Undergrad Robert Adams textbook: Acceleration example

[bigmike94]

TL;DR

Not sure on last step.

I need a little guidance on how they got the last step to derive acceleration, I can follow up till there. Any help would be greatly appreciated as I find it hard to move on unless I have understood.

 

[fresh_42]

$v=\dfrac{dx}{dt}(i + 2xj)$ and $a=\dfrac{dv}{dt}.$ So
\begin{align*}
a&\stackrel{def}{=}\dfrac{d}{dt}\left(\dfrac{dx}{dt}\cdot (i + 2xj)\right)\\
&\stackrel{\text{product rule}}{=}\dfrac{d^2}{dt^2}x \cdot (i + 2xj) + \dfrac{dx}{dt} \dfrac{d}{dt}(i + 2xj)\\
&=\dfrac{d^2}{dt^2}x \cdot (i + 2xj) + \dfrac{dx}{dt}\cdot\underbrace{\dfrac{d}{dt} i}_{=0}+\dfrac{dx}{dt}\cdot 2j\cdot \dfrac{d}{dt}x\\
&=\dfrac{d^2}{dt^2}x \cdot (i + 2xj) + 2j\left(\dfrac{dx}{dt}\right)^2
\end{align*}

 

[bigmike94]

$v=\dfrac{dx}{dt}(i + 2xj)$ and $a=\dfrac{dv}{dt}.$ So
\begin{align*}
a&\stackrel{def}{=}\dfrac{d}{dt}\left(\dfrac{dx}{dt}\cdot (i + 2xj)\right)\\
&\stackrel{\text{product rule}}{=}\dfrac{d^2}{dt^2}x \cdot (i + 2xj) + \dfrac{dx}{dt} \dfrac{d}{dt}(i + 2xj)\\
&=\dfrac{d^2}{dt^2}x \cdot (i + 2xj) + \dfrac{dx}{dt}\cdot\underbrace{\dfrac{d}{dt} i}_{=0}+\dfrac{dx}{dt}\cdot 2j\cdot \dfrac{d}{dt}x\\
&=\dfrac{d^2}{dt^2}x \cdot (i + 2xj) + 2j\left(\dfrac{dx}{dt}\right)^2
\end{align*}

You legend! I know it might be pretty obvious to some but it really can’t be difficult for a textbook to show a few more steps, for slower people like myself. Ron larsons textbook is really good for showing steps I’ve noticed.

Thank you again! I can finally go to bed

 

[Delta2]

I know it might be pretty obvious to some but it really can’t be difficult for a textbook to show a few more steps, for slower people like myself. Ron larsons textbook is really good for showing steps I’ve noticed.

Yes some textbooks are better than others on providing step by step solutions but if every textbook was going to show detailed step by step solutions for every problem and every proof then textbooks would be 3000-5000 pages long instead of the usual 300-500 pages.

 

[bigmike94]

Not every problems but in a worked example it’s not hard to put in brackets on the same line “product rule”. It helps for people with bad memory like me

 

[Delta2]

Well ok don't worry I understand, I also during my youth was complaining about books "this book doesn't explain this topic in full detail" "it omits too many non obvious steps" e.t.c. but oh well growing up I realized this is the world we living and those are the books we get e hehe.

 

[bigmike94]

Well ok don't worry I understand, I also during my youth was complaining about books "this book doesn't explain this topic in full detail" "it omits too many non obvious steps" e.t.c. but oh well growing up I realized this is the world we living and those are the books we get e hehe.

They’re good books don’t get me wrong I love them and I realize it is my own fault, this is why I have started reading multiple books on each chapter. It seems to help a tonne

 

[Delta2]

Yes well some books are much better than others but there isn't the perfect book. Its not exactly your fault if you can't find some step, this step might be not so obvious to you as it is to others and to the book author. Reading multiple books helps on that cause a book might provide some steps and explanation that are omitted by the other.