## Colliding Missles

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

Posts: 84
Joined: Thu Mar 03, 2016 10:58 pm UTC
Location: Somewhere in that country with Big Ben in it.

### Colliding Missles

[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.
For comic writing or short story writing post-haste, contact the off-the peg joke chap. He has the nous.
https://www.upwork.com/freelancers/~01110048bacb43eae6 to hire me at Upwork.

Carlington
Posts: 1588
Joined: Sun Mar 22, 2009 8:46 am UTC
Location: Sydney, Australia.

### Re: Colliding Missles

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.
Kewangji: Posdy zwei tosdy osdy oady. Bork bork bork, hoppity syphilis bork.

Eebster the Great: What specifically is moving faster than light in these examples?
doogly: Hands waving furiously.

Please use he/him/his pronouns when referring to me.

Nitrodon
Posts: 497
Joined: Wed Dec 19, 2007 5:11 pm UTC

### Re: Colliding Missles

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.

ThirdParty
Posts: 351
Joined: Wed Sep 19, 2012 3:53 pm UTC
Location: USA

### Re: Colliding Missles

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.)

Posts: 84
Joined: Thu Mar 03, 2016 10:58 pm UTC
Location: Somewhere in that country with Big Ben in it.

### Re: Colliding Missles

Let's just say they meet each other head-on.
For comic writing or short story writing post-haste, contact the off-the peg joke chap. He has the nous.
https://www.upwork.com/freelancers/~01110048bacb43eae6 to hire me at Upwork.

LaserGuy
Posts: 4585
Joined: Thu Jan 15, 2009 5:33 pm UTC

### Re: Colliding Missles

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.

Xanthir
My HERO!!!
Posts: 5426
Joined: Tue Feb 20, 2007 12:49 am UTC
Contact:

### Re: Colliding Missles

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.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

LaserGuy
Posts: 4585
Joined: Thu Jan 15, 2009 5:33 pm UTC

### Re: Colliding Missles

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.

Xanthir
My HERO!!!
Posts: 5426
Joined: Tue Feb 20, 2007 12:49 am UTC
Contact:

### Re: Colliding Missles

It's not patently absurd, because this isn't a rocket engineering exercise. It's a math puzzle.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

Sizik
Posts: 1260
Joined: Wed Aug 27, 2008 3:48 am UTC

### Re: Colliding Missles

Assume the missiles are travelling in empty space in the absence of a gravity well.
she/they
gmalivuk wrote:
King Author wrote:If space (rather, distance) is an illusion, it'd be possible for one meta-me to experience both body's sensory inputs.
Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

Posts: 84
Joined: Thu Mar 03, 2016 10:58 pm UTC
Location: Somewhere in that country with Big Ben in it.

### Re: Colliding Missles

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.
For comic writing or short story writing post-haste, contact the off-the peg joke chap. He has the nous.
https://www.upwork.com/freelancers/~01110048bacb43eae6 to hire me at Upwork.

ThirdParty
Posts: 351
Joined: Wed Sep 19, 2012 3:53 pm UTC
Location: USA

### Re: Colliding Missles

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?