mass of the rod, the magnitude of the forces applied, and distance moved between measuring point.
Log in to reply
The pull of gravity on the far end of the rod follows the inverse square law for gravity. The force of the pull on the end of the rod from the spaceship is proportional to the reciprocal of the speed of light squared which is an astronomically small number, essentially negligible. Anyways, the other technicalities that one may assume for this problem are all completely valid; however, one must treat this problem as a thought experiment, where all necessary assumptions must be made.
I wonder how long it took to measure a light year with a tape measure?
I learned something new today. This isn't usual in everyday life for the ratios of objects like this are much smaller. One more question: How far does that tape measure even go?!
There is a lot of empty space between the aluminum atoms. It takes time for the previous atom to approach the next, and then time for the next to move. It doesn't happen at the speed of light. To demonstrate, if you have a friend stand 200 m down a train track and hit a hammer to the track, you would see him hit it, then hear it from the track, then you would hear it from the air.
Good example, in my opinion.
Log in to reply
Very well explained. I will use this example with my kids.
that is comparing the speed of sound through the track(hearing in the track) to the speed of sound in air(hearing in air) to the speed of light (seeing)
But If you could push the track, your friend would notice that earlier than he would hear the sound wouldn't he?
Log in to reply
He would notice the vibration in the rail before he notices the vibration in the air, that's always true because the wave propagates faster in the solid rail than in the gaseous air. Remember that you are so far away that there's a noticeable gap between light and vibration.
Here's a way to see that. Imagine the iron rail suspended in the air. Your friend pushes the rail by hitting it on the end with a hammer. Now there is no timing difference between 'pushing' and 'hitting' and both are initialized by the compression wave of the impact, so you can see the effect is not changed regardless of whether you call it a push or a hit.
I believe your confusion is caused by the fact that a push does not start with a loud noise, so your perception of motion is not being timed properly. Sound is just you noticing that air is being pushed around at a given frequency. Same for vibration in solid objects. Try pushing something that is halfway between solid and gas (like a huge jello cube) and you can see the delay in the object moving matches the propagation of the compression wave your push initiated.
Log in to reply
An additional thought - you will notice in some solids a difference between an object vibrating and it "moving". This is because by moving we mean (in casual speech) that the base of the object is now in a different place. That displacement requires overcoming friction at the base, and the push may not have been sufficient. But in both cases, the compression wave travels at the same speed, its just that the energy it contains may not be sufficient to overcome friction at the base. In both cases the object moved, it just doesn't always displace. That's why in the example I provided above, I suspended it in the air, where friction is low enough that if the energy is high enough to travel all the way to you, its also enough for you to visually see the rail move. And that's why they set their example in space where friction is as low as possible, so you wouldn't get confused by the difference between perceived motion (displacement) and actual motion (vibration).
How does that solution compare to electricity which is often mentioned to 'travel' at the speed of light? In electricity an electron is given a 'pressure' (voltage) which delivers an instantaneous push between the source and the destination (current). [even though individual electrons within metals travel VERY slow through the material].
Log in to reply
Electricity doesn't travel at the speed of light, it typically goes about 0.5c to 0.99c depending on the conductor. The mechanism for propagating the electromagnetic wave through a conductor is completely different from physical impulses which use the whole atoms as the wave medium.
Even if you were able to push the object to a speed close to that of light, you would get a push in the opposite direction, which would make it difficult to hop back into the aircraft.
Just boop it with your spacecraft then. This is not meant to be realistic, its a thought experiment where you should wonder about how forces in large scale work.
Log in to reply
Is it just me or is everyone ignoring the fact that this rod weighs 8400kg/km? Seems like pushing it is rather futile.
I understand that's not the point of the exercise but still.
Log in to reply
Any force applied to the end of the rod should cause the rod's overall position to move, in space. That is a fun thought, though.
😂...Nice one
Presumably, you'd be in a space suit with a navigation system which would allow you to correct your position in space after pushing the rod. Anyway, I hope you would.
200 trillion tonnes....
ya, you wouldn't be 'pushing' that rod anywhere. that much mass if poked on the end would just set you drifting backwards in space and any minute compression pulse you put into the rod would be damped out within several meters. so you could hop in your ship and get to the other end of the rod whenever you arrive there, it wouldn't have experienced any overall change from your silly push.
Log in to reply
Your backward drift is in REACTION to the rod's being moved forward. If you are not affecting it, then it can't affect you, so your drift is confirmation that you applied directional energy to it. Given the difference in mass between you and the rod, your backward drift will mean only a tiny forward drift in the rod - you'll need really sensitive measuring instruments.
If you really want to get persnickety: even in low friction vacuum, it would be very difficult for you to transfer enough energy to ultimately move the rod. Much more likely the energy would dissipate into heat energy in the rod, not even reaching the forward end. But the question asserts you did transfer sufficient energy, so instead we must assume you have star trek inertia dampers in your suit that stop you from shooting backwards through space at a significant fraction of light speed.
Very strange question. How is someone suppose d to move an aluminum rod weighing 74,306,520,239,032 Kg, and if so what force can one apply to change its momentum? 74 trillion Kg 74 Billion tons
Strange but interesting... wonderful that you worked out the weight!
... I was intrigued by the tape measure!
Log in to reply
I was also intrigued that he could figure out logistics of his tape measure!!! How did he measure? How did he keep his lighyear long rod! lol.... yeah.. i know, it's a thought process question
I was thinking the same thing!
You apply force to it then it starts moving... it's just veeeeeery slow.
I think you have the wrong mass of the rod: Meter of 10cm diameter rod weighs in at about 20.4Kg and light year is around 9.46 10^15m, gets us about 1.93 10^17Kg. I think you calculated for a 10mm rod. If you were to mass 100kg and manage to push yourself off the end of the rod at 10m/s the rod would start moving away at 1.93e10^-14m/s as a result of the push. Or about 400 to 500 years to travel the width of a human hair.
The question does seem to propose an impossible situation. BUT, it clearly states all the needed assumptions; that you can push the rod with the result that it moves and that you can travel to its far end. You can assume whatever magic tech you want to achieve this, as it doesn't affect the results given the stated assumptions.
I have a bigger problem with questions that contain hidden assumptions, and you have to deduce those assumptions before you can determine which allowed solution is viable. Invariably the questioner has overlooked alternate assumptions that would allow one of the other answers to be viable.
even a small force can change its momentum by a very small amount
I got it right by assuming in space things move at a constant speed unless stopped. So I figured it wouldn't stop and he would have a lot of time to catch up. Is that wrong?
Yes and No. Its wrong in that the goal is to get to the end BEFORE it starts moving, not to catch up to it once it starts moving. But its correct in that you will indeed have lots of time to get to the end before it moves.
Objects don't start moving all at once (unless they are only one molecule thick). The side you push moves forward immediately, compressing the material directly in front of it. That layer of material is compressed, and moves forward passing the force to the next layer and so on. This is the compression wave that moves across the object, until it eventually reaches the far end and the entire object is now moving. Since the compression wave moves at the speed of sound, you will have lots of time to get in your spaceship and accelerate past it.
Compression speed varies for different materials, but is always low enough that its reasonable to assume a spaceship is considerably faster. The real practical issues are: 1. The object will have huge mass, and its unlikely a human could apply sufficient force to overcome its inertia (even in space). The compression wave will likely be translated to heat, long before it reaches the far end. 2. If the compression wave does make it all the way to the far end, the rod will be moving so slowly that you can't tell. In fact when you first give it a shove, the near end may appear to not move at all.
Force exerted on a rod is in longitudinal wave nearly equal to speed of sound, your ship can travel up to the speed of light, technically (speed of light > speed of sound). And theoretically if you pushed the rod equal to the speed of light you will experience a backward force.
lets see the ratio between your space ship and light. presents space ships velocity at maximum is about 50000mile/h. ( this type of speed yet to be generated). we supposed it has been possible to generate this type of speed. now light travels in a year around 6X10^12 mile. at 50000mile/hr speed ur space ship needs time to reach the other end of rod around 4X10^11years. that means ur all generation will be vanished before reach there. it is an impractical question that depends on many "if". practically or even theoretically it is not possible, if u think deeply
Even though you practically wont get to the other end of the rod anytime within your lifespan, you will still be able to catch up with the push-wave, as the propagation speed in aluminium is about 6320 m/s ~ 14137 miles/hr, so you will still be able to go about 3.5 times as fast as that wave, and thus surely catch up with it within your lifespan =)
Log in to reply
And the top speed of the Apollo flights was around 25,000 mph. So existing spaceships could indeed catch up to the push-wave before you starved to death, and get your desiccated corpse to the far end before it started to move.
Fair point - "Trick" or badly conceived questions are a real issue on Brilliant. But I give this one a pass, as it does not require any hidden assumptions to know which physical laws to apply. It states that you ARE capable of moving the rod with your shove and getting back to your spaceship, and you ARE capable of traveling to the far end. So you can suppose whatever future tech you want to achieve that since it does not affect the core question.
There is a potential flaw in the situation that fantasy tech won't paper over: the rod is 10cm by 1 light year and you've shoved it hard enough that it will start moving. That's a lot of directional energy transferred to the rod all at once. Due to limitations of aluminum - won't it just shatter? If you transfer the energy reeeeeally slowly, would it take so long that you can't get to the other end - even assuming instantaneous travel? But again the question states that you push the rod (not slowly accelerate it over time) and states that the only outcome is the rod moving, so you still can deduce one (and only one) correct answer - even if you suspect the situation is not possible in real life.
If your ship can move with the speed of light it can catch up to anything if they start at the same time. Because it's guaranteed that nothing else can be faster.
The movement through the Al rod has the speed of sound, but you can move faster
The longitudinal wave travels at the speed of sound, which is achievable
It takes longer than a year for the other end of the rod to start moving. It takes your spacecraft a year to get to the other end.
According to relativity. Its one year according to the observer. For you, who is traveling in speed of light. Time stops to a stand still. You would simply experience no time at all. Different observers might disagree on the time that took for you to reach the end depending on their initial frame of reference.
The time will be much greater than one year because the speed of the push will be equal to the speed of sound in that object. This is due to the fact that the push creates a pulse of compression followed by rarefaction, which is defined as a longitudinal wave . The speed of this pulse is actually defined by the equation c = ρ Y , where c is the speed, Y is the Young's Modulus of the solid, and ρ is the density of the object.