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Twelve toy cars, each labeled with numbers 1 to 12, are placed on a frictionless circular track in a configuration resembling the numbers on a clock. Initially, the cars are released with a constant speed of one revolution per minute, but the direction of motion for each car is chosen randomly. Whenever two cars collide, they undergo a perfectly elastic collision and reverse their directions while maintaining the same constant speed.

Prove that at the end of every minute, the cars once again form a clock-like arrangement, albeit rotated by some angle. Also, determine the angle of rotation.

this is a problem one of my friends sent me and when i tried to solve it i couldn't find any relation between them that i could prove so i tried observing them: i wrote this python program using numpy and matplotlib you can find it here.

from my observations the rotation should be related to the number of clockwise and counter-clockwise rotating cars(and for something like $2$ counter-clockwise cars and 3 clockwise cars the rotation seems to be $\frac{2}{5}$clockwise (or $\displaystyle\frac{number\space of \space counter-clockwise\space cars}{number\space of\space all\space the\space cars}$)).

i tried assuming the cars are dots with zero length to prove that they will still construct a clock like configuration but when theyre just dots the rotation will always be zero so i think the question meant that cars actually have a length(unbelievable i know :D)

then i tried assuming two groups of cars(based on their motion direction) and we can say that when two cars of diffrent groups collide they'll each jump forward a length of $\displaystyle \frac{length\space of\space the\space car}{2}$ and the number of collisions is the number of members of the bigger group(beside the case where one group has zero members cause then there will be no collision) but i couldn't prove it mathematically.

Thanks in advance.

kn2798
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  • "chosen randomly." - with what distribution? – Henrik supports the community Jan 17 '25 at 12:56
  • @Henriksupportsthecommunity there is no specified distribution in the question so i'd assume its for a general form – kn2798 Jan 17 '25 at 13:00
  • Is my confusion about this question based upon the fact that it's a pure math question and ignores basic principles of physics? – dougp Jan 17 '25 at 21:27
  • @dougp why does it ignore basic principles of physics? – kn2798 Jan 17 '25 at 21:41
  • Revolution implies there is something the cars are travelling around. While the constant speed and the initial direction make up an initial velocity vector, the question includes no indication of an acceleration vector applied by something (gravity? a rope? friction?) that would make the cars travel in a circle. Given initial random direction, many of the cars would immediately leave the track. The ones that collide (elastically) would change direction based on the angle of impact, not simply turn 180 degrees. Then they, too, would leave the track. – dougp Jan 17 '25 at 22:10
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    Perhaps, "...the direction of motion for each car is chosen randomly to be either clockwise or counter-clockwise and each is held on the track by a string attached to the center of the circle." Maybe that's implied, but it is not stated. – dougp Jan 17 '25 at 22:13

1 Answers1

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The perfect 2-body collision acts as if the two bodies pass without any interaction except for exchanging car plates, meaning you have two rigid randomly chosen subsets rotating in opposite directions at constant angular speed

Roland F
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  • if i understand the except for exchanging car plates part correctly you mean the two cars change place in an instance, but the thing is i dont understand why should it mean their speed is constant? cause when they change plates they move more than how much they would have moved without the collision so the speed should change shouldnt it? plus if what i understand from your answer is true then the rotation should always be zero(which is not) – kn2798 Jan 17 '25 at 12:59
  • @kn2798 In an elastic collision, kinetic energy is conserved. There is no speed loss and the direction change happens instantaneously. That's why there is "an exchange of license plates". If one car was moving from twelve o'clock to one o'clock and the other was moving from one o'clock to twelve o'clock, then after the collision, one car is still moving toward twelve o'clock and one car is still moving toward one o'clock. Therefore, if a car starts at twelve o'clock and is moving clockwise, after one minute there will be a car that arrives at twelve o'clock from the same direction. – John Douma Jan 17 '25 at 13:45
  • @JohnDouma the thing is that assume a point $p_1$ and $p_2$ in the center of the two colliding cars, and the distance between each two consecutive numbers on the clock $d$ the distance each of them($p_1$ and $p_2$) moves over in 1 minute is $12d$ so in 5 seconds it'd move the distance $d$ but when the two cars collide lets say from 1 and 12 and then get back to positions 1 and 12 the distance they'll move is $d-r$ where $r$ is the radius of the car but point $p_1$ and $p_2$ move $d$ in 5 seconds so they'll move some extra space that gets them further from points 1 and 12 so they wont end up on – kn2798 Jan 17 '25 at 14:14
  • @JohnDouma the points 1 and 12 by the end of 5 seconds and similarly(ignoring other collisions that happens in the other 55 seconds) they won't end up on 1 and 12. i dont know how to prove that other collisions that happen in the other 55 seconds make sure that the cars will make a clock-like configuration – kn2798 Jan 17 '25 at 14:17
  • @kn2798 The same car doesn't travel all the way around the circle. The point is: for each car that starts in a direction, there will always be a car at the point where the unimpeded car would have been. – John Douma Jan 17 '25 at 14:22
  • @JohnDouma if we assume the cars are just dots i agree with you. but when cars have a length then i fail to see why that should be true, infact i think its false because by your reasoning then the rotation would be zero(which i dont think it is both because of the question saying the configuration of cars rotate by some degree and also from my observations i highly doubt that the rotation is gonna be zero when cars arent dots) its the exact problem i have with RonaldF's answer because the way i understand it it means the rotation is gonna be zero – kn2798 Jan 17 '25 at 14:46