# Mathematical models and numerical methods: "A motorboat starts from rest..."

#### ken_165

##### New member
I have a question and please answer...
19. A motorboat starts from rest (initial velocity v (0)=v sub 0=0). Its motor provides a constant acceleration of 4 ft/s^2, but water resistance causes a deceleration of v^2/400ft/s^2. Find v when t=10s, and also find the limiting velocity as t approaches +infinity(that is, the maximum possible speed of the boat).
Sorry for the alternate symbol, i have no italicized math symbols and infinity symbols, etc. Hope you understand so u can answer. Please, if someone knows the answer, please solve it coz i want to learn how to solve some problems for at least in differential equation.

#### MarkFL

##### Super Moderator
Staff member
So what we have is the IVP:

$$\displaystyle \d{v}{t}=4-\frac{v^2}{400}$$ where $$v(0)=0$$

Are you supposed to solve, or use a numeric method to approximate a solution within a certain tolerance?

#### ken_165

##### New member
I actually dont know the answer. I just wanna know the answer. My question is complete as depending on what ive seen on the book. I hope u can answer the question base on my given question. I hope that u can show the answers so that i might see a little truth about the math, the one of the difficult answers on math. I hope u understand and get my point.

#### MarkFL

##### Super Moderator
Staff member
Your thread title implies that a numerical method is to be used, but the question doesn't indicate which method to use, nor what tolerance.

• topsquark

#### ken_165

##### New member
Ok. Thanks anyway. Its ok if my question isnt answered. Ill just ask another question that can be solved next time. Bye.

#### MarkFL

##### Super Moderator
Staff member
You aren't answering my questions. But, since you have evaded my question regarding whether a numerical method is to be used or not, let's just assume you are to solve the given IVP analytically. Can you see the associated ODE is separable? Can you separate the variables and integrate?

• topsquark

#### MarkFL

##### Super Moderator
Staff member
Okay, let's just solve the given IVP. The ODE may be written:

$$\displaystyle \frac{400}{40^2-v^2}\,\d{v}{t}=1$$

Replace the dummy variables, and integrate on the boundaries:

$$\displaystyle 400\int_{v_0}^{v} \frac{1}{40^2-s^2}\,ds=\int_0^t\,dr$$

$$\displaystyle \ln\left(\frac{(40+v)(40-v_0)}{(40-v)(40+v_0)}\right)=\frac{t}{5}$$

With $$v_0=0$$, this reduces to:

$$\displaystyle \ln\left(\frac{40+v}{40-v}\right)=\frac{t}{5}$$

Or:

$$\displaystyle v(t)=\frac{40\left(e^{\frac{t}{5}}-1\right)}{e^{\frac{t}{5}}+1}$$

Can you now find $$v(10)$$ and $$\displaystyle \lim_{t\to\infty}v(t)$$?

• topsquark

#### topsquark

##### Full Member
Ok. Thanks anyway. Its ok if my question isnt answered. Ill just ask another question that can be solved next time. Bye.
Code words for "My instructor only grades the final answer and I don't have to show any work."

-Dan

• MarkFL

#### HallsofIvy

##### Elite Member
So this is a course on how to ask other people to do your work?

• topsquark

#### ken_165

##### New member
In the first place, sentence spacing were never taught in preschools. That's why we're idiot in mathematics. So, if you will tell me that im just relying on intelligent people, well, my answer is the first sentence(sentence spacing were never taught in preschools). And hey, your race invented the erroneous education, the americans.

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#### MarkFL

##### Super Moderator
Staff member
In the first place, sentence spacing were never taught in preschools. That's why we're idiot in mathematics. So, if you will tell me that im just relying on intelligent people, well, my answer is the first sentence(sentence spacing were never taught in preschools). And hey, your race invented the erroneous education, the americans.
You posted a question, and evaded my questions regarding what was expected. I went ahead to provide help anyway. You did not acknowledge that at all. Nothing was said about sentence spacing, and for what it's worth, "American" is a nationality, not a race.

• topsquark

#### Jomo

##### Elite Member
• HallsofIvy and topsquark

#### HallsofIvy

##### Elite Member
In the first place, sentence spacing were never taught in preschools. That's why we're idiot in mathematics.
"Sentence spacing"? What is the world IS "sentence spacing" and what does it have to do with mathematics?

So, if you will tell me that im just relying on intelligent people, well, my answer is the first sentence(sentence spacing were never taught in preschools). And hey, your race invented the erroneous education, the americans.
First, "Americans" are NOT a "race". Second, "erroneous education" (I would have said just "bad" or "poor" education) existed long before the Americas were discovered by Europe.

(Oh, and since "erroneous education" is singular it should be "was", not "were".)

#### ken_165

##### New member
Please go away, far away.
You go far away javaman. Not me coz i have a home.

• Jomo