### Video Transcript

Which of the following most correctly shows 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?

Here, we have four graphical answer options: (A), (B), (C), and (D). Each one shows us a plot of drag force on the vertical axis against speed on the horizontal axis. We want to identify which one most correctly shows how the drag force exerted by a fluid on an object moving through the fluid varies with the speed of that object.

Imagine, for example, we have some submerged object, like a fish, and this fish moves through the fluid with some speed. When it does, there is a drag force, due to friction, that opposes the motion of the fish. And it turns out that this drag force doesn’t always depend on the fish’s speed in the same way. If the fish is moving along at some speed — we’ll call it 𝑉 — and 𝑉 is low enough that the flow of water past the fish is smooth, then in that case the drag force is proportional to the fish’s speed 𝑉. For example, if the fish doubles its speed, say, and the flow over it is still smooth, then the drag force will double as well. But if the fish’s speed increases enough so that the flow past the fish is no longer smooth but rather turbulent, then under that condition 𝐹 sub 𝐷 is no longer proportional to the speed 𝑉 but rather is proportional to the speed 𝑉 squared.

What we found then is that for relatively lower speeds, the drag force is proportional to that speed. This means we would expect that as speed increases, the drag force linearly increases with it. Notice that three of our four options display a linear relationship between drag force and speed at low speeds. In answer choice (D), we don’t see this relationship at relatively low object speeds. And so we know not to choose this as our final answer. So, at low speeds, drag force is proportional to speed.

But then as we saw, this changes once our object moves past some certain speed. That speed will depend on the object in the fluid it’s moving through. But once the object is moving that fast or faster, the drag force is now proportional to the speed squared. If we set up a horizontal axis and call the values on this axis 𝑋 and a corresponding vertical axis with values 𝑌, then the curve 𝑌 is equal to 𝑋 squared will look something like this. This describes the relationship between drag force and speed at relatively high speeds.

Therefore, we’re looking for this shape to our curve at higher speeds in the correct graph showing the relationship between drag force and speed. Notice that graph (B) over here shows us a linear relationship all through. That means that can’t be correct because it doesn’t account for the drag force being proportional to speed squared. Likewise, graph (C) shows us the drag force leveling out as speeds increase. We expect rather that there’ll be sloping upward as shown in our 𝑌 equals 𝑋 squared graph. We won’t choose answer choice (C) then.

But notice that in answer choice (A), at relatively higher speeds, the curve does take on a shape like our 𝑌 equals 𝑋 squared line. This means that answer option (A) is indeed showing us that drag force is proportional to speed at relatively lower speeds. But then as the speeds increase past some certain value, that drag force is proportional to speed squared. For our answer, we’ll choose answer option (A). This correctly shows how the drag force exerted by a fluid on an object moving through the fluid varies with the object’s speed.