# Thread: differentiating newtons law

1. ## differentiating newtons law

Hi, I was wondering if anyone knows how to differentiate

m(dv/dt)=mg-kv^2

I know partial fractions must be involved however I can't get beyond this point.

thank-you.

2. ## Re: differentiating newtons law

We get

$\frac{d}{dt}(m\frac{dv}{dt})=m\frac{d^2v}{dt^2}$

$\frac{d}{dt}(mg-kv^2)=-2kv\frac{dv}{dt}.$

3. ## Re: differentiating newtons law

Originally Posted by ella745
Hi, I was wondering if anyone knows how to differentiate

m(dv/dt)=mg-kv^2

I know partial fractions must be involved however I can't get beyond this point.

thank-you.
If you expect to have to use partial fractions, you're probably asking how to solve this differential equation (though integration, NOT differentiation).

\displaystyle \begin{align*} m\,\frac{dv}{dt} &= mg - kv^2 \\ \frac{dv}{dt} &= \frac{mg - kv^2}{m} \\ \frac{dt}{dv} &= \frac{m}{mg - kv^2} \\ t &= \int{\frac{m}{mg - kv^2}\,dv} \\ t &= \int{\frac{m}{\left(\sqrt{mg} - \sqrt{k}\,v\right)\left(\sqrt{mg} + \sqrt{k}\,v\right)}\,dv} \end{align*}

and you can now apply Partial Fractions

4. ## Re: differentiating newtons law

Originally Posted by Prove It
If you expect to have to use partial fractions, you're probably asking how to solve this differential equation (though integration, NOT differentiation).

\displaystyle \begin{align*} m\,\frac{dv}{dt} &= mg - kv^2 \\ \frac{dv}{dt} &= \frac{mg - kv^2}{m} \\ \frac{dt}{dv} &= \frac{m}{mg - kv^2} \\ t &= \int{\frac{m}{mg - kv^2}\,dv} \\ t &= \int{\frac{m}{\left(\sqrt{mg} - \sqrt{k}\,v\right)\left(\sqrt{mg} + \sqrt{k}\,v\right)}\,dv} \end{align*}

and you can now apply Partial Fractions

Hi, thank you so much! I actually don't understand it past this point, sorry I didn't make it clear. If you could just show the next few steps that would be perfect because I'm getting something wrong and i dont know what

5. ## Re: differentiating newtons law

Originally Posted by ella745
Hi, thank you so much! I actually don't understand it past this point, sorry I didn't make it clear. If you could just show the next few steps that would be perfect because I'm getting something wrong and i dont know what
Well then you should show what you have tried. A differential equation is an equation that involves derivatives, so the solution of a differential equation is to find the family of functions that will satisfy the differential equation. In this case, you are trying to find a function $\displaystyle v(t)$.

6. ## Re: differentiating newtons law

Originally Posted by ella745
Hi, thank you so much! I actually don't understand it past this point, sorry I didn't make it clear. If you could just show the next few steps that would be perfect because I'm getting something wrong and i dont know what
Why don't you post the original question, as worded in wherever it came from.

CB

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