Motion as a function of time: Resistive force example (part 2)

0:00- [Instructor] A tiny drop of waterfalls vertically from rest through air. It experiences a drag force of this form, and our goal is to find the drops' acceleration, velocity and position as a function of time. In our previous video, we already did some work on it.

0:14We drew the free body diagram. We understood what drag force was. It's a resistive force whenever a particle is moving through any fluid. And notice the form over here is that the magnitude of that force is proportional to the velocity. As the velocity increases, the drag force also increases.

0:30But of course the negative sign represents that the drag force will always be in the opposite direction of the velocity. Since this drop is falling down, the drag force will be upwards. And we figured out the velocity function. We found that the velocity function looks like this.

0:43It's an exponential function. And just to make sense of this one more time, because this function looks a little scary, at time t equals zero, look, the exponent goes to zero. So e power zero is one, one minus one is zero. So the expression is saying at time t equals zero, our velocity of the raindrop is zero.

1:00That makes sense because it's falling from rest. But as time passes by, the exponent increases. But since it's negative, the whole value becomes much smaller than one very quickly. And if you wait long enough, this number will be so small, we can neglect it. The velocity will approach terminal velocity.

1:16And the terminal velocity is given by this expression. And this happens when the two forces are eventually balanced. And so we drew the graph of the velocity time function. This is what the graph looks like. It's an exponential function. It's telling the same story.

1:28It starts with zero and asymptotically approaches the terminal velocity. If you need a refresher, it'll be a great idea to go back and watch our previous video where we talked about this in great detail. But in this video, we're gonna try and figure out the acceleration and the position.

1:46So let's do that. So how do we figure out the acceleration function? Well, we know acceleration is rate of change of velocity. So acceleration is just dv over dt, which means if I can differentiate this function, I'll get the acceleration function. So it'll be a great idea to pause the video and see if you can try and differentiate this function yourself and see what the acceleration function looks like.

2:09Okay, let's do this. So we're gonna differentiate this function. So let's do that. So it's gonna be dv over dt of this thing over here. What would that give us? Well, let's simplify this a little bit. So we can distribute this. So we'll get v terminal minus v terminal times e to the power of this thing.

2:28Now we can take the derivative. V terminal is a constant. It doesn't change with time. You can see that over here. It only depends upon mg and b values, which are constants. They don't change with time. Since this is a constant, the derivative of that with respect to time would be zero.

2:43So this part would be zero. Okay, so that means we have to take the derivative of this term. And over here again, since v terminal is a constant, we can pull it out. So the acceleration function is going to be, let's pull this out first. It's negative, v terminal, times the derivative of this exponential function.

3:02What's the derivative of an exponential function? Well, derivative of e to the power, you know, some constant time t will be c, that constant, times that exponential function. So it's going to be the constant minus b by m times that exponential function, okay?

3:20So the negative and the negative cancels out, so we get we terminal times b over m times the exponential function. What will we do next? Well, we can plug in the value of the terminal velocity over here. So mg over b, if we do that, the m cancels out, the b cancel out.

3:37And look, what we are left with is g times the exponential function. So let's write that over here. So acceleration becomes g times the exponential function, and there you have it. That is the acceleration function. And again, let's try to make sense of this. So the first step will be to draw an acceleration time graph.

4:00And again, it'll be a great idea to pause the video and see and try to do it yourself. Okay, so the way I like to do this is first think about time t equals zero. As t equal to zero, if I plug in t equal zero, this exponent goes to zero. So e power zero is one, which means at time t equal zero acceleration is g.

4:20So it starts at g somewhere over here. So that t equal zero acceleration is g. What happens as time increases? Well, the exponent increases, but there's a negative in front of it, so the value starts reducing very quickly. It becomes smaller than one very quickly.

4:37So if I wait long enough values of time, eventually this number would be so small that this product almost becomes zero. In other words, this acceleration asymptotically approaches zero. And so the graph should look somewhat like this. Look, the acceleration asymptotically approaches zero.

5:00So that's the exponential function. But does this make sense? Does this make intuitive sense? Well, let's see. Let's go back over to our free body diagram and think about this. At time t equal to zero, well, the velocity of the raindrop is zero because starting from rest.

5:17That means at time t equal zero, the drag force is zero. So this is the condition at t equals zero. Since gravity is the only force acting on it at time t equal zero, it's in free fall. Its acceleration is just g. And that's exactly what we got. That makes sense, that's why initially at time t equal zero, the acceleration is g.

5:38Now what happens is time passes by? As time passes by, the drop starts accelerating down. So it gains velocity. But as it gains velocity, the drag force also increases. As the drag force increases, what happens to the net force? Hey, the net force keeps decreasing.

5:55That's why the acceleration keeps decreasing. It keeps decreasing, it keeps decreasing until eventually, if you wait long enough, the two forces become equal to each other. When that happens, terminal velocity is reached and the object will no longer, the raindrop will no longer accelerate.

6:12It'll continue to move with this terminal velocity. And that's exactly what the graph is saying. If you wait long enough, eventually the acceleration goes to zero, terminal velocity is reached and the object will continue moving at a constant velocity. It all makes sense, isn't it?

6:28Okay, so if you put it all together, we have the velocity function and we have the acceleration function. And guess what? There's another way to check if the two graphs make sense. We know geometrically the slope of the velocity should give us acceleration, right?

6:47So let's test that as well. So let's look at the slope over here. The slope initially over here looks somewhat like this, and that slope is very high. That is the initial acceleration. Look, it's pretty high. That's g over here. But now what happens as time passes by, look at the slope, what happens to the slope?

7:05The slope reduces, and that's exactly what we're seeing, acceleration reduces, reduces. The slope is reducing, acceleration is reducing, reducing. And it's asymptotically, the slope is becoming flat, right? It approaching flatness. So asymptotically, the acceleration is approaching zero.

7:22It makes sense, right? Like I love how we can use multiple tools at our disposal to try and make sense of what we have found, which is super, super important, especially if we are dealing with something so involved over here. Okay, so we have found acceleration function.

7:39The last thing for us is to find the position function. How do we find the position function? Well, we know the connection between position and velocity. We can say that velocity is the rate of change of position. So we can plug in for vdy over dt, and then we get another differential equation, and then we can try to solve that differential equation.

8:01So that's one way we can do that. But we won't do that over here, okay? It would be a great idea for you to try and do this yourself. But over here what we're gonna do is we are just going to directly, using all the knowledge we have, all the tools that we have, try to plot the position time function.

8:19So again, it'll be a great idea to see if you can pause and try to draw this yourself. Think about what it'll look like. All right, so at time t equals zero, we already chose our coordinates in such a way that the raindrop position is zero. Okay, that's our coordinate choice.

8:38So it starts with zero. But then what happens as time increases? Well, the drop will keep falling down, it'll keep accelerating until it hits terminal velocity. Once it hits terminal velocity, it doesn't stop. It just keeps moving downwards at a constant, as a constant velocity, right?

8:53So if there was no ground, it'll just keep moving down forever. So its position will just keep increasing forever. But remember, once it hits terminal velocity, its velocity is a constant, meaning once it hits terminal velocity, its slope of this graph must be a constant.

9:11In other words, the slope of this graph should asymptotically approach terminal velocity. And we know the initial slope is zero because the velocity is zero. And so what does the graph look like? It starts out with zero slope and then asymptotically approach the terminal velocity.

9:27So that means the graph would look like this. It'll be another exponential function. And again, we can double check just to be sure. We can draw the slope over here, which represents the velocity. And notice initially the slope is zero, the velocity is zero. And then as time passes by, the velocity increases, increases, increases, the slope increases, increases.

9:48But eventually the slope, look, asymptotically approaches the terminal velocity. Beautiful, right? This is the full story of our raindrop. But here's the last bit of information about the drag force. Over here, we modeled it in such a way that the magnitude of the drag force was proportional to the magnitude of the velocity, right?

10:09That's what we did in this particular case. Turns out that this model only works when we are dealing with very small objects moving through fluids. But for most everyday objects, large objects, like cars or trains or buses or birds, whatever that is, it turns out that a much better model for drag force is the one in which the magnitude of the drag force is proportional to the velocity squared.

10:38In other words, the magnitude of the drag force is called quadratic with velocity. But for very tiny objects like raindrops, we can model it to be linear with velocity. So what that does is it changes everything. The velocity and the acceleration and the position functions will all look very different than what we got over here.

10:57But the way we analyze things, that still stays the same. We draw the free body diagram, and from there calculate the net force, and from there we try to calculate the acceleration where you get some differential equations, and we solve those differential equations to find the velocities and accelerations and position.

11:14All of that stays exactly the same, but the math would be slightly more involved because of the quadratic terms. So in most introductory physics problems, we neglect air resistance, right? That's a great way to start modeling things. The math becomes easier. But of course, if we want it to be more useful in the real world situations, we have to include the drag force.

11:37And when we do that, look, things change. For example, when we neglect the drag force or when we neglect air resistance, the path of a projectile is a parabola, right? However, when we do consider the drag force, the path turns out to be slightly different from the parabola.

11:54Although the map is slightly more involved over there, the idea is the same. So now we have gained tools to create much more realistic models for objects moving through fluids like real projectiles moving through air.