Friday, 3 June 2011

Cannon Balls

A cannon ball is shot out of a cannon and travels some distance before it hits the ground, at the same time a cannon ball is dropped out of the sky the same distance off the ground.
Which Cannon ball will hit the ground first?
I keep seeing this puzzle with a discussion that peters out without result. The distances involved varies, I've seen as low as 4 miles and as high as 10.

Now - a ball dropped and a ball fired horizontally from the same height will both hit the ground at the same time ... but note that they travel different distances. In this puzzle the balls travel the same distance. There is an ambiguity though: is the distance traveled by the fired ball the horizontal distance or the length of it's trajectory. Almost everyone thinks it is the former but I suspect the latter is meant.

Treated as normal ballistics, I can make the fired ball hit first by pointing the cannon directly down. The cannon and the dropped ball start at the same height in this case. I can make the fired ball come last by putting the cannon on the ground and firing directly upwards... the ball reaches a maximum height then falls to the ground, so the dropped ball starts at twice this height in order to travel the same distance. The fired ball then takes 1.4 times longer to hit the ground.

So much for normal ballistics. There are three wrinkles though:
1. air resistance may be significant. The dropped ball may reach it's terminal velocity (for a shot-put this is about 80m/s) before it hits ... over the distances involved, the dropped ball may be delayed enough that it always hits second.

2. gravity depends on height - so the dropped ball will, in general, encounter less initial acceleration. This may make it lag enough to come second.

3. the 10mi distance is suggestive - that is the top of the atmosphere (well, of the really dense part of it). At height, the atmosphere is thinned out, allowing a greater speed to be built up falling ... does the puzzle mean for us to realize that it takes place inside the dense part of the atmosphere and thus air-resistance is important or that the dropped ball starts out high enough to have a consistent advantage? I'm betting on the former.

Without actually doing the math, it looks like these wrinkles slightly favor the shot ball hitting first.

1 comment:

  1. Anonymous9/12/11 14:49

    Well Simon, is this a real site? I have never seen anything quite like this before...To me the whole article is nonsense of course including your opening statement of, "Now - a ball dropped and a ball fired..." was the accepted thought at one time, and that was when the earth was thought to be flat, but gosh, I thought that everyone by now knows that it is kinda round, except maybe you.

    Demented, who me?