Integrating velocity equation problem

[Sho Kano]

I've already completed the first question, but with number two, it's a different case. Here's my attempt:
[itex]\frac { d{ v }_{ y } }{ dt } \quad =\quad -g\quad -\quad \beta { v }_{ y }\\ \frac { d{ v }_{ y } }{ -g\quad -\quad \beta { v }_{ y } } \quad =\quad dt\\ \int { \frac { d{ v }_{ y } }{ -g\quad -\quad \beta { v }_{ y } } } \quad =\quad \int { dt } \\ \frac { -1 }{ \beta } ln(-g-\beta { v }_{ y })\quad =\quad t\\ [/itex]
natural log of a negative?

 

[Sho Kano]

EDIT: I seem to have figured out the answer right after posting this?
[itex]\frac { -1 }{ \beta } ln(-g-\beta { v }_{ y })\quad =\quad t\\ ln(-g-\beta { v }_{ y })\quad =\quad -t\beta \\ -g-\beta { v }_{ y }\quad =\quad { e }^{ -t\beta }[/itex]

 

[Ssnow]

In general ##\int \frac{1}{x} dx= \log{|x|}+c##

 

[Sho Kano]

In general ##\int \frac{1}{x} dx= \log{|x|}+c##

log base e right?

 

[Ssnow]

yes, ##\int \frac{1}{x}=\ln{|x|}+c##

 

[Ssnow]

##−g−βvy=e−tβ\frac { -1 }{ \beta } ln(-g-\beta { v }_{ y })\quad =\quad t\\ ln(-g-\beta { v }_{ y })\quad =\quad -t\beta \\ -g-\beta { v }_{ y }\quad =\quad { e }^{ -t\beta }##

if ##-g-\beta v_{y}>0## this is part of your solution, you must use the condition that ##v_{y0}=0##, but attention that you forgot the integration constant in your solution ...

 

[Sho Kano]

if ##-g-\beta v_{y}>0## this is part of your solution, you must use the condition that ##v_{y0}=0##, but attention that you forgot the integration constant in your solution ...

The integration constant is just the initial condition, which is 0

 

[Ssnow]

mmm, I don't think, assuming ##-g-\beta v_{y}>0## put the condition ##v_{y}=0## with ##t=0## in

##\ln{(-g-\beta v_{y})}=-t\beta +c ##

 

[Ssnow]

You can see the value of ##c## also from ##\ln{|-g-\beta v_{y}|}=-t\beta + c##...

 

[Sho Kano]

Ssnow, not sure I follow you. From post #1, 3rd line of my attempt, the limits of integration are:

left side: 0 to v
(its 0 since initial velocity is given to be 0)
right side: 0 to t

 

[Ssnow]

You forget the constant ##\ln{g}##:

##\int_{0}^{v_{y}}\frac{w}{-g-\beta w}=-\frac{1}{\beta}\ln{|-g-\beta v_{y}|}+\frac{1}{\beta}\ln{g}##

 

[Sho Kano]

You forget the constant ##\ln{g}##:

##\int_{0}^{v_{y}}\frac{w}{-g-\beta w}=-\frac{1}{\beta}\ln{|-g-\beta v_{y}|}+\frac{1}{\beta}\ln{g}##

I see now, thanks!