if two identical orbital paths were at an angle to eachother, and two particles which were initiated at one of the intersections were given different velocities, then what would have to be their relative velocities for them to collide at an intersection? and what would have to be their relative velocities for them not to collide. any formulas for this?

Originally posted by Julixa
if two identical orbital paths were at an angle to eachother, and two particles which were initiated at one of the intersections were given different velocities, then what would have to be their relative velocities for them to collide at an intersection? and what would have to be their relative velocities for them not to collide. any formulas for this?

Lots of formulas. But which one's you use depends on a lot of factors.


A big factor is where in the orbit the velocity is changed. If it changed at the one of the intersection points, then the particle will return to that intersection point every orbit, it will just have a different orbital period. Depending on its new period compared to the period of the other period, eventually the two will find themselves back at the intersection point at the same time. How soon depends on how even the ratio of periods are. The soonest it could be is if the new ratio is 2 to 1, then the particles would collide after two of the shorter orbital periods.

If the velocity change is done while the particlea are not at the intersection point, then just about any large enough change in velocity in either will make it so that they will never collide . (The orbits won't intersect.)

the assumption was that the particles velocities were initiated at one intersection starting out in close proximity (collision) it seems rather obvious that if the orbital velocities were ane even or rational multiple of eachother that would meet at that intersection again. however what if the velocities were an irrational multiple of eachother ? would the ever meet? is there a formula for either of these situations?

What you're looking for is something called "incomensurable" - not sure of the spelling - but for the purposes here I thing having velocities that are relatively prime should be sufficient. Of course, if these are not point particles, then you're just about guaranteed a collision because they will eventually get close enough even with relatively prime velocities.

The question may be answered easier by visualizing an analogous system, an analog clock. Flatten the two orbits on top of each other to make a clock face. The hands of the clock represent the particles, which I’ll assume are point particles. Initially have both hands point to the same number, just as both the particles begin at one intersection. This number is a collision point. On the opposite side (6 hours later) is the other collision point, which I’ll ignore for this analysis.

Suppose the initial intersection is at 12 o’clock. Now ask, at what multiple x of the slower hand’s rate must the faster hand’s rate be, such that they will collide again at 12 o’clock? Or never collide?

We know that a clock’s hands meet at 12 o’clock every 12 hours. So one x that results in a collision is 12; that is, when one particle orbits 12 times faster than the other. Now we ask, is 12 a special integer? No, it could have been any other integer, such as 24 if it were a military clock. The numbers 12 and 24 are arbitrary integers. So any integer x results in a collision.

What if x is a rational non-integer? If x were 1.5, then in (1.5 * 2) = 3 revolutions of the big hand, the faster-moving hand, the hands would collide at 12 o’clock. The number 1.5 is an arbitrary rational number, and the number 2 is the integer that yields an integer when multiplied by x.

Any rational x results in a collision, which occurs at y revolutions of the faster-moving particle, where y is the numerator of x. For example, if x is 4/3, then y is 4, and the collision occurs after 4 revolutions of the faster-moving particle.

What if x is an irrational number? We know that y must be an integer; only a whole number of revolutions brings a particle back to its starting point for a potential collision. When x is irrational, is its numerator an integer? By definition it cannot be; an irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q. So any irrational x results in no collision.

Take this with a grain of salt though as I’m no math whiz.

your analysis of the problem was well laid out and quite coherent. However it seems to prove that rational multiple orbita velocities always collide, however it does not necessarily prove that irrational relative velocity ratios do not ever collide. Am I correct?

You’re right; I didn’t prove that irrational relative velocity ratios never collide. I think I can though:

What if x is an irrational number? If x is (a rational) 4/3, say, then the faster-moving particle makes 4 revolutions while the slower-moving particle makes 3 revolutions. For a collision to occur, x must be expressible as a fraction p/q for any integers p and q, because only an integer number of revolutions for both particles brings both particles back to their mutual starting point for a potential collision. By definition an irrational number cannot be expressed as a fraction p/q for any integers p and q. So if x is irrational then the particles never collide.

your elaboration made a good case for non rational relative velocities as not being inherent in the formulas for rational relative velocities, and therefore you concluded that there are no other formulas or approaches that will prove or show that irrational relative velocities cannot allow collision or coincidince, whatever the term we are using. Is there room for my argument?

I’m not sure what your argument is, so I’ll summarize it this way:

If x is irrational then the particles never collide.

If x is rational, expressed as a fraction p/q for any integers p and q, then the particles always collide after p revolutions of the faster-moving particle.

These conclusions are proven above (not a formal mathematical proof of course).

you have convinced me. thanks for that great analysis.

Is it just me, or is there a subtle mistake with this summarization?

Just because two numbers are both irrational does not mean that their ratio is not rational.

For example, pi and 2*pi are both irrational (as far as we know) right?

So if one particle had a period of pi seconds, and the other, a period of 2*pi seconds, they would collide

Good point; however, it was the ratio being rational or irrational that was analyzed. That is, x mentioned above is the ratio.

ahh, my bad

Just a bit of basic mechanics here, but...
If you increase the angular velocity of a particle, but keeping it in an orbit of the same radius as the first, then the force towards the centre of the orbit must change also, as F=(mv^2)/r.
What exactly is keeping these particles orbiting?

Nothing. I think it's meant to be just a thought experiment with (mathematical) point particles that start from the same point.

it might help for him to think of these points or marbles as orbiting inside of a large hollow sphere