Colliding Missles

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stopmadnessnow
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Colliding Missles

Postby stopmadnessnow » Tue Mar 22, 2016 7:20 am UTC

[math]You've probably seen similar puzzles here, but this precise one isn't present. It was created by Martin Gardner, the annotater to Annotated Alice.

Two missiles are set on a collision course. They start 5,323 miles apart. One travels at 21,000 mph, the other at 9,000 mph. Fortunately the missiles were programmed to miss each other but this reprogramming only took place a minute before impact. How far away were the missiles from each other?

Spoiler:
You have to forget how far they were apart when they started. You need to find out how long each missile travels in a minute-the minute before the explosion.

I worked it out as 205 miles per minute and 105 miles per minute (how many 60s in each number) and got the answer that they were 310 miles apart. But, no. The answer is 500 miles. What am I doing wrong?
[/math]

EDIT: I think I'll re-edit this once the madness is over.
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Carlington
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Re: Colliding Missles

Postby Carlington » Tue Mar 22, 2016 11:33 am UTC

Unless the numbers in your post are all addled, the missiles are approaching each other at the sum of their velocities, which is twenty-five times ten to the third power mph. If the distance between them shrinks by twenty-five times ten to the third power miles every hour, then this number divided by sixty will be the distance covered every minute. I think that number is four hundred and sixteen and two-thirds (if that gets filtered, twelve-hundred and forty-eight divided in thirds). I'm not sure where 500 comes from.
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Re: Colliding Missles

Postby Nitrodon » Tue Mar 22, 2016 2:38 pm UTC

You seem to have made some arithmetic errors in dividing by 60. The correct speeds are 350 and 150 miles per minute, which do indeed add up to 500.

The comma in 9000 (along with the initial distance, which is either 1323 or 5323) allowed the filters to change the first digit.

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Re: Colliding Missles

Postby ThirdParty » Mon Mar 28, 2016 10:58 pm UTC

I agree with the other posters that a minute before collision, the first missile is (21,000 miles per hour)*(1 hour per 60 minutes)*(1 minute) = 350 miles from the point of collision, and the second is (9,000 miles per hour)*(1 hour per 60 minutes)*(1 minute) = 150 miles from the point of collision. However:

We've been told that the missiles were "set on a collision course", but have not been told the angle at which they were set to collide. It matters. If they're on opposite paths (i.e. set to have a nose-to-nose collision), they are (350 miles) + (150 miles) = 500 miles from each other. If they are on identical paths (i.e. set to have a nose-to-tail collision), they are (350 miles) - (150 miles) = 200 miles from each other.

More generally, the distance between them a minute before collision can be found using the Law of Cosines. If θ is the angle of collision, the distance a minute before collision is ((350 miles)2 + (150 miles)2 - 2(350 miles)(150 miles)(cos θ))0.5 = 100*(14.5 - 10.5(cos θ))0.5 miles.

(All that assumes, of course, that they're following linear trajectories. Since their starting distance from one another implies that they are ICBMs, this is not a good assumption; it is more likely that they are following elliptical trajectories, which further complicates the situation. However, "500 miles" will still be the upper bound.)

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Re: Colliding Missles

Postby stopmadnessnow » Wed Mar 30, 2016 4:50 pm UTC

Let's just say they meet each other head-on.
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Re: Colliding Missles

Postby LaserGuy » Wed Mar 30, 2016 5:58 pm UTC

Well, you really ought to consider the ballistic trajectories rather than straight line ones. If they start at different latitudes, then you should probably also take into account the Earth's rotation. Getting two missiles to collide in midair is actually really hard. Without adjusting course midflight, you would expect them to miss.

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Re: Colliding Missles

Postby Xanthir » Thu Mar 31, 2016 7:25 pm UTC

Orrrrr, since this is a math puzzle and not a rocket engineering exercise, you take the ordinary assumptions that things generally travel in straight lines and obey Newtonian physics in a flat Euclidean universe.
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Re: Colliding Missles

Postby LaserGuy » Thu Mar 31, 2016 9:57 pm UTC

Xanthir wrote:Orrrrr, since this is a math puzzle and not a rocket engineering exercise, you take the ordinary assumptions that things generally travel in straight lines and obey Newtonian physics in a flat Euclidean universe.


Sure, you can do that, but then you'll get an answer that's patently absurd. Just getting this problem to have a solution that is not "both rockets crash into the ground long before they meet" already requires quite a bit of handwaving if you aren't willing to treat their paths properly.

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Re: Colliding Missles

Postby Xanthir » Thu Mar 31, 2016 10:03 pm UTC

It's not patently absurd, because this isn't a rocket engineering exercise. It's a math puzzle.
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Re: Colliding Missles

Postby Sizik » Thu Mar 31, 2016 10:38 pm UTC

Assume the missiles are travelling in empty space in the absence of a gravity well.
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Re: Colliding Missles

Postby stopmadnessnow » Fri Apr 01, 2016 4:43 pm UTC

stopmadnessnow wrote:You've probably seen similar puzzles here, but this precise one isn't present. It was created by Martin Gardner, the annotater to Annotated Alice.


Martin Gardner never studied rocket science, as far as I know.
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Re: Colliding Missles

Postby ThirdParty » Fri Apr 01, 2016 6:05 pm UTC

Xanthir wrote:since this is a math puzzle and not a rocket engineering exercise, you take the ordinary assumptions that things generally travel in straight lines and obey Newtonian physics in a flat Euclidean universe.
It would still be a math puzzle if we tried to figure out the answer for elliptical paths rather than linear paths. It would just be a more interesting math puzzle.

(It's not like we'd need to go out and do experiments to find out how ellipses work. Calculus should suffice, but mine's a little too rusty to be up for the task so I'm hoping someone else will do it.)

Above I gave the answer for what will happen if the rockets are traveling in straight lines; their distance a minute before collision will be 100*(14.5 - 10.5(cos θ))0.5 miles, where θ is the angle at which they are set to collide with one another. I don't think there's anything more to say about the linear case, is there? If not, then why shouldn't we look for a more general solution that doesn't assume linearity?


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