Dropping food from airplanes -- without parachutes?

(minor changes made 10/21/01)
© 2001 Richard A. Muller

According to news reports, the U.S. is dropping food and medicine from airplanes, from altitudes perhaps as high as 30,000 feet. And they are not using parachutes! Isn't the food crushed from the tremendous speed of the impact?

Why not use parachutes? It is a completely new idea to drop humanitarian aid into an area controlled by your enemy. In the past, the danger has always been that the enemy military will take the food and medicine, and use it to help their soldiers. Thus, instead of helping the people, we wind up helping the oppressors.

The prior method of dropping food has been to put it into big containers, and air drop those into the cities and towns using a big parachute. But the military does not want to do that if the town is controlled by the enemy. They see the parachute, and send some soldiers to follow it until it lands. If, instead, you break the aid up into small individual packages, and disperse them widely, then the military will not have the resources to go out and retrieve the packages. In contrast, a poor family will go out and spend an hour, if necessary, to retrieve a pound of flour from down in a ravine. It is a better return on their effort than the same amount of time spent farming. Once they have that food, then the military can try to take it from them, but will usually not succeed. (The people will hide the food.)

Isn't the food crushed? The answer, which seems to surprise many reporters, is "not necessarily." The physics behind this is very interesting, and has important implications for other issues, such as the gasoline efficiency of automobiles. So let's look at the physics.

When an object falls, the force of gravity accelerates it, so it falls faster and faster. The force of gravity is called the "weight." It is typically measured in pounds, or in kilograms. But there is another force on the falling object: that of the air resistance. The force of the air becomes greater and greater as the object moves faster and faster. You can measure this yourself by putting your hand outside of the window of a car. The faster you go, the stronger is the force. Not only that, but the force increases with the square of the velocity. So if you double your speed, the force of the air increases by a factor of 4.

For physics majors only: In the MKS system, a kilogram is a unit of mass. However, in common usage, we say the "weight" of a kilogram of mass is also a kilogram. Even physics professors do this. In Europe, if you buy a kilogram of potatoes, you get potatoes that have a mass of a kilogram, which weighs 2.2 lb. The force that this kilogram exerts on the table that is holding it is, technically speaking, 9.8 "Newtons".

As an object falls faster and faster, the force of air resistance continues to increase, until it matches that of the weight. When that happens, there is no longer any net force on the object. Of course, it doesn't come to a stop. When there is no force on it, it just continues on at a constant speed.

What is that speed? We can measure it, or we can calculate it. The answer depends on how big the object is, but for small packages of food, it is about 100 miles per hour. This is called the terminal velocity.

Isn't that enough to crush food? It would certainly smash an egg. But with a little bit of careful packaging, most food would survive. A strong bag of flour would survive. 100 mph is a little faster than a baseball pitch, and a little slower than a tennis serve (when done by professionals). You can imagine a person catching something thrown at that speed, and not breaking it open, if the bag is made out of tough canvas. (And maybe the canvas bag is surounded by a little bit of padding.)

Why not use small parachutes? You could, and then the velocity of the fall would be even less. But why bother?

Unrequired calculation. For those who are interested, I'll calculate the terminal velocity for a sphere that has a radius of 10 cm. I'll assume the sphere has a density of 1 gm per cubic centimeter, i.e. it is similar to that of water. The physics equation for the force on an object depends on the shape of the object, and so we pick the sphere because the shape has a simple equation. For the sphere, the force is given by

     F = (1/2)A r v²

In this equation, F is the force, A is the area of a circle (that's what the sphere looks like to the wind: A =p r²), r is the density of air = 0.001 gm per cubic cm, and v is the velocity.

The force of gravity is given by F = mg, where m is the mass and g is the gravitational constant. For typical physics units, we will use A in square centimeters, p in grams per cubic centimeter, v in centimeters per second. For these units, g = 980 = 1000 (approximately). The object falls faster and faster until the force of gravity equals that of the air. So we take the equation above and set it equal to F = mg. This gives

     (1/2)A r v² = mg

Now we substitute m = (volume)*(density), take the density of food = 1 gram per cubic centimeter, and use volume of sphere = (4/3)pr³. This gives the following equation:

     (1/2)(p r²)r v² = (4/3)pr³g

Plugging in the numbers, and solving for the velocity v, gives v = 5000 cm/sec. Using the fact that there are 3600 seconds in an hour, 100000 cm in a kilometer, and 1.6 km in a mile, we convert this velocity, and get that it is approximately equal to 100 miles per hour.

That's fast, but not too fast. If there air weren't there, an object falling from a height of 30,000 ft would reach a velocity close to 1000 miles per hour, i.e. ten times faster. (And at 10x the speed, it would carry 100x the energy. And if the ground is hard, most of that energy will go into crushing the food.)

How does a parachute work? A parachute is light, so it doesn't add much weight. But when it opens, it is very big. So it has a very large air resistance. The air resistance is proportional to the area. So even at a relatively slow fall velocity, the air resistance is equal to that of the weight of the person using it. The person then falls at that slower velocity. (A person also falls more slowly because of his shape, which is usually not a sphere.)

Another unrequired calculation. How big a parachute do you need to fall at a velocity of 3 meters per second? If you followed the previous unrequired calculation, you realize that you just use the same equations. But now the mass is the same (the object hanging on the parachute), and the area is larger (the area of the parachute). See if you can do it, and whether the answer seems reasonable.

Nice cartoon. This was sent to me by one of our students.