Friday, September 21, 2007

Terminal Velocity

I wrote a while ago about the paper on how physics is handled in Hollywood blockbusters and the limited lessons that they can give. Strangely enough, in the current online issue of The Answer Man (September 20, 2007) at the famous film critic Roger Ebert's website, there was a rather interesting lesson on terminal velocity between a person and a bullet after both have left an airplane.

Q. It is foolish of me to wonder about the physics of a movie that contains skull-piercing carrots and bullet-propelled merry-go-rounds, but in "Shoot 'em Up" would there be any point to shooting down at Mr. Smith when he is falling from the plane? He should be traveling at terminal velocity and wouldn't the bullets also be going that fast, too? Hence, they couldn't catch him?
Alex Kincade, St. Joseph, Mich.

A. According to, "If an object falls with a larger surface area perpendicular to the direction of motion, it will experience a greater force and a smaller terminal velocity. On the other hand, if the object fell with a smaller surface area perpendicular to the direction of motion, it will experience a smaller force and a greater terminal velocity." A skydiver has a larger surface area than a bullet; also, the skydiver is falling, but the bullet is propelled by an explosive charge.

While certainly the cross-sectional geometry would dictate a amount of drag force exerted on the object, one also cannot ignore the mass of the object here, no? For example, take 2 object of the same shape by different mass. While they would fall at the same rate in vacuum, the one with the large mass has a larger gravitational force, and requires a larger drag force (i.e. at a higher velocity) to cause it to reach a terminal velocity.

But the other aspect of it is also interesting. While it is true that the bullet was propelled out of the gun (i.e. initial velocity is some value) while the person fell out of the plane (initial velocity is approximately zero), would the bullet still reaches the same terminal velocity? I say it would, because if it is moving faster than the terminal velocity, the drag force is larger than the gravitational force on it, and will slow it down until they both are equal. So whether the bullet will catch up to the person or not depends how far that person has started his fall.

In other words, what happened as described in the movie isn't impossible physically. But it isn't necessarily what can happen all the time since a few other facts are involved.

So what do you think? Did I analyze this correctly myself?



Anonymous said...

if I had the option of being shot on the street or being shot while falling down very fast I would obviously take the later choice as not only are the factors you raise important - but when I am standing the body of mine will absorb all of the bullets mass but when in free fall the bullet will not enjoy such power and hopefully, since silk used to be used for bullet proofing and there is silk and other strong fibers in a sky divers suit you will be much less inclined to be hurt this way by a bullet, the shooter would also have to recalculate the aim depending on the rate of speed of the plane since again the mass of the bullet and it's shape is going to travel differently when hit by a ? 300 mile per hour side steam of air but is dense but the jumper is so different in this situation that ya hollywood is full of it

Mason barge said...

Let's give the person a terminal fall rate of 150 feet per second. The type of gun is not mentioned and is very important. Also important is how long it has been since he jumped.

If you shot an AK-47 at a man who had jumped from a plane 3 seconds earlier, the man would have fallen less than 200 feet. The bullet's muzzle velocity would be over 2000 fps. (An M15 would be almost 5000 fps, iirc.) Given the very heavy density and near-optimal drag coefficient of the bullet, the impact of the bullet would be reduced by about 5%.

In other words, this scenario would be almost the same as being shot while standing in a field. Of course the airplane would have traveled farther forward than the man, etc.

With much longer delays (say 10 seconds) and this time using a handgun with a muzzle velocity of 800 fps, let us imagine that the bullet actually has time to slow to terminal velocity of roughly 200-250 fps, giving it a bit extra since the spin will hold it in a superior aerodynamic profile. With the man traveling at 150 fps, it would hit him at something like 50-100 fps, which is in the marginal range for penetration. It could fail to penetrate (although it would sting, especially if it were large caliber) or it could penetrate and do some damage. A serious injury or fatality would be extremely unlikely, a preposterous fluke, such as if he were turned sideways and it managed to cut a juglar vein.

The real problem here is that the likelihood of hitting him with a bullet which has reached terminal velocity is practically 0. Trained gunners can hardly hit another airplane at shorter distances; and here, you have to add in Wind gusts that will affect a terminal velocity bullet greatly, and differently than the man. Not to mention that they will be traveling in different -- and in the man's case, unpredictable -- directions.

In short, if the shooter could hit the jumper, he could kill him. But even a quick shot from a high powered rifle by an expert marksman would be extremely difficult.