### Video Transcript

Which of the following most correctly describes how the drag force exerted by a fluid on an object moving through the fluid varies with the speed at which the object moves through the fluid? (A) The drag force is proportional to the square root of the speed. (B) The drag force is proportional to the square of the speed. (C) The drag force is proportional to the speed. (D) Below a certain speed, the drag force is proportional to the square root of the speed, but above the speed, the drag force is proportional to the speed. And (E) below a certain speed, the drag force is proportional to the speed, but above the speed, the drag force is proportional to the square of the speed.

In this example, we’re thinking about an object that is moving through a fluid. An example of this could be, say, a submarine moving through water. Whenever an object moves through a fluid, there is a drag force on that object. This is a force, we’ll call it 𝐹 sub 𝐷, that opposes the motion of the object. The drag force is caused by friction.

In this example, we want to identify the correct relationship between the drag force on an object moving through a fluid and that object’s speed through the fluid. If we call the speed of an object moving through a fluid 𝑣, then answer option (A) can be summarized as saying the drag force is proportional to the square root of object’s speed.

Option (B) claims that the drag force is proportional to the square of the speed 𝑣. Option (C) says that the drag force is proportional to the speed 𝑣. And option (D) said that below a certain speed, the drag force is proportional to the square root of 𝑣, but above that certain speed, the drag force is proportional to 𝑣. And then of course, we have answer choice (E) to keep in mind as well.

It’s not so unusual to have some sort of practical experience of the drag force, for example, any time a person is riding quickly on a bicycle or, say, traveling in a car with the windows down with their hand out the window. In each of these cases, we experience a drag force, a resistance to forward motion on an object moving through a fluid. It’s hard though to get an intuitive sense for how the drag force relates to the object’s speed mathematically.

It turns out that if an object is moving relatively slowly through a fluid, that is, if the fluid flows smoothly past the object, then under those smooth flow conditions, drag force is indeed proportional to the object speed 𝑣 through the fluid. This ends up not being true though for all speeds 𝑣. If the speed of our object increases enough so that the fluid now flows past the object, not smoothly, but turbulently, then under those conditions, the drag force is proportional to the square of the speed.

Notice something interesting about this. It means that if we’re moving fairly slowly through a fluid and the flow is smooth, it won’t take much additional energy to overcome the drag force, which at that point is proportional to our speed and add to our forward speed. On the other hand, if we’re already moving fast enough that the flow around us is turbulent as we move through a fluid, then to overcome that drag force and increase our speed by some increment takes much more energy. We find then that it takes less energy to speed up at slower speeds, and more energy to speed up at higher speeds.

In any case, this description of the drag force is matched by answer choice (E). Below a certain speed, and that speed depends on both the object and the fluid the object is moving through, drag force is proportional to speed, but above this speed, the drag force is proportional to the square of the speed.