Cannon Duel
Science brain teasers require understanding of the physical or biological world and the laws that govern it.
Two identical cannons are aimed at each other, one from the top of a cliff and the other from the ground below. They fire their cannons simultaneously at exactly the same speeds. Assuming they are in range of each other and ignoring wind resistance for a moment, will the cannon balls hit each other or miss?
Answer
They will hit each other.
The cannon balls start out in a straight line towards each other, but then each ball is pulled off this line by the force of gravity. However, each ball will fall from this line of fire by the same amount as the other, and so a collision is inevitable. Inevitable, as noted, unless we take wind resistance or the cannons are out of range of each other.
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Comments
jimbo
Jun 05, 2004
 Great teaser. I don't know about the logic part, though. We can sometimes deceive ourselves with apparently logical suppositions. However, three quarters of a page of calculus has left me with the surprising conclusion that you are absolutely right! Loved it! 
humoeba
Jun 05, 2004
 This is a cool teaser! Shouldn't it be in science though??? 
ThisIsLynne
Jun 05, 2004
 I liked this one. I knew that gravity would have some effect, but didn't know what or how much. The answer was very interesting, I thought. 
fishmed
Jun 08, 2004
 When it was being editted, i requested it be moved to science, but it was apparently voted by editors to still be logic. 
rose
Jun 08, 2004
 this is a nice one...got some food for thought....though i didnt find the answer i enjoyed trying to figure out the answer 
canu
Jul 08, 2004
 This is not a problem of logic, but of math, and I don't agree with the explanation given. A cannon does not fire in a straight line to its target, but along a parabola (assuming no air resistance). Since the two cannons are at different altitudes, but the muzzle velocity of the projectiles is the same, the two travel along different parabolas. 
fishmed
Jul 08, 2004
 You are right about that, but based on their locations and their aim, those parabolas would intersect, wind resistance not withstanding. 
I_am_the_Omega
Jul 13, 2004
 Just doesn't seem right to me. Wouldn't gravity be working for the descending ball, and totally against the ascending ball? I think they'd miss, given enough distance between them (but still within range). I certainly can't do the math, I'm 17 and haven't taken any calculus.. 
fishmed
Jul 13, 2004
 That would seem to be the case, but it would pull each one down from their direction, which is towards the other cannon and as such, they would still meet. It would pull the lower one closer to horizontal and the upper one closer to vertical. Ignoring wind, the gravity affects them both the same and so at some point, as long as they are not out of range and are fired at the same time, they will intersect. Maybe not exactly in the middle, but I saw the written demo and I was surprised too. 
biztycl
Jul 30, 2004
 I disagree with the logic too. They will definitely not hit each other because of different altittude when they crisscross each other. One will fly over the other even if they are on the same straight line. 
fishmed
Aug 11, 2004
 I will go back to where I found it and see if I can find the explanation again and PM anyone who disagrees. I thought they would not intersect either, but the explanation given showed me otherwise. 
shandela
May 02, 2005
 It doesn't take calculus to figure this out, just simple algebraic maipulation of some physics equations. I didn't do the math but I think it would work out, not counting the resisstance. The greatthing about physics calss was everything occured in an airfrictionless world. 
fishmed
May 03, 2005
 And most parts of a frictionless world would be cool. Not all, but most. 
curtiss82
Aug 31, 2005
 I did the algebra, and this does work out. If anybody would like to see the algebra, I'll post it. Simply stated, if there were no gravity, the cannon balls would stay on the line of sight and hit. And with gravity, they are pulled off the line of sight by the same amount since gravity only acts in the vertical direction, thus they still hit. 
fishmed
Sep 03, 2005
 Thank you curtis. Glad I have backup. 
vikingboy
Sep 29, 2005
 curtis, I would disagree in that gravity is overcome by speed. Thus, the downward ball would retain more speed and be affected less by gravity than the upward ball. Saw this on a Roadrunner cartoon, the balls missed! 
fishmed
Sep 30, 2005
 Vikingboy. Do you really think that Roadrunner or Coyote do the math before confirming whether the cannonballs would hit or not?
Though contreversial, this does work. Glad you liked it nonetheless. 
curtiss82
Sep 30, 2005
 vikingboy, once you take an advanced mechanics class, as I have, you'll understand the physics that make this teaser possible. Get a physics book and look in the chapters under constant acceleration, you'll see that gravity will act on an object the same regardless of it's initial velocity. 
achnid
Nov 21, 2005
 I would have to dissagree with this one. Personaly I belive if done right math can prove almost anything.
The big variable is how much of a hieght difference and how much of a horizontal difference. assuming both are within line of sight of eachother. The one thats lower will have to fire a lot higher then where the other cannon actually is to even hit it.while the one thats up way higher would have to aim lower then the other cannons trajectory to hit.
the last thing is most older cannons didn't have very high accuract rates. It would be more of a general aim and hope to hit. 
beegum
Dec 12, 2005
 The qualifier "within range of each other is inadequate i think...Since they are pointed at each other neither is assured to be within range...it may be possible to fiddle with the angle and get a case in which both balls cross the ground level before colliding 
Warrior_Poet
Jan 19, 2006
 If we neglect wind resistance, there are always two parabolas which will result in hitting a target, except for the rare condition of firing at exactly 45 degrees, whose complement is also 45 degrees. This, along with the flawed logic in the answer leave me to rate it poorly. 
dantak
Jun 17, 2006
 I agree that it is flawed, but I do not claim that they do not collide. I think that the initial velocity is crucial. And for you physics geeks I mean velocity in the true sense(speed and direction). For instance, if the speed were great enough the top cannon could fire at the lower cannon in a near straight line (presumably) and the lower cannon could lob a high altitude shot at the higher cannon. Clearly, these paths do not interesect. Also, bear in mind that it is not simply a matter of intersecting parabolas, but at what each cannonball is at the intersection. 
woooo
Jan 30, 2008
 what about the recoil? won't that affect the cannon? 
vg674
Mar 30, 2008
 If yo don't believe the math, take the problem and change it a bit. Place one cannon directly above the other (assume it hovers there to make it simple) and then shoot. They will hit eachother Take the cannon on top and slowly slide it in one direction while still maintaining the parameters of the puzzle. It still works. Because both inputs are the same throughout each launch, the should collide with eachother. 
myhalcyondaze
Jun 05, 2008
 The answer is correct. It sounds like some people are confused as to where the cannons are aimed. The teaser isn't stated incredibly clearly, but the two cannons are pointed directly at each other (along the straight line connecting both cannons). They are NOT pointed in such a way that the parabolic trajectory of their respective cannonballs will hit the other cannon. For example, if the teaser is changed so that the two cannons are on the same level ground, the cannons are pointed level to the ground. Obviously, if just one cannon were fired, it would not hit the other because the cannonball decreases in altitude as it travels. Hope this clarifies the answer. 
BOBdaMAN
Aug 09, 2011
 i have this riddle its so easy 
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