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  Grandpa -

My friends and I have been debating about what happens when you fire a gun straight up in the air.  Some of us think the bullet will come down as fast as it went up, and others think it's safe when it comes down.  Can you help us settle this.  The Fourth of July is coming soon.

- Thanks - Kimberly

 

Kimberly -

Different guns have different "muzzle velocities."  That's the speed that the bullet is going when it leaves the barrel.  Many high-powered rifles have muzzle velocities that are 3000 feet per second or more.  That's more than 2000 miles per hour.

When such a gun is fired straight up into the air, this is what happens:

The bullet leaves the gun at 3000 feet per second.  There are two things now that slow the bullet down as it goes up.  One is gravity - which slows the bullet 32 feet per second every second.  If this were all that slowed it down, then after 10 seconds, the bullet would still be going (3000 - 320), or 2680 feet per second.  This is not even close to the true velocity after 10 seconds,  because air friction is slowing the bullet also.  That friction is much greater while the bullet is going fast.  In the first second, it may slow the bullet by 1000 feet per second.  Then in the second second, maybe another 500, and then less and less, until finally, the gravitational slowing is more significant than the friction.  The bullet will continue to slow down until it stops at the top.

Now let's talk about when the bullet heads back down:

The same two forces are present, but now the gravity part is speeding up the bullet - friction is still trying to slow it down.  Gravity will be speeding up the bullet 32 feet per second every second, but friction will be increasing more and more as the bullet speeds up, and this friction will subtract from that 32.  There is a certain velocity when the friction is equal to the gravity in the opposite direction.  At this point, the frictional acceleration is 32 feet per second every second upward (or -32), while the gravitational acceleration is 32 feet per second every second downward (+32).  From now on, the bullet will have no acceleration at all (32 - 32 = 0).  When this happens, the velocity of the bullet is called its "terminal velocity."  The bullet will continue at the same velocity until it hits the ground or some object.

This terminal velocity is different depending mostly on two things.  1. The density of the thing that is falling.  2. The shape of the thing that is falling.  Bullets are very dense (usually lead), and they have a shape to allow relatively low friction.  The terminal velocity of a feather may be only 2 feet per second because of its shape and density, but the terminal velocity of a bullet could be 300 or 400 feet per second.  This could be about 300 miles per hour.

Now about your debate: 

1. That 400 feet per second is much slower than the 3000 feet per second that the bullet had when first fired. If you or a friend argued that the bullet is going slower on the way down, then you or that friend was correct on the physics.

2. It is still dangerous.  A three hundred mile per hour piece of lead can still do harm.  There is no question that it is not as effective as a bullet fired directly at something.  The bullet that is fired at something has another very important feature, however, a feature the bullet fired into the air does not have.  It was aimed - there was an intended target.  The bullet fired into the air falls randomly and could hit anything.  If it hit someone right on top of the head, it's possible that it could kill.  It's going about three times as fast as a professionally hit tennis serve, and it's much smaller than a tennis ball, and harder - more penetrating.  When you triple the speed, the energy goes up nine times (it's proportional to the square of the speed), and I'll bet there's no-one who wants to be hit by a tennis ball with nine times the energy of a fast serve.  Then, because the bullet is small and hard, it is hundreds of times more penetrating than that tennis ball would be.  Put a tennis ball in your palm and put twenty pounds on top of it.  No problem.  But now put the blade of a screw-driver on your palm and put twenty pounds on that.  That will hurt.  Same twenty pounds - smaller cross section - more penetrating.  If you or a friend argued that the falling bullet is dangerous, then you or that friend was correct on that.

3.  Ultimate winner of the debate:

If you or a friend argued that the bullet falls much slower than the muzzle velocity, is still dangerous, but not as dangerous, then you or that friend won the debate and had everything right. 

 

Bottom line:  Grandpa does not recommend firing guns straight up into the air, but he would rather have a hundred of these than just one aimed and fired at him.  Guns are particularly dangerous in our country, because we take them far too much for granted.  They are familiar to us, and we have too much comfort concerning them.  They are often treated as toys.  Sometimes very large children play with them.

 

Note:  If this gun were fired into a vacuum - no air - then the bullet would fall to the same speed as it left the muzzle.  Many times in physics classes, friction is ignored.  This is because the problems become much more difficult when friction is included, and the principles can be taught easier with this assumption.  The assumption is, of course, false.  When we do real physics in the real world, we cannot ignore friction.  Our results would be ridiculous.  For example, in this vacuum, the feather would fall as fast as the bullet.  You would be very surprised on the moon if you held a feather in one hand and a rock in the other and then dropped them at the same time.  Both would hit the ground together.  Though Grandpa would expect this result, it would still be a surprise to see it.

 

- Grandpa

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