Problem 69 Hard Difficulty

In Chapter 9 we will be able to show, under certain assumptions, that the velocity $ v(t) $ of a falling raindrop at time $ t $ is $$ v(t) = v^*(1 - e^{-gt/v^*}) $$ where $ g $ is the acceleration due to gravity and $ v^* $ is the $terminal $ $ velocity $ of the raindrop.

(a) Find $ \displaystyle \lim_{t \to \infty} v(t) $.

(b) Graph $ v(t) $ if $ v^* = 1 m/s $ and $ g = 9.8 m/s^2 $. How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity?

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04:52

Daniel J.

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this problem. We need to find first the limit of the function B. The limited when t tends to infinity of the of team. So we simply put the functioning there and we can use properties of limits. We know God, we multiply this for each element and we know that the limit of a song is the limit of each term. So we will have the limit of the first. Sure minus the limit of the second one. We also know that the limit of a constant. The constant can be take out of the limit so we will have this limit of one minus the limit. Um we're just gonna take out also the the terminal is speed the limit of d attempts to infinity of this potential. No, we know that the limit of the constant is just against them because it doesn't depend on the variable team on the other hand or dysfunction I'm going to graph deal. The essential function just to see is the tendency when he tells to infinity Oh this is the function uh this name show of team when we not that the the argument of the function is negative which is simply and this is the function minus the exponential of minus 60. So in this case the is sending to infinity. So this part that is the positive are of ads or well in this case we call a team Then in this case 10 is he is tending to infinity So that the function just approaches to zero and then we can say that this term is zero. So we will have yes, B just speak which is the german all his speed. No or party of these problems. We are asked two graph the function need to graph if I'm shun the complete function. Okay. When um terminal speed is one m square and we take the acceleration of gravity as 9.8 m per second square. Putting these values in the function we have that you can hear This is just one m square 1- in the exponential. We will have minus T. Nine point a major square over one Teacher Square one. Meet 1 meter per second. So to do this to do the function. One thing that we can do is just to take some bodies of D. And and graph these points and just connect the points in the graph so that we will obtain these functions. So that's what we're going to do. Taking some somebody's off T. We also know how we also can determine the behaviour of this function at infinity or minus infinity. With a limit. So that we can also rough that function for um great values of T. So or tea. And we will have in here B of T. So we're gonna try 00.10.25 and zero point. And the value of one The equal to one. So 14th. Would you simply evaluate the function for all of these values of T. So.40. We obtain a value of being of one of zero because we know that The exponential function at zero is equal to one. So 1 -1 is zero. So we obtain zero for 0.1 we obtained you're a point sits theme two 47. Okay. were so 40 equal to zero 25. We obtain zero .91. 30 seven. And for one we obtain 0999. Not self. What we are going to do right now is to grab this point and then connect them. So we will have something like this in here. We have The zero in here we have zero point one and here we have zero point when you find any here we have the point corresponding to one. Now we connect this. We can also see how is the behavior of this to function of this function at the infinity and at the minus infinity. So we already uh we already found that when t tends to infinity deem the function tends to the value Of the terminal speed and in this case is one. So we know that the function will take will tend to do To the value one when it is going to infinity. So now if we take the limit to t when the times tend to minus infinity of the function we will have that we will have to put in minus and fainting here. Damn minus sign canceling shorter. So that we will have an exponential, that is turned into infinity. So we will have these read function. And that red function when when it tends to infinity it tends to infinity so that we will have one minus infinity and that is minus infinity. So in this case the function is just decreasing. Right then. If you connect all the points and considering the behavior uh t equal detained into minus infinity and infinity we'll have something off the form where the function two d turned into infinity. Uh The function tends to the value of the Terminal speed. In this case it is one and when the function tends to minus infinity, the function just decrease to minus infinity. Now, the final question of this problem And how long does it take for the velocity of the raindrop to reach 90 9% of his terminal velocity? We want to find when this happened. So we know that this can also be written a 0.99 the terminal velocity. So this is the value that we want to calculate. So it is the value of B. T. So that we're gonna like right again, the function we know that this is is volume here. Um So now we just simply put volumes