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Q & A: bullets dropped and fired

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Q:
We just finished a 100km charity walk in which much time was spent discussing the outcome if a bullet was dropped and fired from exactly the same height above the earthís surface which would hit the ground first. Can you please clear up a 24 hour discussion.
- Jason Smith (age 35)
Sydney, Australia
A:
There are several levels to answer your question at, depending on how much precise realism you want vs. how much working out the implications of simple physical laws.

First, let's think of the simple case where there's a uniform gravitational field and no air friction. The downward gravitational acceleration is determined completely by the gravitational field, so firing the bullet exactly horizontally (presumably what you mean, otherwise the answer is easy!) has no effect on the vertical motion, and hence no effect on the time to fall.

Now letís consider a level of complication. The Earth is a round ball. As you go sideways, you start to get farther from the surface, so you have farther to fall. That effect is enhanced a bit because the gravitational field also gets weaker as you get farther away. So that would make the bullet fired sideways take longer to fall. If it were fired fast enough (around 27,000 km/hr) it would actually be in orbit, and not fall to the surface, ignoring air friction.

Now letís quit ignoring air friction. For a simply dropped bullet, that slows the fall. I guess, since the friction grows faster than linearly in the velocity, that horizontal motion increases the vertical friction as well, so the horizontally fired bullet would fall slower than the dropped one.

It seems that both these effects lead to the fired bullet falling more slowly, so I guess the combination of them does the same.


Real bullets these day are not spherical, so the orientation affects the air friction. If the fired bullet is oriented differently, that might change the result we reached, but if it does thatís too complicated for me to figure out reliably now. Anyway the fired bullet would be long-wise horizontal, which would probably also tend to increase the falling time, so Iíd bet that it hits the  ground a little later.

Mike W.

It's conceivable there might even be a little bit of aerodynamic lift possible in a bullet going so fast.  They come out of guns spinning on their axes, and large forces and torques are applied during the time the bullet accelerates in the gun.  If the bullet during its flight points its axis not quite along its axis of travel, the lift could point up or down.  Of course the average expected value of the lift is zero if the system is symmetric, but even a tiny random flaw in the bullet can cause some nonzero lift, and if there's a crosswind, the Magnus effect may be not very small either.  For these reasons you don't expect every bullet fired from the same gun pointed in the same direction to land in the same spot every time; there'll be some scatter.  It wouldn't surprise me if some of these effects were bigger than the gravitational force on the bullet.

Tom

(published on 08/29/06)

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