Stardate 20030417.1942

(On Screen): John Hawkins asks my opinion of this piece which proposes the idea of using pennies from heaven as a cheap battlefield weapon. I'm afraid it's not practical. (Another great idea slain by ugly facts.)

Different objects have different terminal velocities as they fall. Terminal Velocity is where the force of gravity is balanced by air resistance. The force of gravity is pretty much a constant for any object (it increases by a negligible amount as the object falls), but air resistance rises significantly as the object increases its speed. Eventually they balance, and the net force on the object is zero, which means it ceases to accelerate.

For sky divers, it's about 140 miles per hour. Once any falling object reaches terminal velocity, whatever it is for that object, it won't get any faster no matter how far it falls. In the case of skydivers they reach terminal velocity in less than 1000 feet, even when they're jumping from 10,000 feet up.

The value of terminal velocity for any given object is a function of its mass and its shape (and the characteristics of the atmosphere of whatever planet you happen to be on). If the object is heavy and streamlined it's going to be really fast. For a feather, it's a less than a foot per second. Drop feathers from five miles up and it's going to take them a long time to hit the ground, and they're not going to do anything beyond annoy people once they do.

The impact kinetic energy of an object is proportional to the square of the velocity, so if you're relying only on the kinetic energy to cause damage (for instance, with a concrete bomb) and not on release of chemical energy from explosives in the projectile, then it's important that it be going as fast as possible when it hits. Throw a bullet at someone and they'll get angry when it bounces off. Shoot it out of a gun, and they'll die when it penetrates. And that's why the muzzle velocity of a gun is much more important when it's firing a sabot round than when it's firing HESH or HEAT.

There are two problems with dropping pennies from high altitude. First, they're going to tumble, which will make them drift. Drop them from 4 km up and they'll land over an area a kilometer across. But worse is that because of their shape and their density, with little mass and a large cross-section even when edge-on, they're not going to have anything like an acceptable velocity to actually seriously harm anything. John's picture of pennies embedded in a corpse is a fantasy; the most it's going to do is cause a slight bruise.

It would be about like being caught in a hail storm with cherry-sized hail. It hurts like hell. It doesn't cause much actual damage and no single such hail stone is capable of killing. The reason is the same: hail that size also doesn't achieve a high enough velocity to actually pack enough of a punch to do that kind of damage.

Hail has a better shape than a penny but less density. It's more or less spherical. As to ball bearings, it depends enormously on how large they are. Unlike a penny, a metal sphere has a moderately reasonable shape, though it's far from being ideal.

But again it's a question of air resistance versus the force of gravity. The air resistance is proportional to the cross section of the sphere, which rises as the square of the diameter. But the mass, and thus the force of gravity, rises as the cube of the diameter.

Which means that the terminal velocity for spheres of different sizes, all made of the same kind of metal, will be different as a function of their sizes. The big ones will have a higher terminal velocity than the small ones. Drop BBs from 4 km up and you've just created yet another artificial simulation of a hail storm, doing about as much damage. I don't have the math to calculate the actual size they'd need to be, but I think you're probably talking about objects the size of a cannon ball before they'd actually be able to achieve a high enough velocity, and thus kinetic energy, to actually start to really damage things. (Among other things, it depends on where you set the threshold for "damage".) Generally speaking, larger objects are going to cause more damage than small ones will.

Of course, a ball is still not really an ideal shape, and if they're just dropped loose there's still going to be a tendency for them to tumble, to start to spin. It's not going to be rotational around the axis of motion, as if it had been fired out of a rifled weapon aimed straight down. Rather, it's likely to be turning around some other axis, pumped by the air resistance, and that's going to make it curve like a pitched baseball, or a ball fired out of a smoothbore musket. Drop a huge number of ball bearings from 4 km up and they won't scatter as badly as pennies or feathers, but they're going to spread out a lot by the time they hit the ground. And the smaller they are, the further they'll spread and the less damage they'll do.

John suggests balls weighing about 1 pound. I'm not sure that's large enough, but there comes a point where you have to ask just what it is you're trying to accomplish. Remember, jets are expensive and each sortie is a risk and they don't have infinitely large capacity to carry tonnage. Are you better off carrying 50,000 pounds of metal sph