Video: Forces that Resist Motion

In this lesson, we will learn how to compare friction and drag forces, and also explain how the variable drag force on an object results in a terminal velocity.

16:55

Video Transcript

In this video, we’re talking about forces that resist motion. Of all the forces that we can encounter, these are some of the most common because any time an object is in motion, these forces are present. In the case of this truck driving down the road, two of these resistive forces are involved.

To understand these forces better, we could begin with literally any moving object. Just to pick one, let’s say that we want to move this box from its current spot on the floor to another location. And say we do that by pushing it so it slides across the floor. When we do this, we notice that it takes a force in order to keep the box moving. We can’t just start at moving and then let go of it and have it continue to slide along the floor. Physically, what’s going on is that there are forces that resist the motion of this box.

The first one of these two forces is known as the force of friction. And this arises whenever there are two surfaces in contact with one another and the surfaces are moving past one another. We can say that friction is a force that resists motion. Whatever the direction of our object’s motion, friction points in the exact opposite direction. For example, if we were to switch sides and start pushing right to left on the box so that its motion was towards the left, then, in that case, the force of friction would point to the right. It opposes the way the box is moving. So whichever way our box is moving, indicated by the blue arrow, friction is a force that acts opposite that motion.

But as it turns out, friction isn’t the only force opposing motion in this scenario. We’re going to assume that this box is in an environment where there’s air. That is, there’s atmosphere. That means that there are all these air molecules, very small particles, that our box is running into as it moves from left to right. What’s essentially going on is millions and millions of tiny collisions between the box and the air molecules that’s moving out of the way. The net effect of all those collisions is also a force that opposes the motion of the box. That is, it’s a force that acts in the opposite direction to the box’s motion.

When this force is caused by air molecules colliding with an object in motion, the force is sometimes called air resistance. But there’s an even more general term for this, and it applies to any kind of fluid, whether air or a liquid such as water. The general name for this force is drag. And that name really does convey what effect this force has. Just like friction, the force of drag always tends to slow moving objects down because it opposes their motion. Now there are important differences between these two forces, even though they both are responsible for resisting object motion.

Friction occurs whenever there are two surfaces, which are in contact with one another and are in motion relative to each other. The example here is the bottom of the box and the floor it’s sliding across. Unlike friction, drag doesn’t require two surfaces that are in contact. Rather, the drag force requires a single surface to be encountering a fluid while there’s relative motion between the fluid and the surface. So these terms do indeed refer to two different forces. But it’s worth pointing out that there’s a connection between them.

If we think on a microscopic scale and start to consider the interactions of individual air molecules with our box as it moves. Then we would see that each of these collisions involves some friction between that particular molecule and the box. And as we said, it’s the cumulative or the net effect of all those many, many collisions that we called drag. So there is a sense that drag at its root is caused by friction. At the same time, though, not all of the effects of drag are due to friction. For example, if we had an individual air molecule collide with our box moving exactly in the opposite direction of the box, then some of the force that interaction would exert on the box is not caused by friction but is instead caused by changing the momentum of that air molecule as it bounces off the box.

So these two forces that resist motion are indeed distinct. We can tell the difference between friction and drag. Keeping that distinction in mind, say we hold out a baseball in our hand, and then we release it so it starts to fall to the ground. Which, if any, of these forces that resist motion does the baseball encounter? Well, we know that as the baseball falls, its surface isn’t encountering any other surfaces. So we can say that friction is not present here. But we do know that as the ball descends, it’s pushing air molecules out of the way. So it is subject to the force of drag.

But then, on the other hand, imagine a situation like this. Say that we have an inclined plane and a block on that plane. And then say that we enclose that whole system in a chamber that can evacuate the air from its interior. So the idea is, at first there’s all this air inside the chamber, like we would normally expect. But then, using some kind of pump, we pump all the air molecules outside of the chamber so that now all that’s left inside is a vacuum, along with the block in the incline. In this case, as our block starts to slide down the incline, the force of friction is acting on it because we have these two surfaces in contact, and the surfaces are in motion relative to one another. But since the block doesn’t encounter any air molecules as it slides, the force of drag isn’t present here. We’ve removed that possibility by removing the air from this chamber.

When it comes to object motion, then it’s possible for both friction and drag to be involved or simply one or the other of these two forces. Now we’ve already mentioned one way that friction and drag are not like one another. We’ve said that friction requires two interacting surfaces, while drag just requires one surface and a fluid. But there’s another way that we treat these two forces as different. Let’s consider again our block that’s sliding down the plane but not experiencing any drag.

We would say that the frictional force experienced by the block — this arrow in pink here — is constant as the block slides down. Regardless of how fast the block is sliding, we would say that as soon as it’s in motion, this frictional force is a constant value against that motion. Drag, on the other hand, works differently. One way to think about it is like this. Say that we’re sitting in the passenger seat of a car that’s moving along down the road. If we roll down that window and put our hand out the window, we can start to feel the air against our arm. Many of us know from experience that the faster the car moves, the more we feel the air pushing on our hand and arm.

Now, the force responsible for this push is the drag force. It has to do with all the air molecules that are colliding with our hand and arm that are outside the window. And the faster the car travels, the greater the drag force is. We can feel it. It turns out that this is true in general. That the faster an object is moving, the more drag force it experiences. This fact has a fascinating physical consequence that we can illustrate using our dropped baseball from earlier.

To see how this works, let’s consider our baseball from the moment that it’s released by our hand. That is, the moment it’s dropped. And along with considering the ball’s motion, let’s also consider its velocity 𝑣 against the time 𝑡, after which it’s been dropped. We can say that when 𝑡 equals zero is the moment that we release the ball, and so at that time, the velocity of the ball is also zero. Now, from a forced perspective, what’s happening to the ball when we release it? We know the baseball experiences a gravitational attraction to the center of the earth. And the magnitude of that attraction is equal to the mass of the ball times the acceleration due to gravity. At this initial moment, there are no other forces on the ball. Remember we’ve released it. So it starts to accelerate downward.

Now, if we say motion down towards the earth is motion in the positive direction, then we can see that after a little bit of time has elapsed, the velocity of our ball will have increased. It’s speeding up from rest down towards the center of the earth. But now consider this. As soon as the ball starts moving, we have a moving object. And what is a moving object experience? It experiences forces that resist its motion. In this case, just as we saw earlier, the drag force is at play, resisting the falling motion of the ball.

At this early stage, with the velocity of the ball still at a relatively small value, our drag force is correspondingly small. So we can say that that force just points upward like this. We’ll call it 𝐹 sub 𝐷. And it’s still being dominated by the gravitational force on the ball. So even though there is now a drag force on the ball, the ball is still falling towards the earth and it’s still speeding up. But because of that drag force, it’s not speeding up as fast as it was over this initial time interval. It is still speeding up. Its velocity is still increasing, but not at the same rate as before, thanks to the drag force that’s now acting on the ball.

So if we were to plot our next velocity and time point, that point might look something like this. The velocity of the ball is still going up, but it’s just not going up as fast as it was before. But since our ball is speeding up, remember that we saw that the drag force also goes up with object’s speed. So now let’s say our drag force does this. It increases against the motion of the ball. It’s still not as great of a force as the gravitational force in the ball, but it is increasing in strength. Since the force of gravity though is still stronger than the drag force, the ball continues to accelerate speed up downward.

But just like before, if we compare the velocity increase over the previous time interval to the velocity increase over this current time interval. What we’ll find is that the velocity increase isn’t as great. Say our point might be here. And that’s thanks to the fact that our drag force is greater than it was before. And then, the cycle happens again. Because our velocity continues to increase, the drag force does as well. But so long as that drag force is still weaker than the gravitational force, our ball continues to speed up as it gets towards earth. But notice that it is speeding up at a slower rate. After this goes on for some time, something very interesting happens. Eventually, the velocity of the ball gets high enough that the drag force acting on the ball increases in magnitude to the point that it is equal to the magnitude of the gravitational force on the ball.

In other words, when the drag force reaches this point, we can write that now, finally, the drag force is equal to the gravitational force. And when that’s true, it means that the forces in the vertical direction on this ball balance one another out. The net force on this baseball then is zero. And by Newton’s second law of motion, the object stops accelerating.

Looking at our graph of velocity versus time, let’s see that we get one more data point where the velocity is still increasing before our drag force becomes equal to our gravitational force. Then, at this moment in time, these two forces become equal and the ball stops accelerating. And that means that even if the ball continues to fall through the air, it won’t be speeding up anymore. From this point on, its velocity will stay at the same exact value. It’s not that the ball has stopped falling or that it stopped moving very fast, but it’s only that its motion is in some sense frozen. It keeps moving at the same speed.

We can say then that this particular value of the ball’s velocity is the fastest it can ever go in free fall. No matter how long it descends, it will never exceed the speed. This is called the ball’s terminal velocity. Terminal because it’s the final velocity this object will achieve. In general, for any object in free fall, that is under only the influence of the force of gravity and the drag force, the point at which its velocity flattens out, at which it stops increasing, is known as the point that it achieves terminal velocity. Knowing what we know now about forces that resist motion, let’s get some practice with these ideas through an example.

Which of the graphs a, b, c and d most correctly shows how the velocity of an object changes with time if the object is subject to a constant force while moving through a fluid that exerts a drag force on it, bringing it to its terminal velocity?

Okay, as we get started here, we’re told that we have some kind of object. And let’s say that this here is our object. We’re told further that our object is subject to a constant force. We can sketch that in on our object, say acting to the right. And we’ll simply call that force 𝐹. We don’t know what it is, but it’s just some force. In addition to this, we know that our object is moving through a fluid. A fluid recall is a liquid or a gas. And as a result, as the object moves, it experiences a drag force. That’s a force resisting its motion. Lastly, we know that the effect of this drag force is such that eventually our object is brought to its terminal velocity. We want to know which of these four graphs — a, b, c and d — most correctly represents how the velocity of our object changes in time.

To begin figuring that out, let’s clear a bit of working space on screen. Okay, so here again is our object, and it’s being acted on by this constant force we’ve called 𝐹. We know our object is moving through a fluid. And what if we just say that that fluid is water, that our object is moving through a tank of water? Well, in that case, we know what will happen as our object starts to move under the influence of the force 𝐹. As it begins to move to the right, there will be a drag force that emerges acting in the opposite direction. We can call it 𝐹 sub 𝐷 that opposes this motion.

And there’s an important difference between the drag force and this force we’ve called 𝐹. Recall that 𝐹 is a constant force. It never changes, whereas the drag force does change as the velocity of our object changes. In particular, the drag force will increase the faster our object moves. Now, before we get too far ahead of ourselves. Let’s notice that in all four cases for all four of our answer options, the initial velocity of our object is said to be zero.

So let’s do this. Let’s let our object in this tank of water start from rest. Beginning at that point, our object speed will clearly be zero. And if it has no speed, then there’s no force resisting its motion, since it has no motion. At the outset then, at the initial moment, we can say that there is no drag force on our object. The only force acting on it is this constant force 𝐹. Under the influence of that force, though, because it is a net force acting on our object, our object starts to accelerate. And the instant our object speeds up, the moment that it has a speed above zero, a small drag force appears opposing that motion. This drag force is caused by the interaction between this object and the water in the tank.

So our object’s velocity started at zero. And it’s increasing thanks to the fact that there are unbalanced forces acting on the object. And therefore, it accelerates. But here is where we want to recall that the drag force, unlike our constant force 𝐹, doesn’t stay the same. It’s generally the case that drag force increases as object velocity increases. And indeed, thanks to the fact that there’s an overall or net force on our object to the right, its velocity will increase. But then, as its velocity increases, so does the drag force. We see, though, that at this point the drag force is still less than the constant force 𝐹. So our object continues to speed up. And as it does so, the drag force grows and grows.

So long as these forces — our drag force and our constant force — are unbalanced, our object will continue to speed up. But the more it does so, the greater the drag force until eventually the two forces are equal. At this point, we can recall that when the net force on a projectile such as our object is zero, that means that object has reached what’s called terminal velocity. This is the point at which a projectile is no longer speeding up, but it’s reached its maximum speed. Because an object’s terminal velocity is its maximum speed, that means we can eliminate a couple of our possible answer options.

We see both for graph a as well as for graph d that the far-right portion of the graph in each case does not indicate a maximum velocity. In the case of graph a, that’s because we’ve already achieved a higher velocity earlier on. So graph a doesn’t correctly show an object reaching terminal velocity. Then, in the case of graph d, our velocity is still increasing as we get to the right most portion of the curve. In this case also, we’re not at a maximum value. We’re not at terminal velocity. If we look then at graph c, this graph shows us an object that is speeding up initially at an increasing rate. In other words, not only was the object getting faster, but the rate at which it got faster was growing.

But in the case of our object, acted on by the constant force 𝐹 and opposed by the drag force, what we saw instead was that while the object’s velocity did increase, it was increasing at a decreasing rate. That is, as the object’s velocity approached its maximum value, its velocity changed by smaller and smaller amounts. So it’s the shape of this part of the curve of graph c, which tells us that this is not a correct representation of our object’s velocity versus time. Yes, at first, our object was speeding up. But the rate at which it was speeding up was not increasing. It was decreasing. So option c can’t be our choice either.

When we look at the graph for option b, we see that this graph has the right shape to it. Velocity is increasing but increasing at a decreasing or smaller rate over time. And eventually, the velocity does level out to a set value, the object’s terminal velocity. It’s graph b then that correctly represents the relationship we’re looking for.

Let’s take a moment now to summarize what we’ve learned in this lesson. We’ve seen here that friction and drag are two distinct forces that both oppose object motion. Friction involves two surfaces that are in contact with one another and in motion relative to each other, while drag forces involve one surface that’s in motion relative to a fluid environment. And lastly, when the net force on a projectile that’s moving through a fluid is zero, that means the projectile has reached terminal velocity, its maximum speed.

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