An object of mass m is moving horizontally through a...

An object of mass $m$ is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is, $$m \frac{d^{2} s}{d t^{2}}=m \frac{d v}{d t}=f(v)$$ where $v=v(t)$ and $s=s(t)$ represent the velocity and position of the object at time $t,$ respectively. For example, think of a boat moving through the water. \begin{equation} \begin{array}{l}{\text { (a) Suppose that the resisting force is proportional to the }} \\ {\text { velocity, that is, } f(v)=-k v, k \text { a positive constant. (This }} \\ {\text { model is appropriate for small values of } v . \text { Let } v(0)=v_{0}} \\ {\text { and } s(0)=s_{0} \text { be the initial values of } v \text { and } s . \text { Determine } v} \\ {\text { and } s \text { at any time } t . \text { What is the total distance that the object }} \\ {\text { travels from time } t=0 ?}\\{\text { (b) For larger values of } v \text { a better model is obtained by sup- }} \\ {\text { posing that the resisting force is proportional to the square }} \\ {\text { of the velocity, that is, } f(v)=-k v^{2}, k>0 \text { . (This model }} \\ {\text { was first proposed by Newton.) Let } v_{0} \text { and } s_{0} \text { be the initial }}\\{\text { values of } v \text { and } s . \text { Determine } v \text { and } s \text { at any time } t . \text { What is }} \\ {\text { the total distance that the object travels in this case? }}\end{array} \end{equation}

Transcript

0:00 All right, hello.

00:01 In this question we're told that we have this equation, our force is equal to our mass times our acceleration, and our acceleration could take these two forms, where s is the position function and v is the speed function.

00:14 And we're asked in part a to suppose our drag force is negative kv.

00:20 Using this we want to go ahead and find an expression for v of t and s of t, and then find the total distance that the object will travel from the start.

00:28 So in order to do this, well we know our force is going to be negative kv, and we know that force is equal to mass times dv dt.

00:37 So we just want to solve this differential equation.

00:40 To do that i'm going to multiply both sides by dt and divide both sides by v.

00:45 So i'm going to get negative k m dt equals dv over v, and i'm going to integrate both of those sides.

00:52 So i'm going to get negative k m times t is equal to the natural log of the absolute value of v of t at some time t.

01:01 And i'm going to have some plus some constant of integration here.

01:04 So in order to get rid of everything and solve for v of t, i'm going to raise the entire equation to the base e here.

01:12 So i'm going to have e to the negative k m t times one equals v of t, and this is going to be times e to some constant c.

01:24 Well i can divide that over and i'm just going to rename this some value a naught here, and i'm going to move it over and since it's i don't know what this constant is i can just write v of t equals a naught e to the negative k m t, because i just want to figure out what this the value of this constant is.

01:42 And i can do that by saying well if i have time equals zero i know that i'm going to be at v naught, and so if i plug in zero for my time here, well this is going to be zero here.

01:54 E to the zero is one, so v naught is going to be a naught.

01:57 So my ultimate speed equation is going to be v naught e to the negative k m t.

02:04 If i want to find the position function, i'm going to just say that my position function is equal to the integral of my speed function with respect to time.

02:14 So i'm going to have some s of t is equal to v naught divided by negative k m times e to the negative k m t plus some constant c.

02:32 Well what is that constant c going to be? if i plug in t equals zero i know that s of zero is going to be s naught, and that's when i plug in v equals zero, or t equals zero rather.

02:43 And so this is just going to be a value of one, so s naught, or i guess i want to solve for c here actually, c is going to be equal to s naught plus v naught over k m.

02:57 So i have my velocity function and my position function is going to be v naught over negative k m times e to the negative k m t plus c, which is v naught over k m plus s naught.

03:16 So that will be my position function.

03:19 And then i want to figure out what my position is at time equals infinity.

03:25 What is my total distance that i travel? so how far am i from x equals zero at time equals infinity? so i want to set, i want to find this limit as time goes to infinity of s of t.

03:37 So that's going to be the limit.

03:38 If i plug in time equals infinity to this, well my exponential term is going to be one over a massive number.

03:44 So this is going to be zero plus v naught over k m plus s naught...

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