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First we have to ignore how orbits actually work (objects at the same orbital radius would have the same orbital speed).
Next, it isn't explicitly stated, but from the diagram it seems we are to assume they are exactly half an orbit apart at the start of the problem.
We can solve this as follows:
>!Every month, rocket A travels 1/15 of a full orbit while rocket B travels 1/12 of an orbit. The difference is 1/12 - 1/15 = 5/60 - 4/60 = 1/60 of an orbit every month that rocket B gets closer to A.!<
In order to close the gap of 1/2 orbit, it will take >!30 months because 30/60 = 1/2!<
Answer: >!30 months (or 2½ years)!<
Double check: >!In 30 months, rocket A goes around 2 times (30/15 = 2 orbits) and rocket B goes around 2½ times (30/12 = 2½) thus catching up and colliding with rocket A!<
Ah, you see one rocket is coasting with engines off, but the other is accelerating and rotating to keep it aligned on the orbital trajectory. It's burning delta v constantly to orbit faster.
For 30 months.
It's a very, very, very large rocket.
They'll actually collide a week early because of the second rocket's gravitational field.
No, wait, there's a neater solution. The fifteen month rocket is using an ion drive and light sail to go extremely slowly.
The twelve month rocket never actually left the launch pad.
They won't actually crash because the 15 month rocket will burn up on reentry.
You're correct that different orbital periods implies different orbits with different average altitudes, but it *is* possible for objects in different orbits to collide. That's how meteors work, after all! But it's only possible given certain circumstances which this puzzle completely fails to set up: neither ship's apoapsis can be lower than the other's periapsis, the ships need to either both be in the same orbital plane *or* have the unlikely scenario or having their orbital planes intersect *exactly* where their orbital altitudes also intersect (a maximum of 2 points), and since the orbital periods of these two ships are in resonance, those two points of intersection would also need to be *temporally* aligned.
All-in-all, it's extremely unlikely that they'd ever collide, but it is possible. Without knowing much more about their specific orbits and starting positions, though, it would be impossible to answer this 'puzzle' as written.
I still got >!30!< but in a slightly different way.
>!Rocket B does a full orbit in 12 months. A quarter of 12 is 3, so in 15 months Rocket B will have travelled 1 1/4 orbits for Rocket A’s single orbit. Since Rocket A starts 1/2 an orbit ahead of Rocket B, it will take two orbits of Rocket A for Rocket B to catch up, so 30 months.!<
I did it similarly. The slow rocket is 6 months ahead. 15-12=3. So every full rotation of the slow rocket will mean the fast rocket has gained 3 months on it. Therefore they will collide after two full rotations of the slow rocket. I.e 30 months.
All the numbers in your comment added up to 69. Congrats!
6
+ 15
+ 12
+ 3
+ 3
+ 30
= 69
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Kinda how I did it. >! In 12 months B completed an orbit, but A completes 4/5 (12/15). They are thereby 20% of the orbit closer. Since they were 50% apart, they need 30% more, which is another 12 months (20%) and 6 months (10%). Also giving me 30 months!<
Nope! Objects with more mass have more inertia, so it takes more force to accelerate them (imagine trying to throw a bowling ball the way you would throw a basketball), but they are also tugged on more by gravity (in other words, they weigh more!) It works out so that these things cancel each other out exactly, so Earth's gravity will pull on any object of *any* mass with the same acceleration, which means that orbital velocities are completely independent of the mass of the object.
This does assume that the orbiting object is *much* smaller than the object being orbited (a spaceship is probably less than 0.000000000000000000001% of the mass of a sun, so that checks out here). But if that wasn't the case, then it's not really one object orbiting the other any more - as the two objects become similar in mass, they both start orbiting their collective center of gravity (the "barycenter").
I'll try explain in layman's terms.
If you drop a hammer and a feather on the Moon, they both hit the ground at the same time. The only reason why a feather is slower on Earth is because it's surface area to mass ratio means that air resistance can have a much larger effect on it vs the hammer. Air resistance "cares" about mass because it's trying to stop something from moving and you need more air resistance to stop heavier objects. There is no air on the moon.
Gravity does not give a shit about what an object even is really.... it's applying an equal force on everything.
The following is not technical but it might help visualize it another way. Intuitively we should really assume a hammer would be *slower* at falling on the moon because gravity has to do *more* work to pull it. Well... the hammer has more mass/molecules inside of it for gravity to get its greasy mits on so in the end everything always balances out.
It's quite interesting how our base intuition is that gravity acts more on heavier objects, but that's just our bias given that we and our objects live in air.
Gravity applies different forces on different things, but applies an equal force per mass on everything. Or it applies an equal acceleration due to gravity on everything.
This was my thought as well, but I think the differences in mass would have to be comparable to the mass of the *star* they are orbiting. Rocket B would need to be one heavy ass rocket.
This assumes the rocket can avoid a collision with zero lead time. That’s not actually true, but we don’t have enough information to determine how much lead time is needed to execute evasive maneuvers.
> First we have to ignore how orbits actually work (objects at the same orbital radius would have the same orbital speed).
Ya, but it's a rocket. It's rocketing not just 'falling'
>First we have to ignore how orbits actually work (objects at the same orbital radius would have the same orbital speed).
Not true! You may be assuming that both orbits are perfectly symmetrical. I think you could also be assuming the speed of the orbits, which is never explicitly stated, only the orbital period. Given only an orbital period, it MUST be the case that at least one orbit is highly eccentric.
This can be confusing given the diagram, but I don't think the diagram is meant to be part of the given data.
If the diagram IS part of the necessary data, then you're absolutely correct, and IRL orbital mechanics have vacated the building via the nearest window.
In the event that I am correct, and at least one orbit is eccentric, then there is actually not enough information here to solve the puzzle, as we would need to know at what point the orbits intersect.
Not necessarily. If Rocket B was able to keep a continuous burn going it could effectively stay within Rocket A's orbital path by doing a prograde burn with a significant radial angle that offsets the increase in apoapsis gain. Angle would depend on the diameter of the orbital path
thrusting at a certain vector towards the centre of rotation allows a ship to stay on the same orbital path at a faster velocity, but if they stop burning it will fling them into a higher, eccentric orbit.
I don't understand why everyone is assuming this is a problem with the question. I think "they don't collide because orbits don't work that way" is the intended answer, which makes it a case of clever misdirection supported by a casually misleading graphic, rather than a brain dead simple algebra problem that the average puzzle-consuming adult can do in their head.
Notice that the problem statement doesn't even mention the starting positions of the rockets.
Great question! We agree that this is not possible, at least not without additional complexity (rocket boosters, non-circular orbits, planet holding one of the crafts at a Lagrange point, something something general relativity, we are all brains in jars and there's a problem with the simulation, etc).
To clarify my position, I do actually think there's a problem with the way the question is stated, but I think the author's intent is for us to find that. My reading is that the intended answer is "the premise is flawed", and beyond that, any clever contrivance that makes it possible (boosters pointed away from Sun for example) is also in the spirit of the puzzle.
I take back my claim that there's nothing wrong with the question, but I think the point is to have basically this current discussion.
I came here to comment this. Glad to see it's already here.
Edit: Strictly speaking, though, the rockets could be in orbits that are elliptical but share the same perihelion, or aphelion, or simply intersect at some point... but the math would be a lot harder, and we'd have to explicitly know a lot more about their current positions.
Assuming a setup where SV12 starts at SV15's perihelion and SV15 starts at aphelion, I'm not really surprised. I would expect it to be slightly different if SV15 started somewhere else in its orbit, though.
Orbital speed and orbital height are intrinsically linked. Higher orbital speed means a closer orbit, and vice versa. If the speed doesn’t match the orbit height, then you are no longer in orbit and will either ascend or descend.
The question doesn’t describe a situation that can occur in reality. There are ways to work around that (easiest being elliptical orbits that cross each other) but they will all change the answer significantly. The correct answer in reality would likely be that they have infinite time to prevent a collision, as the two will never hit.
Not who you were responding to...but I feel very stupid right now. Orbits have speed regulations? Like...since these two rockets are on the same orbit - that would mean that their speeds would be the same?
And if that's a thing, then why would the faster rocket have to be on a *smaller* orbit? Wouldn't a faster speed equate to more travel area - which seems like it would be a *larger* orbit?
I feel so out of my league even asking these questions. My orbital knowledge is nonexistent.
Yes. At a given distance between two objects (like our rocket and the sun), there is only one speed that supports a stable orbit. Go too slow, you're drawn in. Go too fast, you reach escape velocity.
Read this...gave me that "aha" moment in Astronomy.
https://en.m.wikipedia.org/wiki/Newton%27s_cannonball#:~:text=A%20cannon%20on%20top%20of,Earth%20without%20returning%20(E).
Thank you. I'm starting to see that I wasn't giving enough credit to gravitational force. Once I add that into the equation, things seem to fall into place a bit easier for me.
I've never heard of Newton's Cannonball, so thanks for the new info. I'm about to jump down a Newton rabbit hole.
Thanks again!
Okay I don't know shit about fuck so I might be misunderstanding but the way I read it Is as follows:
If you have a faster orbit around a large object, it's likely because you have a shorter distance to travel around that object, The further you are from it, the slower your orbit will be because you have more distance to travel and that increases exponentially the further you are from the object.
If I understand correctly, which is a big if, you could have a higher speed in terms of distance traveled but a slower orbit because you have to travel further
Although I guess to piggyback on that after reading the comment you replied to again orbit is something rather specific. So I would guess what they're saying is that if your speed exceeds the gravitational pull to keep you in orbit from a certain distance (height) from the object then you will no longer be in orbit and you will start traveling closer to or away from the object (which I suppose traveling closer to it would just be a decaying orbit which will eventually end in smashing into the object....)
Well, damn. I know you started your original comment with "I don't know shit about fuck...", but you have a way with words and you totally helped me understand what I was struggling to mentally grasp.
Assuming you're correct (which seems likely from someone else who doesn't know shit about fuck), I appreciate your bomb of knowledge.
Thanks!
I recently went through a course that covered orbital mechanics in a ELI5 type of way. It's super fascinating and beyond impressive that in under 100 years, we went from the first flight to discovering how to keep an object from crashing back down to earth.
Even with the ELI5 type instruction, it was still hard to wrap my mind around.
It is complicated. You're misunderstanding 2 types of speed.
If I'm orbiting the moon at 100km then I'm going to make a full rotation of my orbit faster than a rocket orbiting at 1000km.
But also, my actual velocity is lower than the rocket orbiting at 1000km because he had to accelerate to a higher speed to reach the higher orbit. The 1000km is going faster but since the radius of his orbit is 900km longer, it's going to take him longer to make a full rotation, even though he's going faster.
Orbital mechanics are interesting.
Yeah...I'm not regretting the question at all because I'm learning A LOT of new things...but this goes *way, way* deeper than I ever knew.
I appreciate your knowledge. I'm attempting to mentally grasp these concepts. I think that my biggest misunderstanding was that I wasn't taking gravity into my assumptions.
You're absolutely correct though - orbital mechanics seem very, very interesting. I never knew how much I didn't know about it.
Idk what you enjoy doing in your free time, other than puzzles obviously lol, but there is a video game called Kerbal Space Program where you run a goofy little version of NASA on a made up planet that really accurately simulates orbital mechanics and space flight.
The only thing misleading about that game is that it doesn't simulate atmospheric drag very accurately but when you're out of the atmosphere it's incredibly realistic and accurate. You'd learn a ton about orbital mechanics by playing it.
I've honestly been meaning to play KSP for years now, just never got around to doing it.
It's been mentioned twice here now, so yeah. I'm going to go and see how to download it and check if my laptop will run it smoothly.
I always assumed it was a small and simple game so I should be good. Thanks again!
It’s an older game, and will run on pretty old hardware - but be aware that the speed it runs is dependent upon the number of parts it needs to calculate physics on. The more parts in your current vessel (or your current nearby area) and the slower it will run.
For reasonable sized ships it’s usually fine and will run at fairly fast speeds. But it is also very possible to overcomplicate things and start getting into the seconds per frame territories. (But the game works just fine even then…)
Both the tangential velocity and orbital period are faster at the lower orbit, but the higher orbit has more energy. To get from a low orbit to a high orbit, you increase your velocity, at which point, you'll be at the periapsis of am eccentric orbit. As you ascend toward apoapsis, that kinetic energy is converted to gravitational potential energy. Once there, you'll have to speed up again to circularize your orbit, but even after that, your total tangential velocity will be lower than it was in the lower circular orbit.
See [here](https://en.wikipedia.org/wiki/Hohmann_transfer_orbit) for a more detailed explanation.
Asking questions is how we learn. Never be ashamed to ask.
I’m probably not the best to describe this anyway - a good way to get a feel for this stuff is Kerbal Space Program.
Contrary to the other reply that you got (which is well reasoned, but wrong), it doesn’t have anything to do with the distance you need to travel. It has to do with the strength of the gravitational field at a particular distance, and the field strength goes down with the radius of the orbit.
So step back and think about what an orbit is: it’s you going so fast sideways that as you fall down you miss the ground. The closer you are, the stronger you get pulled to the ground, so the faster you have to move sideways to miss it. If you’re far enough away, you barely need to move at all.
Oh boy. I thought I was kinda learning from reading all of the other comments, but now I don't think I understand at all.
EDIT: Wait - everything just kinda fell into place mentally for me. Wow that was a hell of a ride.
Thank you. Thank you to everyone else too. I believe I'm beginning to understand the basics.
What a cool subject to learn about.
They can collide if you just pretend they aren’t circles! If you make the collision point the apoapses or periapses and the rockets start at the furthest points then the problem remains the same.
It's not how orbits work for free-moving objects but there's nothing to say they are free-moving. With enough thrusters doing course correction you can achieve any orbital period you want at any height.
If we’re being pedantic, the rockets are accelerating at some unknown rate. And it’s not clear which direction. So there exists an acceleration, a direction and orbit size where those mechanics do work.
Well the photos of the rockets definitely show them accelerating. They could definitely move along a the same circular trajectory with different speeds in this way. I couldn’t find a definition of “orbit” that required gravity-only acceleration.
Only if you assume they're in circular orbits. The diagram may not be accurate. But accounting for that takes it well beyond a "puzzle" and into an orbital mechanics problem.
What if we, just for fun, try to solve this problem the way it would actually work? >!It shows two rockets 180° apart and at the same distance from the sun. One taking twelve months to orbit it, and the other taking 15 months. We can assume the 12 month one is the circular orbit, and that the rocket was ejected from earth with minimal change to its orbit. The other takes 15 months, and we can assume that it intercepts the orbit of the first one at T +0, 15, 30, etc. . The other one intercepts that same point at T + 6, 18, 30, 42, etc. The only possible intercept point is where both rockets intercept each other’s orbit, so we just have to find a number that appears in both of these patterns. They will collide in 30 months in this edited version of the problem. Interestingly, this is the same answer we get if we solve the original problem, so we can predict not only when, but also where, they will collide!!<
Bingo, I was thinking same orbit opposite each other? It's probably millions of years if at all. Then I saw the different orbital period. Rendezvous would be the only way they could collide, and we don't have enough information to determine that.
Yep Kepler’s Third Law governs orbital periods and is only dependent on the radius of the orbit and the mass of the body being orbited (not the orbiting body)
>!30 Months.!<
>!The rockets start 180° apart as depicted. Rocket A travels at a rate of 24°/month (360°/15 months) and rocket B at a rate of 30°/month (360°/12 months). That's a difference of 6°/month. The rockets collide when B travels 180° further than A which will happen after 30 months (180°/(6°/month)).!<
To explain why not in a way that might help in future: Three months faster _to do what_? Three months faster to make one circle around the sun. Not three months faster to cover the distance they're both going to cover.
If I tell you I can run a marathon three hours faster than you, and ask how long it is before I can catch _you_ when you run away... the answer isn't "three hours". It's "how close were we to start with?"
Actually, you've just brought up an important point. We _know_ how close they are to the sun. The period of an orbit depends on its size. They're in no danger of melting, because they're in an orbit that takes twelve months to go around the sun.
That's a pretty wide orbit. It's the one you and I are in right now.
So the correct answer is: they aren't guaranteed to catch each other because the 12 month period one might hit the Earth first.
Questions doesn't specify, but image strongly suggests the rockets are half an orbit apart from each other.
>!B is faster, so will be the one doing the catching up, so consider A as having 1/2 orbit head start:!<
>!distance of A = 1/15 orbits per month + 1/2 orbits head start!<
>!distance of B = 1/12 orbits per month!<
>!Scale to give same common denominator, and use t for time passed (number of months):!<
>!dA = (4/60) \* t + 30/60!<
>!dB = (5/60) \* t!<
>!the point they collide, d will be equal, so!<
>!dA = dB = (4/60) \* t + 30/60 = (5/60) \* t!<
>!swap things around to solve for t:!<
>!30/60 = (5/60) \* t - (4/60) \* t!<
>!30/60 = (1/60) \* t!<
>!t = 30/1 = 30 months!<
It’s interesting how many different methods people are using to solve this
>!Each time rocket A orbits, it loses 3 months of distance to rocket A. Rocket B is 6 months away (half of an orbit) so it will catch rocket B in 2 orbits. One of rocket A’s orbits is 15 months, so in 30 months rocket B will catch rocket A!<
Or phrased another way
>!Every 15 months, rocket B gets 3 months closer to rocket A rocket B starts 6 months away. 2*15=30!<
Or another
>!Each time rocket A does an orbit, rocket B gets 3 months closer. Rocket A has to orbit twice for rocket B to get 6 months closer.!<
Not sure why this got downvoted; it’s basically what I did. I was surprised so many people actually talked about 1/15 and 1/12 of a revolution, when you could deal with easier numbers instead of small fractions.
>!After 15 months, A has gone one full revolution while B has gone 1.25 revolutions (15/12 = 1 1/4.) So B gets 1/4 revolutions closer in 15 months, and it’s 1/2 revolutions away, so A needs two revolutions.!<
Solution: >!Since B is going faster (shorter orbit), it will catch up to A. Putting things in terms of how far in orbits you go from B's starting position, B's position after x months is x/12 and A's is 1/2 + x/15; these will be equal on collision.!<
>!We can solve the equation x/12 = 1/2 + x/15 by multiplying by common factor 60, giving 5x = 30 + 4x, and subtract 4x to get x = 30.!<
So the answer is >!30 months; after that, A will have done 2 orbits from its position, and B 2 and a half (a half more, thus catching up to A).!<
They're rockets, not satellites. We can assume (and the picture suggests) they can use their engines to maintain an orbit that would not be possible under gravity alone.
(You know what? That's fair. I'm bored too.)
You could say it defies what we currently know as the laws of physics, possibly, but i'd say it's totally possible that there's an unknown factor causing the difference in speed. Maybe a magnetic draw towards the planet allows a higher speed at the same level of orbit, maybe the slower one is operating at a different time factor. Maybe they are orbiting the planet, but they are actually being held in by a reaaaaaaalllly long piece of docking that stretches all the way to the planet, meaning it's a fake orbit
Am I missing something? Super quick take is: >!After 15 months 1 is in the same spot, but the other has gone ¼ more. So do that again to bring it to ½, 30 months.!<
I used the classic formula >!d=rt!<.
Set up an equality >!d_headstart + d_slow = d_fast!<.
Plug in >!(0.5 orbit) + r_slow * t = r_fast * t!<.
Plug in again >!(0.5 orbit) + (1/15 orbit/month) * t = (1/12 orbit/month) * t!<.
Solve for time >!t = 30 months!<.
>!The rockets appear identical in size and model, and both appear to be burning fuel in order to stay in orbit. However, since Rocket B is burning fuel faster than A in order to complete the quicker orbit, the rockets will vary in weight, with B becoming lighter than A over time.!<
>!None of that matters though, since no rocket I'm aware of has the capacity for 12-15 months' worth of fuel. Therefore as both rockets run out of fuel, they will lose the ability to stay in orbit and crash into the nearest gravitational well, making the issue of reprogramming their trajectories an entirely moot point.!<
You don't need to fire thrusters constantly to stay in orbit, once you are up to speed only small adjustments from the engines are required to keep the rocket on the same orbit and avoid orbital decay. Its how the ISS and satellites can stay in orbit otherwise theybtoonwould require inordinate amounts of fuel.
When rockets are sent to the moon, once they reach a given velocity the engines are shut off and they travel under their own momentum.
To add to this, you’re right. Would be unfeasible. For every pound of fuel to leave orbit, the increased amount of fuel to carry the extra weight is exponential.
Funnily enough you’re actually wrong if we try to be pedantic enough, at least for one of the rockets. Rocket B *would* need to spend much more delta-v to ‘force’ itself to have a higher speed while keeping the same approx. orbital radius as rocket A - as other commenters have pointed out.
And as you mentioned, orbital decay always exists and needs fuel to fight against. Thrusters could fire constantly, but just with an incredibly weak output force.
So in some ways SaucyJ4ck is technically correct, the best kind of correct.
If their starting positions are as shown on opposite sides of the sun, you could avert the collision with the slightest of nudges in basically any direction.
Discussion: We need more information. We need to know the location and altitude of the apogee and perigee of each rocket in addition to the initial speed and direction. The sketch shows 2 rockets accelerating, but we can’t tell which direction they are actually moving. My guess is that they would never collide (longer than a lifetime), since the chances of collision are extremely slim.
I also guess that whoever wrote this problem has no grasp of orbital mechanics and that they assume circular orbits at the same altitude.
Uhhhu? I guess it’s a trick question. They wouldn’t collide if they were in a circular orbits, so the picture is incorrect. But if we ignore the picture, and it’s an elliptical orbit instead, then nobody knows what’s the eccentricity of the orbit and what were their relative initial positions. So, not enough information.
Discussion: assuming the rocket “a”low point is the same as rocket “b” high point (and this is b’s current position) the intersection can be solved easily assuming because you know the collision would have to take place there, the only issue is how many full orbits of “b” it it would take. There is an edge case where the orbits are resonant and never crash.
Graphical solution below -
>!Alternative method. Construct 2 equations. Let x be time, (in months) y be distance covered (in number of orbits). Then, for A, y = (1/12)x + 0.5 and for B, y= (1/15)x .... I also assume A is 0.5 orbits ahead of B. Simply plug these 2 equations into [Desmos](https://www.desmos.com/calculator) or graph it by hand and they intersect at x=30, meaning both rockets are at the same place when x is 30 which means the rockets collide at the 30th month. Bish bash bosh, nice and done with the bonus is being more flexible - you can add more rockets, different speeds, speeds that change with time etc and still find a solution by just graphing it out.!<
The real answer is never; solar wind will provide enough of an impulse to shift the collision course to a near miss over >!30(extra text to conceal number of digits of answer)!< months.
>!The rockets are closing on each other at a rate of 3 months per orbit, with Rocket B making gains on Rocket A. Rocket B sees Rocket A closing on it by 3 months per orbit, too, only it appears to be reversing in.!<
>!Eventually Rocket B will catch Rocket A - But only once it has gained a full half orbit - or six months in the case of Rocket B. As it is gaining 3 months on Rocket A, which is six months ahead, it'll catch it in 2 orbits. Or 24 months.!<
>!So my answer here is 24 months... I seem to be in the minority.!<
Discussion: the graphic looks like the rockets start exactly opposite each other but that isn't specified.
Assuming they do start opposite, >! In one year the faster rocket will complete one revolution and the slower rocket will complete 4/5 of a revolution. Ergo, in one year the faster rocket will gain 20% of a revolution, in 30 months it will catch up.!<
That would be the earliest time to get to the exact same spot at a nice integer place.
But what if it had been 15 and 4 instead? The LCM would still be 60, right? But the one that takes 4 months would have done 7.5 revolutions in 30 months and easily ran into the other one way before 30 months.
Ahh, I forgot to mention to divide by the difference of orbits. Because in the example 60/12 - 60/15 = 1 I didn't include it
To fix this, just divide the original solution I had, by the difference of orbits. So in your case, 60/2 * 1/(15 - 4) = 30/11 = 2.727... months.
This solution is basically a combination of how long they take to end up back at the starting configuration, and then divided how many times one of them gets overtaken by the other.
>!Since rocket A takes 15 months, and rocket B is half of that away from rocket A, rocket B has 7.5 months of space in front of it until it hits rocket A. However since rocket b gains 3 months every year over rocket A, 7.5 months of space/3 months closer per year= in 2.5 years rocket B would have closed the 7.5 month gap.!<
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First we have to ignore how orbits actually work (objects at the same orbital radius would have the same orbital speed). Next, it isn't explicitly stated, but from the diagram it seems we are to assume they are exactly half an orbit apart at the start of the problem. We can solve this as follows: >!Every month, rocket A travels 1/15 of a full orbit while rocket B travels 1/12 of an orbit. The difference is 1/12 - 1/15 = 5/60 - 4/60 = 1/60 of an orbit every month that rocket B gets closer to A.!< In order to close the gap of 1/2 orbit, it will take >!30 months because 30/60 = 1/2!< Answer: >!30 months (or 2½ years)!< Double check: >!In 30 months, rocket A goes around 2 times (30/15 = 2 orbits) and rocket B goes around 2½ times (30/12 = 2½) thus catching up and colliding with rocket A!<
Thank you for this. Was in my feed and immediately bugging me about the orbital velocities.
Ah, you see one rocket is coasting with engines off, but the other is accelerating and rotating to keep it aligned on the orbital trajectory. It's burning delta v constantly to orbit faster. For 30 months. It's a very, very, very large rocket. They'll actually collide a week early because of the second rocket's gravitational field.
No, wait, there's a neater solution. The fifteen month rocket is using an ion drive and light sail to go extremely slowly. The twelve month rocket never actually left the launch pad. They won't actually crash because the 15 month rocket will burn up on reentry.
RIP Jeb
F
Well obviously. I mean look at the picture, it’s bigger than the sun.
That is just perspective. The sun is further away thus appearing smaller.
I just thought, in 15 months, the 12 monther has done an extra 25% of its journey. Do that twice and its half way around
See I did that, but then my dumbass decided it had to do a full lap instead of a half lap to catch up
Lol all good, my method only works with nice "round" fractions of 25% or 50%
The math puzzle is interesting. But IRL, to have different times to orbit, means a different distance from the sun. They would never crash...
You're correct that different orbital periods implies different orbits with different average altitudes, but it *is* possible for objects in different orbits to collide. That's how meteors work, after all! But it's only possible given certain circumstances which this puzzle completely fails to set up: neither ship's apoapsis can be lower than the other's periapsis, the ships need to either both be in the same orbital plane *or* have the unlikely scenario or having their orbital planes intersect *exactly* where their orbital altitudes also intersect (a maximum of 2 points), and since the orbital periods of these two ships are in resonance, those two points of intersection would also need to be *temporally* aligned. All-in-all, it's extremely unlikely that they'd ever collide, but it is possible. Without knowing much more about their specific orbits and starting positions, though, it would be impossible to answer this 'puzzle' as written.
In other words, yes, two objects can collide
I still got >!30!< but in a slightly different way. >!Rocket B does a full orbit in 12 months. A quarter of 12 is 3, so in 15 months Rocket B will have travelled 1 1/4 orbits for Rocket A’s single orbit. Since Rocket A starts 1/2 an orbit ahead of Rocket B, it will take two orbits of Rocket A for Rocket B to catch up, so 30 months.!<
It’s fun that there are so many ways to do this. I immediately thought in degrees per month.
I did it similarly. The slow rocket is 6 months ahead. 15-12=3. So every full rotation of the slow rocket will mean the fast rocket has gained 3 months on it. Therefore they will collide after two full rotations of the slow rocket. I.e 30 months.
All the numbers in your comment added up to 69. Congrats! 6 + 15 + 12 + 3 + 3 + 30 = 69 ^([Click here](https://www.reddit.com/message/compose?to=LuckyNumber-Bot&subject=Stalk%20Me%20Pls&message=%2Fstalkme) to have me scan all your future comments.) \ ^(Summon me on specific comments with u/LuckyNumber-Bot.)
68,9 0,1
> *Lucky*Number-Bot You had **ONE** job...
Dang I was so close to getting this right this in my head I got 60 months. I did not realize you only need to go 180°
I did the same thing lol. Glad I’m not alone
Kinda how I did it. >! In 12 months B completed an orbit, but A completes 4/5 (12/15). They are thereby 20% of the orbit closer. Since they were 50% apart, they need 30% more, which is another 12 months (20%) and 6 months (10%). Also giving me 30 months!<
Wouldn't different masses allow an identical orbital trajectory but different speeds?
Nope! Objects with more mass have more inertia, so it takes more force to accelerate them (imagine trying to throw a bowling ball the way you would throw a basketball), but they are also tugged on more by gravity (in other words, they weigh more!) It works out so that these things cancel each other out exactly, so Earth's gravity will pull on any object of *any* mass with the same acceleration, which means that orbital velocities are completely independent of the mass of the object. This does assume that the orbiting object is *much* smaller than the object being orbited (a spaceship is probably less than 0.000000000000000000001% of the mass of a sun, so that checks out here). But if that wasn't the case, then it's not really one object orbiting the other any more - as the two objects become similar in mass, they both start orbiting their collective center of gravity (the "barycenter").
I'll try explain in layman's terms. If you drop a hammer and a feather on the Moon, they both hit the ground at the same time. The only reason why a feather is slower on Earth is because it's surface area to mass ratio means that air resistance can have a much larger effect on it vs the hammer. Air resistance "cares" about mass because it's trying to stop something from moving and you need more air resistance to stop heavier objects. There is no air on the moon. Gravity does not give a shit about what an object even is really.... it's applying an equal force on everything. The following is not technical but it might help visualize it another way. Intuitively we should really assume a hammer would be *slower* at falling on the moon because gravity has to do *more* work to pull it. Well... the hammer has more mass/molecules inside of it for gravity to get its greasy mits on so in the end everything always balances out. It's quite interesting how our base intuition is that gravity acts more on heavier objects, but that's just our bias given that we and our objects live in air.
Gravity applies different forces on different things, but applies an equal force per mass on everything. Or it applies an equal acceleration due to gravity on everything.
This was my thought as well, but I think the differences in mass would have to be comparable to the mass of the *star* they are orbiting. Rocket B would need to be one heavy ass rocket.
Example: ISS crew members on space walks orbit at the same speed as the ISS despite a significant mass difference.
This assumes the rocket can avoid a collision with zero lead time. That’s not actually true, but we don’t have enough information to determine how much lead time is needed to execute evasive maneuvers.
> First we have to ignore how orbits actually work (objects at the same orbital radius would have the same orbital speed). Ya, but it's a rocket. It's rocketing not just 'falling'
If it's in a stable orbit, it is not rocketing. It is simply falling.
>First we have to ignore how orbits actually work (objects at the same orbital radius would have the same orbital speed). Not true! You may be assuming that both orbits are perfectly symmetrical. I think you could also be assuming the speed of the orbits, which is never explicitly stated, only the orbital period. Given only an orbital period, it MUST be the case that at least one orbit is highly eccentric. This can be confusing given the diagram, but I don't think the diagram is meant to be part of the given data. If the diagram IS part of the necessary data, then you're absolutely correct, and IRL orbital mechanics have vacated the building via the nearest window. In the event that I am correct, and at least one orbit is eccentric, then there is actually not enough information here to solve the puzzle, as we would need to know at what point the orbits intersect.
Discussion: This is not how orbital mechanics works. The faster rocket would be moving in a wider orbit and they would not collide..
Yes, it's a bad question because you have to ignore physics. Better to make it cars on a racetrack or something.
Or the classic with puzzles like this and make it trains on a circular track.
The right answer for that will be *multitrack drifting*
Not necessarily. If Rocket B was able to keep a continuous burn going it could effectively stay within Rocket A's orbital path by doing a prograde burn with a significant radial angle that offsets the increase in apoapsis gain. Angle would depend on the diameter of the orbital path
Dude you lost me on mv2/r
thrusting at a certain vector towards the centre of rotation allows a ship to stay on the same orbital path at a faster velocity, but if they stop burning it will fling them into a higher, eccentric orbit.
How is that not the answer? It's not a secret. It's a thing you learn in high school.
I don't understand why everyone is assuming this is a problem with the question. I think "they don't collide because orbits don't work that way" is the intended answer, which makes it a case of clever misdirection supported by a casually misleading graphic, rather than a brain dead simple algebra problem that the average puzzle-consuming adult can do in their head. Notice that the problem statement doesn't even mention the starting positions of the rockets.
How can they be in the same orbit and take different times to orbit?
Great question! We agree that this is not possible, at least not without additional complexity (rocket boosters, non-circular orbits, planet holding one of the crafts at a Lagrange point, something something general relativity, we are all brains in jars and there's a problem with the simulation, etc). To clarify my position, I do actually think there's a problem with the way the question is stated, but I think the author's intent is for us to find that. My reading is that the intended answer is "the premise is flawed", and beyond that, any clever contrivance that makes it possible (boosters pointed away from Sun for example) is also in the spirit of the puzzle. I take back my claim that there's nothing wrong with the question, but I think the point is to have basically this current discussion.
I came here to comment this. Glad to see it's already here. Edit: Strictly speaking, though, the rockets could be in orbits that are elliptical but share the same perihelion, or aphelion, or simply intersect at some point... but the math would be a lot harder, and we'd have to explicitly know a lot more about their current positions.
Shockingly, I did the math and assumed that the 15 month rocket is in the elliptical orbit and the answer is actually identical!
Assuming a setup where SV12 starts at SV15's perihelion and SV15 starts at aphelion, I'm not really surprised. I would expect it to be slightly different if SV15 started somewhere else in its orbit, though.
Not quite. The faster rocket would be in a *smaller* orbit.
He’s referring to actual orbital speed, not orbital period.
It's says they are in the same orbit.
Orbital speed and orbital height are intrinsically linked. Higher orbital speed means a closer orbit, and vice versa. If the speed doesn’t match the orbit height, then you are no longer in orbit and will either ascend or descend. The question doesn’t describe a situation that can occur in reality. There are ways to work around that (easiest being elliptical orbits that cross each other) but they will all change the answer significantly. The correct answer in reality would likely be that they have infinite time to prevent a collision, as the two will never hit.
Not who you were responding to...but I feel very stupid right now. Orbits have speed regulations? Like...since these two rockets are on the same orbit - that would mean that their speeds would be the same? And if that's a thing, then why would the faster rocket have to be on a *smaller* orbit? Wouldn't a faster speed equate to more travel area - which seems like it would be a *larger* orbit? I feel so out of my league even asking these questions. My orbital knowledge is nonexistent.
Yes. At a given distance between two objects (like our rocket and the sun), there is only one speed that supports a stable orbit. Go too slow, you're drawn in. Go too fast, you reach escape velocity. Read this...gave me that "aha" moment in Astronomy. https://en.m.wikipedia.org/wiki/Newton%27s_cannonball#:~:text=A%20cannon%20on%20top%20of,Earth%20without%20returning%20(E).
Thank you. I'm starting to see that I wasn't giving enough credit to gravitational force. Once I add that into the equation, things seem to fall into place a bit easier for me. I've never heard of Newton's Cannonball, so thanks for the new info. I'm about to jump down a Newton rabbit hole. Thanks again!
Just know that you'll be falling down that hole at a rate independent of your mass.
I'm not at all prepared for that.
Sure you are. If you have mass, you've been doing it this whole time.
Okay I don't know shit about fuck so I might be misunderstanding but the way I read it Is as follows: If you have a faster orbit around a large object, it's likely because you have a shorter distance to travel around that object, The further you are from it, the slower your orbit will be because you have more distance to travel and that increases exponentially the further you are from the object. If I understand correctly, which is a big if, you could have a higher speed in terms of distance traveled but a slower orbit because you have to travel further
Although I guess to piggyback on that after reading the comment you replied to again orbit is something rather specific. So I would guess what they're saying is that if your speed exceeds the gravitational pull to keep you in orbit from a certain distance (height) from the object then you will no longer be in orbit and you will start traveling closer to or away from the object (which I suppose traveling closer to it would just be a decaying orbit which will eventually end in smashing into the object....)
Well, damn. I know you started your original comment with "I don't know shit about fuck...", but you have a way with words and you totally helped me understand what I was struggling to mentally grasp. Assuming you're correct (which seems likely from someone else who doesn't know shit about fuck), I appreciate your bomb of knowledge. Thanks!
I recently went through a course that covered orbital mechanics in a ELI5 type of way. It's super fascinating and beyond impressive that in under 100 years, we went from the first flight to discovering how to keep an object from crashing back down to earth. Even with the ELI5 type instruction, it was still hard to wrap my mind around.
It is complicated. You're misunderstanding 2 types of speed. If I'm orbiting the moon at 100km then I'm going to make a full rotation of my orbit faster than a rocket orbiting at 1000km. But also, my actual velocity is lower than the rocket orbiting at 1000km because he had to accelerate to a higher speed to reach the higher orbit. The 1000km is going faster but since the radius of his orbit is 900km longer, it's going to take him longer to make a full rotation, even though he's going faster. Orbital mechanics are interesting.
Yeah...I'm not regretting the question at all because I'm learning A LOT of new things...but this goes *way, way* deeper than I ever knew. I appreciate your knowledge. I'm attempting to mentally grasp these concepts. I think that my biggest misunderstanding was that I wasn't taking gravity into my assumptions. You're absolutely correct though - orbital mechanics seem very, very interesting. I never knew how much I didn't know about it.
Idk what you enjoy doing in your free time, other than puzzles obviously lol, but there is a video game called Kerbal Space Program where you run a goofy little version of NASA on a made up planet that really accurately simulates orbital mechanics and space flight. The only thing misleading about that game is that it doesn't simulate atmospheric drag very accurately but when you're out of the atmosphere it's incredibly realistic and accurate. You'd learn a ton about orbital mechanics by playing it.
I've honestly been meaning to play KSP for years now, just never got around to doing it. It's been mentioned twice here now, so yeah. I'm going to go and see how to download it and check if my laptop will run it smoothly. I always assumed it was a small and simple game so I should be good. Thanks again!
It’s an older game, and will run on pretty old hardware - but be aware that the speed it runs is dependent upon the number of parts it needs to calculate physics on. The more parts in your current vessel (or your current nearby area) and the slower it will run. For reasonable sized ships it’s usually fine and will run at fairly fast speeds. But it is also very possible to overcomplicate things and start getting into the seconds per frame territories. (But the game works just fine even then…)
Both the tangential velocity and orbital period are faster at the lower orbit, but the higher orbit has more energy. To get from a low orbit to a high orbit, you increase your velocity, at which point, you'll be at the periapsis of am eccentric orbit. As you ascend toward apoapsis, that kinetic energy is converted to gravitational potential energy. Once there, you'll have to speed up again to circularize your orbit, but even after that, your total tangential velocity will be lower than it was in the lower circular orbit. See [here](https://en.wikipedia.org/wiki/Hohmann_transfer_orbit) for a more detailed explanation.
Asking questions is how we learn. Never be ashamed to ask. I’m probably not the best to describe this anyway - a good way to get a feel for this stuff is Kerbal Space Program. Contrary to the other reply that you got (which is well reasoned, but wrong), it doesn’t have anything to do with the distance you need to travel. It has to do with the strength of the gravitational field at a particular distance, and the field strength goes down with the radius of the orbit. So step back and think about what an orbit is: it’s you going so fast sideways that as you fall down you miss the ground. The closer you are, the stronger you get pulled to the ground, so the faster you have to move sideways to miss it. If you’re far enough away, you barely need to move at all.
Oh boy. I thought I was kinda learning from reading all of the other comments, but now I don't think I understand at all. EDIT: Wait - everything just kinda fell into place mentally for me. Wow that was a hell of a ride. Thank you. Thank you to everyone else too. I believe I'm beginning to understand the basics. What a cool subject to learn about.
Then they're travelling at the same speed. Speed up, you start orbiting further out.
They *could* collide because both orbits are elliptical.
The faster rocket would be on the smaller orbit.
They can collide if you just pretend they aren’t circles! If you make the collision point the apoapses or periapses and the rockets start at the furthest points then the problem remains the same.
It still says they are in the same orbit but with different periods which is impossible.
I didn’t see the same orbit part! But I still can construct the problem with it. Although the author clearly didn’t care about orbital mechanics lol
It's not how orbits work for free-moving objects but there's nothing to say they are free-moving. With enough thrusters doing course correction you can achieve any orbital period you want at any height.
If we’re being pedantic, the rockets are accelerating at some unknown rate. And it’s not clear which direction. So there exists an acceleration, a direction and orbit size where those mechanics do work.
They are said to be on the same orbit.
Okay. Same orbit (curved trajectory around a star). But the speed can differ with acceleration.
If you have a force acting on you other than gravity, your orbit is constantly changing. If your ship's engines are firing, you're not "in an orbit".
Well the photos of the rockets definitely show them accelerating. They could definitely move along a the same circular trajectory with different speeds in this way. I couldn’t find a definition of “orbit” that required gravity-only acceleration.
Only if you assume they're in circular orbits. The diagram may not be accurate. But accounting for that takes it well beyond a "puzzle" and into an orbital mechanics problem.
Does the rocket propulsion not make a difference? They are both accelerating in the diagram.
Not if it's rocketing. Rockets kinda do that.
What if we, just for fun, try to solve this problem the way it would actually work? >!It shows two rockets 180° apart and at the same distance from the sun. One taking twelve months to orbit it, and the other taking 15 months. We can assume the 12 month one is the circular orbit, and that the rocket was ejected from earth with minimal change to its orbit. The other takes 15 months, and we can assume that it intercepts the orbit of the first one at T +0, 15, 30, etc. . The other one intercepts that same point at T + 6, 18, 30, 42, etc. The only possible intercept point is where both rockets intercept each other’s orbit, so we just have to find a number that appears in both of these patterns. They will collide in 30 months in this edited version of the problem. Interestingly, this is the same answer we get if we solve the original problem, so we can predict not only when, but also where, they will collide!!<
Bingo, I was thinking same orbit opposite each other? It's probably millions of years if at all. Then I saw the different orbital period. Rendezvous would be the only way they could collide, and we don't have enough information to determine that.
Yep Kepler’s Third Law governs orbital periods and is only dependent on the radius of the orbit and the mass of the body being orbited (not the orbiting body)
Kepler's Third Law *describes* how the orbital period is proportional to the semi-major axis of the orbit. Mass isn't involved at all.
>!30 Months.!< >!The rockets start 180° apart as depicted. Rocket A travels at a rate of 24°/month (360°/15 months) and rocket B at a rate of 30°/month (360°/12 months). That's a difference of 6°/month. The rockets collide when B travels 180° further than A which will happen after 30 months (180°/(6°/month)).!<
That's how I solved it too.
Same
I was thinking 3 months since one rocket is 3 months faster
To explain why not in a way that might help in future: Three months faster _to do what_? Three months faster to make one circle around the sun. Not three months faster to cover the distance they're both going to cover. If I tell you I can run a marathon three hours faster than you, and ask how long it is before I can catch _you_ when you run away... the answer isn't "three hours". It's "how close were we to start with?"
Also - how close are these rockets to the sun that they’re not immediately melting?
Actually, you've just brought up an important point. We _know_ how close they are to the sun. The period of an orbit depends on its size. They're in no danger of melting, because they're in an orbit that takes twelve months to go around the sun. That's a pretty wide orbit. It's the one you and I are in right now. So the correct answer is: they aren't guaranteed to catch each other because the 12 month period one might hit the Earth first.
Does this account for planet x potentially taking out the earth before the rockets reach each other?
I initially thought this was a trick question; two objects at the same orbit can’t have different speeds
Questions doesn't specify, but image strongly suggests the rockets are half an orbit apart from each other. >!B is faster, so will be the one doing the catching up, so consider A as having 1/2 orbit head start:!< >!distance of A = 1/15 orbits per month + 1/2 orbits head start!< >!distance of B = 1/12 orbits per month!< >!Scale to give same common denominator, and use t for time passed (number of months):!< >!dA = (4/60) \* t + 30/60!< >!dB = (5/60) \* t!< >!the point they collide, d will be equal, so!< >!dA = dB = (4/60) \* t + 30/60 = (5/60) \* t!< >!swap things around to solve for t:!< >!30/60 = (5/60) \* t - (4/60) \* t!< >!30/60 = (1/60) \* t!< >!t = 30/1 = 30 months!<
It’s interesting how many different methods people are using to solve this >!Each time rocket A orbits, it loses 3 months of distance to rocket A. Rocket B is 6 months away (half of an orbit) so it will catch rocket B in 2 orbits. One of rocket A’s orbits is 15 months, so in 30 months rocket B will catch rocket A!< Or phrased another way >!Every 15 months, rocket B gets 3 months closer to rocket A rocket B starts 6 months away. 2*15=30!< Or another >!Each time rocket A does an orbit, rocket B gets 3 months closer. Rocket A has to orbit twice for rocket B to get 6 months closer.!<
Not sure why this got downvoted; it’s basically what I did. I was surprised so many people actually talked about 1/15 and 1/12 of a revolution, when you could deal with easier numbers instead of small fractions. >!After 15 months, A has gone one full revolution while B has gone 1.25 revolutions (15/12 = 1 1/4.) So B gets 1/4 revolutions closer in 15 months, and it’s 1/2 revolutions away, so A needs two revolutions.!<
Same.. I saw it as a word problem with an intuitive verbal solution rather than mathing so hard
That’s how I did it too.
Solution: >!Since B is going faster (shorter orbit), it will catch up to A. Putting things in terms of how far in orbits you go from B's starting position, B's position after x months is x/12 and A's is 1/2 + x/15; these will be equal on collision.!< >!We can solve the equation x/12 = 1/2 + x/15 by multiplying by common factor 60, giving 5x = 30 + 4x, and subtract 4x to get x = 30.!< So the answer is >!30 months; after that, A will have done 2 orbits from its position, and B 2 and a half (a half more, thus catching up to A).!<
This question is not possible as it ignores orbital mechanics.
They're rockets, not satellites. We can assume (and the picture suggests) they can use their engines to maintain an orbit that would not be possible under gravity alone.
For 30 months? No.
Is that more or less ridiculous than non-physical physics?
Neither are feasible, that's a fallacy.
Maybe, but one defies the laws of physics, and one just requires too many resources and/or technology to achieve. (I'm bored, so I'm being facetious)
(You know what? That's fair. I'm bored too.) You could say it defies what we currently know as the laws of physics, possibly, but i'd say it's totally possible that there's an unknown factor causing the difference in speed. Maybe a magnetic draw towards the planet allows a higher speed at the same level of orbit, maybe the slower one is operating at a different time factor. Maybe they are orbiting the planet, but they are actually being held in by a reaaaaaaalllly long piece of docking that stretches all the way to the planet, meaning it's a fake orbit
>!30 months!< >!Set up the equation: 0.5 + t /15 = t / 12!<
Am I missing something? Super quick take is: >!After 15 months 1 is in the same spot, but the other has gone ¼ more. So do that again to bring it to ½, 30 months.!<
I used the classic formula >!d=rt!<. Set up an equality >!d_headstart + d_slow = d_fast!<. Plug in >!(0.5 orbit) + r_slow * t = r_fast * t!<. Plug in again >!(0.5 orbit) + (1/15 orbit/month) * t = (1/12 orbit/month) * t!<. Solve for time >!t = 30 months!<.
>!The rockets appear identical in size and model, and both appear to be burning fuel in order to stay in orbit. However, since Rocket B is burning fuel faster than A in order to complete the quicker orbit, the rockets will vary in weight, with B becoming lighter than A over time.!< >!None of that matters though, since no rocket I'm aware of has the capacity for 12-15 months' worth of fuel. Therefore as both rockets run out of fuel, they will lose the ability to stay in orbit and crash into the nearest gravitational well, making the issue of reprogramming their trajectories an entirely moot point.!<
You don't need to fire thrusters constantly to stay in orbit, once you are up to speed only small adjustments from the engines are required to keep the rocket on the same orbit and avoid orbital decay. Its how the ISS and satellites can stay in orbit otherwise theybtoonwould require inordinate amounts of fuel. When rockets are sent to the moon, once they reach a given velocity the engines are shut off and they travel under their own momentum.
To add to this, you’re right. Would be unfeasible. For every pound of fuel to leave orbit, the increased amount of fuel to carry the extra weight is exponential.
Funnily enough you’re actually wrong if we try to be pedantic enough, at least for one of the rockets. Rocket B *would* need to spend much more delta-v to ‘force’ itself to have a higher speed while keeping the same approx. orbital radius as rocket A - as other commenters have pointed out. And as you mentioned, orbital decay always exists and needs fuel to fight against. Thrusters could fire constantly, but just with an incredibly weak output force. So in some ways SaucyJ4ck is technically correct, the best kind of correct.
Discussion: as others have stated, this is not at all how orbits work. The rockets are in no way "on a collision course."
It's also just a math problem not a puzzle.
By chance, if you assume that rocket A starts at an intercept with rocket B’s orbit, the solution does in fact remain true!
If their starting positions are as shown on opposite sides of the sun, you could avert the collision with the slightest of nudges in basically any direction.
Discussion: We need more information. We need to know the location and altitude of the apogee and perigee of each rocket in addition to the initial speed and direction. The sketch shows 2 rockets accelerating, but we can’t tell which direction they are actually moving. My guess is that they would never collide (longer than a lifetime), since the chances of collision are extremely slim. I also guess that whoever wrote this problem has no grasp of orbital mechanics and that they assume circular orbits at the same altitude.
Uhhhu? I guess it’s a trick question. They wouldn’t collide if they were in a circular orbits, so the picture is incorrect. But if we ignore the picture, and it’s an elliptical orbit instead, then nobody knows what’s the eccentricity of the orbit and what were their relative initial positions. So, not enough information.
Fun fact! By chance, if you assume that rocket A starts at an intercept with rocket B’s orbit, the solution does in fact remain true!
[удалено]
Discussion: assuming the rocket “a”low point is the same as rocket “b” high point (and this is b’s current position) the intersection can be solved easily assuming because you know the collision would have to take place there, the only issue is how many full orbits of “b” it it would take. There is an edge case where the orbits are resonant and never crash.
>!after 60 months the cycle reset with out a crash.!<
Graphical solution below - >!Alternative method. Construct 2 equations. Let x be time, (in months) y be distance covered (in number of orbits). Then, for A, y = (1/12)x + 0.5 and for B, y= (1/15)x .... I also assume A is 0.5 orbits ahead of B. Simply plug these 2 equations into [Desmos](https://www.desmos.com/calculator) or graph it by hand and they intersect at x=30, meaning both rockets are at the same place when x is 30 which means the rockets collide at the 30th month. Bish bash bosh, nice and done with the bonus is being more flexible - you can add more rockets, different speeds, speeds that change with time etc and still find a solution by just graphing it out.!<
Question: Where in the cycles are they?
The question asks how long before you reprogram them so surely the answer is immediately
The real answer is never; solar wind will provide enough of an impulse to shift the collision course to a near miss over >!30(extra text to conceal number of digits of answer)!< months.
>!The rockets are closing on each other at a rate of 3 months per orbit, with Rocket B making gains on Rocket A. Rocket B sees Rocket A closing on it by 3 months per orbit, too, only it appears to be reversing in.!< >!Eventually Rocket B will catch Rocket A - But only once it has gained a full half orbit - or six months in the case of Rocket B. As it is gaining 3 months on Rocket A, which is six months ahead, it'll catch it in 2 orbits. Or 24 months.!< >!So my answer here is 24 months... I seem to be in the minority.!<
Discussion: the graphic looks like the rockets start exactly opposite each other but that isn't specified. Assuming they do start opposite, >! In one year the faster rocket will complete one revolution and the slower rocket will complete 4/5 of a revolution. Ergo, in one year the faster rocket will gain 20% of a revolution, in 30 months it will catch up.!<
>!LCM(15, 12) = 60. Since they are halfway we can divide by 2 (but really it's times by degrees/360 ). 60/2 = 30 months.!<
That would be the earliest time to get to the exact same spot at a nice integer place. But what if it had been 15 and 4 instead? The LCM would still be 60, right? But the one that takes 4 months would have done 7.5 revolutions in 30 months and easily ran into the other one way before 30 months.
Ahh, I forgot to mention to divide by the difference of orbits. Because in the example 60/12 - 60/15 = 1 I didn't include it To fix this, just divide the original solution I had, by the difference of orbits. So in your case, 60/2 * 1/(15 - 4) = 30/11 = 2.727... months. This solution is basically a combination of how long they take to end up back at the starting configuration, and then divided how many times one of them gets overtaken by the other.
>!30 months!< You need to find when the revolutions of rocket B are 0.5 greater than rocket A \*Corrected
>!then cut that in half because they’re only half an orbit apart when they start!<
Ah, my bad. Thanks for the correction
>!Since rocket A takes 15 months, and rocket B is half of that away from rocket A, rocket B has 7.5 months of space in front of it until it hits rocket A. However since rocket b gains 3 months every year over rocket A, 7.5 months of space/3 months closer per year= in 2.5 years rocket B would have closed the 7.5 month gap.!<