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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?
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Here, we have four graphical answer options: (A), (B), (C), and (D).
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Each one shows us a plot of drag force on the vertical axis against speed on the horizontal axis.
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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.
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Imagine, for example, we have some submerged object, like a fish, and this fish moves through the fluid with some speed.
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When it does, there is a drag force, due to friction, that opposes the motion of the fish.
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And it turns out that this drag force doesn’t always depend on the fish’s speed in the same way.
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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.
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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.
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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.
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What we found then is that for relatively lower speeds, the drag force is proportional to that speed.
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This means we would expect that as speed increases, the drag force linearly increases with it.
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Notice that three of our four options display a linear relationship between drag force and speed at low speeds.
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In answer choice (D), we don’t see this relationship at relatively low object speeds.
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And so we know not to choose this as our final answer.
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So, at low speeds, drag force is proportional to speed.
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But then as we saw, this changes once our object moves past some certain speed.
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That speed will depend on the object in the fluid it’s moving through.
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But once the object is moving that fast or faster, the drag force is now proportional to the speed squared.
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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.
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This describes the relationship between drag force and speed at relatively high speeds.
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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.
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Notice that graph (B) over here shows us a linear relationship all through.
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That means that can’t be correct because it doesn’t account for the drag force being proportional to speed squared.
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Likewise, graph (C) shows us the drag force leveling out as speeds increase.
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We expect rather that there’ll be sloping upward as shown in our 𝑌 equals 𝑋 squared graph.
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We won’t choose answer choice (C) then.
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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.
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This means that answer option (A) is indeed showing us that drag force is proportional to speed at relatively lower speeds.
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But then as the speeds increase past some certain value, that drag force is proportional to speed squared.
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For our answer, we’ll choose answer option (A).
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This correctly shows how the drag force exerted by a fluid on an object moving through the fluid varies with the object’s speed.